Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Percentage Accurate: 98.5% → 98.5%

Time: 16.4s

Alternatives: 28

Speedup: 1.0×

• 47.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (log a) (+ t -1.0))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + (log(a) * (t + (-1.0d0)))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + (Math.log(a) * (t + -1.0))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + (math.log(a) * (t + -1.0))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(log(a) * Float64(t + -1.0))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Final simplification 98.9%

   \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \]
4. Add Preprocessing

Alternative 2: 37.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;t\_1 \leq 10^{+75}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (log a) (+ t -1.0))) b))) y)))
   (if (<= t_1 -5e+62)
     (/ (/ x y) a)
     (if (<= t_1 1e+75) (/ x (* a (fma b y y))) (/ (/ x a) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
	double tmp;
	if (t_1 <= -5e+62) {
		tmp = (x / y) / a;
	} else if (t_1 <= 1e+75) {
		tmp = x / (a * fma(b, y, y));
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(log(a) * Float64(t + -1.0))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e+62)
		tmp = Float64(Float64(x / y) / a);
	elseif (t_1 <= 1e+75)
		tmp = Float64(x / Float64(a * fma(b, y, y)));
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+62], N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e+75], N[(x / N[(a * N[(b * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5.00000000000000029e62

   1. Initial program 99.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 68.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 58.4

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 58.4%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 39.8

          \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{1}{a}} \]

       2. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

       4. /-lowering-/.f64 39.8

          \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{a} \]

   13. Applied egg-rr 39.8%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{a}} \]

   if -5.00000000000000029e62 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 9.99999999999999927e74

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 76.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 66.5

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 66.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. accelerator-lowering-fma.f64 48.2

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 48.2%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]

   if 9.99999999999999927e74 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 69.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 57.4

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 57.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
   10. Step-by-step derivation

   11. Simplified 36.7%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 10^{+75}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := {a}^{t} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 35:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot 0.5, y\right), y\right)}\\ \mathbf{elif}\;t\_1 \leq 544.5:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{elif}\;t\_1 \leq 200000000000:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))) (t_2 (* (pow a t) (/ x y))))
   (if (<= t_1 -1000000000.0)
     t_2
     (if (<= t_1 35.0)
       (/ x (* a (fma b (fma b (* y 0.5) y) y)))
       (if (<= t_1 544.5)
         (/ (* x (exp (- b))) y)
         (if (<= t_1 200000000000.0) (/ (/ x a) y) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = pow(a, t) * (x / y);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 35.0) {
		tmp = x / (a * fma(b, fma(b, (y * 0.5), y), y));
	} else if (t_1 <= 544.5) {
		tmp = (x * exp(-b)) / y;
	} else if (t_1 <= 200000000000.0) {
		tmp = (x / a) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = Float64((a ^ t) * Float64(x / y))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 35.0)
		tmp = Float64(x / Float64(a * fma(b, fma(b, Float64(y * 0.5), y), y)));
	elseif (t_1 <= 544.5)
		tmp = Float64(Float64(x * exp(Float64(-b))) / y);
	elseif (t_1 <= 200000000000.0)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, t], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$2, If[LessEqual[t$95$1, 35.0], N[(x / N[(a * N[(b * N[(b * N[(y * 0.5), $MachinePrecision] + y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 544.5], N[(N[(x * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 200000000000.0], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 2e11 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 74.2

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 74.2%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around inf

      \[\leadsto {a}^{\color{blue}{t}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

   11. Simplified 74.2%

       \[\leadsto {a}^{\color{blue}{t}} \cdot \frac{x}{y} \]

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 35

   1. Initial program 97.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 65.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 73.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right) + y\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot y\right) + y\right)} + y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\color{blue}{\left(b \cdot y\right) \cdot \frac{1}{2}} + y\right) + y\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(\color{blue}{b \cdot \left(y \cdot \frac{1}{2}\right)} + y\right) + y\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)} + y\right) + y\right)} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot y\right) + y, y\right)}} \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot y, y\right)}, y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{y \cdot \frac{1}{2}}, y\right), y\right)} \]

       9. *-lowering-*.f64 67.7

          \[\leadsto \frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{y \cdot 0.5}, y\right), y\right)} \]

   11. Simplified 67.7%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot 0.5, y\right), y\right)}} \]

   if 35 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 544.5

   1. Initial program 99.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 86.0

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 86.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 69.4

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 69.4%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   if 544.5 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2e11

   1. Initial program 97.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 87.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 89.6

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 89.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
   10. Step-by-step derivation

   11. Simplified 83.6%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -1000000000:\\ \;\;\;\;{a}^{t} \cdot \frac{x}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 35:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, y \cdot 0.5, y\right), y\right)}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 544.5:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 200000000000:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;{a}^{t} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -200:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;t\_1 \leq 200000000000:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0)))
        (t_2 (/ (* x (exp (- (* t (log a)) b))) y)))
   (if (<= t_1 -1000000000.0)
     t_2
     (if (<= t_1 -200.0)
       (/ (* x (exp (- (* y (log z)) b))) y)
       (if (<= t_1 200000000000.0) (/ (/ x (* y (exp b))) a) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = (x * exp(((t * log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -200.0) {
		tmp = (x * exp(((y * log(z)) - b))) / y;
	} else if (t_1 <= 200000000000.0) {
		tmp = (x / (y * exp(b))) / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(a) * (t + (-1.0d0))
    t_2 = (x * exp(((t * log(a)) - b))) / y
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-200.0d0)) then
        tmp = (x * exp(((y * log(z)) - b))) / y
    else if (t_1 <= 200000000000.0d0) then
        tmp = (x / (y * exp(b))) / a
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.log(a) * (t + -1.0);
	double t_2 = (x * Math.exp(((t * Math.log(a)) - b))) / y;
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -200.0) {
		tmp = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	} else if (t_1 <= 200000000000.0) {
		tmp = (x / (y * Math.exp(b))) / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.log(a) * (t + -1.0)
	t_2 = (x * math.exp(((t * math.log(a)) - b))) / y
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_2
	elif t_1 <= -200.0:
		tmp = (x * math.exp(((y * math.log(z)) - b))) / y
	elif t_1 <= 200000000000.0:
		tmp = (x / (y * math.exp(b))) / a
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = Float64(Float64(x * exp(Float64(Float64(t * log(a)) - b))) / y)
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= -200.0)
		tmp = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y);
	elseif (t_1 <= 200000000000.0)
		tmp = Float64(Float64(x / Float64(y * exp(b))) / a);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = log(a) * (t + -1.0);
	t_2 = (x * exp(((t * log(a)) - b))) / y;
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= -200.0)
		tmp = (x * exp(((y * log(z)) - b))) / y;
	elseif (t_1 <= 200000000000.0)
		tmp = (x / (y * exp(b))) / a;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$2, If[LessEqual[t$95$1, -200.0], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 200000000000.0], N[(N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 2e11 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot t - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot t - b}}{y} \]

      5. rem-exp-log 92.8

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

   5. Simplified 92.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -200

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 83.7

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 83.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   if -200 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2e11

   1. Initial program 97.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 82.7

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 77.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y \cdot e^{b}}}}{a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{y \cdot e^{b}}}}{a} \]

      5. exp-lowering-exp.f64 83.2

         \[\leadsto \frac{\frac{x}{y \cdot \color{blue}{e^{b}}}}{a} \]

   10. Applied egg-rr 83.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -1000000000:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -200:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 200000000000:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t + -1\right)}\\ t_2 := \log a \cdot \left(t + -1\right)\\ \mathbf{if}\;t\_2 \leq -700:\\ \;\;\;\;x \cdot \frac{t\_1}{y}\\ \mathbf{elif}\;t\_2 \leq -600:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;t\_2 \leq 10^{+81}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (+ t -1.0))) (t_2 (* (log a) (+ t -1.0))))
   (if (<= t_2 -700.0)
     (* x (/ t_1 y))
     (if (<= t_2 -600.0)
       (* x (/ (pow z y) y))
       (if (<= t_2 1e+81) (/ (/ x (* y (exp b))) a) (/ (* x t_1) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t + -1.0));
	double t_2 = log(a) * (t + -1.0);
	double tmp;
	if (t_2 <= -700.0) {
		tmp = x * (t_1 / y);
	} else if (t_2 <= -600.0) {
		tmp = x * (pow(z, y) / y);
	} else if (t_2 <= 1e+81) {
		tmp = (x / (y * exp(b))) / a;
	} else {
		tmp = (x * t_1) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t + (-1.0d0))
    t_2 = log(a) * (t + (-1.0d0))
    if (t_2 <= (-700.0d0)) then
        tmp = x * (t_1 / y)
    else if (t_2 <= (-600.0d0)) then
        tmp = x * ((z ** y) / y)
    else if (t_2 <= 1d+81) then
        tmp = (x / (y * exp(b))) / a
    else
        tmp = (x * t_1) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t + -1.0));
	double t_2 = Math.log(a) * (t + -1.0);
	double tmp;
	if (t_2 <= -700.0) {
		tmp = x * (t_1 / y);
	} else if (t_2 <= -600.0) {
		tmp = x * (Math.pow(z, y) / y);
	} else if (t_2 <= 1e+81) {
		tmp = (x / (y * Math.exp(b))) / a;
	} else {
		tmp = (x * t_1) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t + -1.0))
	t_2 = math.log(a) * (t + -1.0)
	tmp = 0
	if t_2 <= -700.0:
		tmp = x * (t_1 / y)
	elif t_2 <= -600.0:
		tmp = x * (math.pow(z, y) / y)
	elif t_2 <= 1e+81:
		tmp = (x / (y * math.exp(b))) / a
	else:
		tmp = (x * t_1) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t + -1.0)
	t_2 = Float64(log(a) * Float64(t + -1.0))
	tmp = 0.0
	if (t_2 <= -700.0)
		tmp = Float64(x * Float64(t_1 / y));
	elseif (t_2 <= -600.0)
		tmp = Float64(x * Float64((z ^ y) / y));
	elseif (t_2 <= 1e+81)
		tmp = Float64(Float64(x / Float64(y * exp(b))) / a);
	else
		tmp = Float64(Float64(x * t_1) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t + -1.0);
	t_2 = log(a) * (t + -1.0);
	tmp = 0.0;
	if (t_2 <= -700.0)
		tmp = x * (t_1 / y);
	elseif (t_2 <= -600.0)
		tmp = x * ((z ^ y) / y);
	elseif (t_2 <= 1e+81)
		tmp = (x / (y * exp(b))) / a;
	else
		tmp = (x * t_1) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -700.0], N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -600.0], N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+81], N[(N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -700

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 72.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 69.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 69.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot {a}^{\left(t + -1\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot {a}^{\left(t + -1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{-1 \cdot -1}}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-1 \cdot -1\right) \cdot {a}^{\left(t + -1\right)}}{y}} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1} \cdot {a}^{\left(t + -1\right)}}{y} \cdot x \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{y} \cdot x \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}} \cdot x \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{y} \cdot x \]

      12. +-lowering-+.f64 81.5

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y} \cdot x \]

   10. Applied egg-rr 81.5%

       \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y} \cdot x} \]

   if -700 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -600

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 84.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 84.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 67.9

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y}} \cdot x \]

      5. pow-lowering-pow.f64 67.9

         \[\leadsto \frac{\color{blue}{{z}^{y}}}{y} \cdot x \]

   10. Applied egg-rr 67.9%

       \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

   if -600 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 9.99999999999999921e80

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 76.5

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y \cdot e^{b}}}}{a} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{y \cdot e^{b}}}}{a} \]

      5. exp-lowering-exp.f64 78.7

         \[\leadsto \frac{\frac{x}{y \cdot \color{blue}{e^{b}}}}{a} \]

   10. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{y \cdot e^{b}}}{a}} \]

   if 9.99999999999999921e80 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 74.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 88.4

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 88.4%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}} \]

      10. +-lowering-+.f64 95.4

          \[\leadsto \frac{1}{\frac{y}{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}} \]

   10. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}{y} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}{y} \]

       5. +-lowering-+.f64 95.4

          \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}{y} \]

   12. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -700:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -600:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 10^{+81}:\\ \;\;\;\;\frac{\frac{x}{y \cdot e^{b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := {a}^{t} \cdot \frac{x}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 75:\\ \;\;\;\;\frac{x}{\left(y \cdot a\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;t\_1 \leq 200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot \frac{x}{y}, \frac{x \cdot \mathsf{fma}\left(0.5, b, -1\right)}{y}\right), \frac{x}{y}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))) (t_2 (* (pow a t) (/ x y))))
   (if (<= t_1 -1000000000.0)
     t_2
     (if (<= t_1 75.0)
       (/
        x
        (* (* y a) (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))
       (if (<= t_1 200000000000.0)
         (/
          (fma
           b
           (fma
            (* b b)
            (* -0.16666666666666666 (/ x y))
            (/ (* x (fma 0.5 b -1.0)) y))
           (/ x y))
          a)
         t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = pow(a, t) * (x / y);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 75.0) {
		tmp = x / ((y * a) * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (t_1 <= 200000000000.0) {
		tmp = fma(b, fma((b * b), (-0.16666666666666666 * (x / y)), ((x * fma(0.5, b, -1.0)) / y)), (x / y)) / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = Float64((a ^ t) * Float64(x / y))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_2;
	elseif (t_1 <= 75.0)
		tmp = Float64(x / Float64(Float64(y * a) * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (t_1 <= 200000000000.0)
		tmp = Float64(fma(b, fma(Float64(b * b), Float64(-0.16666666666666666 * Float64(x / y)), Float64(Float64(x * fma(0.5, b, -1.0)) / y)), Float64(x / y)) / a);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, t], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000000.0], t$95$2, If[LessEqual[t$95$1, 75.0], N[(x / N[(N[(y * a), $MachinePrecision] * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 200000000000.0], N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(-0.16666666666666666 * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1e9 or 2e11 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 74.2

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 74.2%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around inf

      \[\leadsto {a}^{\color{blue}{t}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

   11. Simplified 74.2%

       \[\leadsto {a}^{\color{blue}{t}} \cdot \frac{x}{y} \]

   if -1e9 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 75

   1. Initial program 97.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 68.2

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 75.1

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 68.5

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 68.5%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot y\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + 1\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + 1\right) \cdot \left(a \cdot y\right)}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + 1\right) \cdot \left(a \cdot y\right)}} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(b, b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1, 1\right)} \cdot \left(a \cdot y\right)} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, b \cdot \frac{1}{6} + \frac{1}{2}, 1\right)}, 1\right) \cdot \left(a \cdot y\right)} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{6}, \frac{1}{2}\right)}, 1\right), 1\right) \cdot \left(a \cdot y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \color{blue}{\left(y \cdot a\right)}} \]

       8. *-lowering-*.f64 68.5

          \[\leadsto \frac{x}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\left(y \cdot a\right)}} \]

   13. Applied egg-rr 68.5%

       \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(y \cdot a\right)}} \]

   if 75 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2e11

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 82.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 73.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 40.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{b \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2} \cdot x}{y} + \frac{x \cdot \left(\frac{1}{2} \cdot b - 1\right)}{y}\right) + \frac{x}{y}}{a}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{b \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2} \cdot x}{y} + \frac{x \cdot \left(\frac{1}{2} \cdot b - 1\right)}{y}\right) + \frac{x}{y}}{a}} \]

   14. Simplified 58.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b \cdot b, \frac{x}{y} \cdot -0.16666666666666666, \frac{x \cdot \mathsf{fma}\left(0.5, b, -1\right)}{y}\right), \frac{x}{y}\right)}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -1000000000:\\ \;\;\;\;{a}^{t} \cdot \frac{x}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 75:\\ \;\;\;\;\frac{x}{\left(y \cdot a\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b \cdot b, -0.16666666666666666 \cdot \frac{x}{y}, \frac{x \cdot \mathsf{fma}\left(0.5, b, -1\right)}{y}\right), \frac{x}{y}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;{a}^{t} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -61:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 18:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot \left(t + -1\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -61.0)
     t_1
     (if (<= y 18.0) (/ (* x (exp (- (* (log a) (+ t -1.0)) b))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (y <= -61.0) {
		tmp = t_1;
	} else if (y <= 18.0) {
		tmp = (x * exp(((log(a) * (t + -1.0)) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (y <= (-61.0d0)) then
        tmp = t_1
    else if (y <= 18.0d0) then
        tmp = (x * exp(((log(a) * (t + (-1.0d0))) - b))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (y <= -61.0) {
		tmp = t_1;
	} else if (y <= 18.0) {
		tmp = (x * Math.exp(((Math.log(a) * (t + -1.0)) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if y <= -61.0:
		tmp = t_1
	elif y <= 18.0:
		tmp = (x * math.exp(((math.log(a) * (t + -1.0)) - b))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y)
	tmp = 0.0
	if (y <= -61.0)
		tmp = t_1;
	elseif (y <= 18.0)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * Float64(t + -1.0)) - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (y <= -61.0)
		tmp = t_1;
	elseif (y <= 18.0)
		tmp = (x * exp(((log(a) * (t + -1.0)) - b))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -61.0], t$95$1, If[LessEqual[y, 18.0], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -61 or 18 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 94.7

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 94.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   if -61 < y < 18

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]

      2. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{\left(e^{\log a}\right)} \cdot \left(t - 1\right) - b}}{y} \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\log \left(e^{\log a}\right)} \cdot \left(t - 1\right) - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot \left(t - 1\right) - b}}{y} \]

      5. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\log a \cdot \color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)} - b}}{y} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\log a \cdot \left(t + \color{blue}{-1}\right) - b}}{y} \]

      7. +-lowering-+.f64 98.1

         \[\leadsto \frac{x \cdot e^{\log a \cdot \color{blue}{\left(t + -1\right)} - b}}{y} \]

   5. Simplified 98.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t + -1\right)} - b}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -6.8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -6.8)
     t_1
     (if (<= y 1.65) (/ (* x (pow a (+ t -1.0))) (* y (exp b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (y <= -6.8) {
		tmp = t_1;
	} else if (y <= 1.65) {
		tmp = (x * pow(a, (t + -1.0))) / (y * exp(b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (y <= (-6.8d0)) then
        tmp = t_1
    else if (y <= 1.65d0) then
        tmp = (x * (a ** (t + (-1.0d0)))) / (y * exp(b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (y <= -6.8) {
		tmp = t_1;
	} else if (y <= 1.65) {
		tmp = (x * Math.pow(a, (t + -1.0))) / (y * Math.exp(b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if y <= -6.8:
		tmp = t_1
	elif y <= 1.65:
		tmp = (x * math.pow(a, (t + -1.0))) / (y * math.exp(b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y)
	tmp = 0.0
	if (y <= -6.8)
		tmp = t_1;
	elseif (y <= 1.65)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / Float64(y * exp(b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (y <= -6.8)
		tmp = t_1;
	elseif (y <= 1.65)
		tmp = (x * (a ^ (t + -1.0))) / (y * exp(b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6.8], t$95$1, If[LessEqual[y, 1.65], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.79999999999999982 or 1.6499999999999999 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 94.7

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 94.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   if -6.79999999999999982 < y < 1.6499999999999999

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 88.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= b -4.5e+17)
     t_1
     (if (<= b 2.2e+28) (/ 1.0 (/ y (* x (pow a (+ t -1.0))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((y * log(z)) - b))) / y;
	double tmp;
	if (b <= -4.5e+17) {
		tmp = t_1;
	} else if (b <= 2.2e+28) {
		tmp = 1.0 / (y / (x * pow(a, (t + -1.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((y * log(z)) - b))) / y
    if (b <= (-4.5d+17)) then
        tmp = t_1
    else if (b <= 2.2d+28) then
        tmp = 1.0d0 / (y / (x * (a ** (t + (-1.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((y * Math.log(z)) - b))) / y;
	double tmp;
	if (b <= -4.5e+17) {
		tmp = t_1;
	} else if (b <= 2.2e+28) {
		tmp = 1.0 / (y / (x * Math.pow(a, (t + -1.0))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((y * math.log(z)) - b))) / y
	tmp = 0
	if b <= -4.5e+17:
		tmp = t_1
	elif b <= 2.2e+28:
		tmp = 1.0 / (y / (x * math.pow(a, (t + -1.0))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(y * log(z)) - b))) / y)
	tmp = 0.0
	if (b <= -4.5e+17)
		tmp = t_1;
	elseif (b <= 2.2e+28)
		tmp = Float64(1.0 / Float64(y / Float64(x * (a ^ Float64(t + -1.0)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((y * log(z)) - b))) / y;
	tmp = 0.0;
	if (b <= -4.5e+17)
		tmp = t_1;
	elseif (b <= 2.2e+28)
		tmp = 1.0 / (y / (x * (a ^ (t + -1.0))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[b, -4.5e+17], t$95$1, If[LessEqual[b, 2.2e+28], N[(1.0 / N[(y / N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5e17 or 2.19999999999999986e28 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 95.2

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 95.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   if -4.5e17 < b < 2.19999999999999986e28

   1. Initial program 97.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 76.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 71.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}} \]

      10. +-lowering-+.f64 78.0

          \[\leadsto \frac{1}{\frac{y}{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}} \]

   10. Applied egg-rr 78.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 72.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.5e+63)
   (fma
    b
    (*
     x
     (fma
      -0.16666666666666666
      (/ (* b b) (* y a))
      (/ (fma 0.5 b -1.0) (* y a))))
    (/ x (* y a)))
   (if (<= b 3e+28)
     (/ 1.0 (/ y (* x (pow a (+ t -1.0)))))
     (/ x (* a (* y (exp b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.5e+63) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= 3e+28) {
		tmp = 1.0 / (y / (x * pow(a, (t + -1.0))));
	} else {
		tmp = x / (a * (y * exp(b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.5e+63)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= 3e+28)
		tmp = Float64(1.0 / Float64(y / Float64(x * (a ^ Float64(t + -1.0)))));
	else
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.5e+63], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e+28], N[(1.0 / N[(y / N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.50000000000000017e63

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 62.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 85.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 89.2

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 89.2%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -4.50000000000000017e63 < b < 3.0000000000000001e28

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 71.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}} \]

      10. +-lowering-+.f64 76.9

          \[\leadsto \frac{1}{\frac{y}{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}} \]

   10. Applied egg-rr 76.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}} \]

   if 3.0000000000000001e28 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.7

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 81.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5.5e+63)
   (fma
    b
    (*
     x
     (fma
      -0.16666666666666666
      (/ (* b b) (* y a))
      (/ (fma 0.5 b -1.0) (* y a))))
    (/ x (* y a)))
   (if (<= b 7.2e+28)
     (/ (* x (pow a (+ t -1.0))) y)
     (/ x (* a (* y (exp b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5.5e+63) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= 7.2e+28) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = x / (a * (y * exp(b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5.5e+63)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= 7.2e+28)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(x / Float64(a * Float64(y * exp(b))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5.5e+63], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+28], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.50000000000000004e63

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 62.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 85.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 89.2

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 89.2%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -5.50000000000000004e63 < b < 7.1999999999999999e28

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 71.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}} \]

      10. +-lowering-+.f64 76.9

          \[\leadsto \frac{1}{\frac{y}{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}} \]

   10. Applied egg-rr 76.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}{y} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}{y} \]

       5. +-lowering-+.f64 76.8

          \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}{y} \]

   12. Applied egg-rr 76.8%

       \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

   if 7.1999999999999999e28 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.7

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 81.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -7.5e+62)
   (fma
    b
    (*
     x
     (fma
      -0.16666666666666666
      (/ (* b b) (* y a))
      (/ (fma 0.5 b -1.0) (* y a))))
    (/ x (* y a)))
   (if (<= b 3.3e+28) (/ (* x (pow a (+ t -1.0))) y) (/ (* x (exp (- b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -7.5e+62) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= 3.3e+28) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = (x * exp(-b)) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -7.5e+62)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= 3.3e+28)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(-b))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -7.5e+62], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.3e+28], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.49999999999999998e62

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 62.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 85.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 89.2

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 89.2%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -7.49999999999999998e62 < b < 3.3e28

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 71.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot -1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot -1}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{y}{{a}^{\left(t + -1\right)} \cdot x}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{y}{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}} \]

      10. +-lowering-+.f64 76.9

          \[\leadsto \frac{1}{\frac{y}{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}} \]

   10. Applied egg-rr 76.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x \cdot {a}^{\left(t + -1\right)}}}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot {a}^{\left(t + -1\right)}}}{y} \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t + -1\right)}}}{y} \]

       5. +-lowering-+.f64 76.8

          \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}{y} \]

   12. Applied egg-rr 76.8%

       \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

   if 3.3e28 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 92.8

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 92.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 81.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 81.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4e+63)
   (fma
    b
    (*
     x
     (fma
      -0.16666666666666666
      (/ (* b b) (* y a))
      (/ (fma 0.5 b -1.0) (* y a))))
    (/ x (* y a)))
   (if (<= b 2.3e+28) (* x (/ (pow a (+ t -1.0)) y)) (/ (* x (exp (- b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4e+63) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= 2.3e+28) {
		tmp = x * (pow(a, (t + -1.0)) / y);
	} else {
		tmp = (x * exp(-b)) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4e+63)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= 2.3e+28)
		tmp = Float64(x * Float64((a ^ Float64(t + -1.0)) / y));
	else
		tmp = Float64(Float64(x * exp(Float64(-b))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4e+63], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e+28], N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.00000000000000023e63

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 62.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 85.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 89.2

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 89.2%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -4.00000000000000023e63 < b < 2.29999999999999984e28

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 71.9

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 71.9%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot {a}^{\left(t + -1\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot {a}^{\left(t + -1\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{\color{blue}{-1 \cdot -1}}{y} \cdot {a}^{\left(t + -1\right)}\right) \cdot x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-1 \cdot -1\right) \cdot {a}^{\left(t + -1\right)}}{y}} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1} \cdot {a}^{\left(t + -1\right)}}{y} \cdot x \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{y} \cdot x \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}} \cdot x \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{y} \cdot x \]

      12. +-lowering-+.f64 73.7

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y} \cdot x \]

   10. Applied egg-rr 73.7%

       \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y} \cdot x} \]

   if 2.29999999999999984e28 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 92.8

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 92.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\mathsf{neg}\left(b\right)}}}{y} \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{e^{\mathsf{neg}\left(b\right)}}}{y} \]

      3. neg-lowering-neg.f64 81.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 81.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq -2.25 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y \cdot \mathsf{fma}\left(b, t\_1 \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right)}{\mathsf{fma}\left(b, t\_1, -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b -2.25e+55)
     (fma
      b
      (*
       x
       (fma
        -0.16666666666666666
        (/ (* b b) (* y a))
        (/ (fma 0.5 b -1.0) (* y a))))
      (/ x (* y a)))
     (if (<= b 2e+28)
       (* x (/ (pow z y) y))
       (if (<= b 1.05e+103)
         (/
          x
          (*
           a
           (/
            (*
             y
             (fma
              b
              (* t_1 (fma b (* b (fma b 0.16666666666666666 0.5)) b))
              -1.0))
            (fma b t_1 -1.0))))
         (/ x (* a (* y (* (* b b) (fma 0.16666666666666666 b 0.5))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (b <= -2.25e+55) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= 2e+28) {
		tmp = x * (pow(z, y) / y);
	} else if (b <= 1.05e+103) {
		tmp = x / (a * ((y * fma(b, (t_1 * fma(b, (b * fma(b, 0.16666666666666666, 0.5)), b)), -1.0)) / fma(b, t_1, -1.0)));
	} else {
		tmp = x / (a * (y * ((b * b) * fma(0.16666666666666666, b, 0.5))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (b <= -2.25e+55)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= 2e+28)
		tmp = Float64(x * Float64((z ^ y) / y));
	elseif (b <= 1.05e+103)
		tmp = Float64(x / Float64(a * Float64(Float64(y * fma(b, Float64(t_1 * fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), b)), -1.0)) / fma(b, t_1, -1.0))));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(Float64(b * b) * fma(0.16666666666666666, b, 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, -2.25e+55], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+28], N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(x / N[(a * N[(N[(y * N[(b * N[(t$95$1 * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(N[(b * b), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.24999999999999999e55

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 61.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 85.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 81.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 87.6

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 87.6%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -2.24999999999999999e55 < b < 1.99999999999999992e28

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 53.4

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 53.4%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 50.0

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y}} \cdot x \]

      5. pow-lowering-pow.f64 50.0

         \[\leadsto \frac{\color{blue}{{z}^{y}}}{y} \cdot x \]

   10. Applied egg-rr 50.0%

       \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

   if 1.99999999999999992e28 < b < 1.0500000000000001e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 55.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 72.7

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 29.8

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 29.8%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + 1\right) \cdot y\right)}} \]

       2. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\frac{\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}} \cdot y\right)} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1\right) \cdot y}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1\right) \cdot y}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}}} \]

   13. Applied egg-rr 67.5%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right) \cdot y}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -1\right)}}} \]

   if 1.0500000000000001e103 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 78.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 82.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 82.6

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left({b}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot {b}^{3}\right)}\right)} \]

       2. cube-mult N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot b\right) \cdot {b}^{2}\right)}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)} \cdot {b}^{2}\right)\right)} \]

       6. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot b + \left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b\right)} \cdot {b}^{2}\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right)}\right) \cdot {b}^{2}\right)\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \frac{1}{2} \cdot \color{blue}{1}\right) \cdot {b}^{2}\right)\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \color{blue}{\frac{1}{2}}\right) \cdot {b}^{2}\right)\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)} \cdot {b}^{2}\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot b\right) \cdot {b}^{2}\right)}\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)} \cdot {b}^{2}\right)\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, b, \frac{1}{2}\right)} \cdot {b}^{2}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\mathsf{fma}\left(\frac{1}{6}, b, \frac{1}{2}\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

       15. *-lowering-*.f64 82.6

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

   14. Simplified 82.6%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \left(b \cdot b\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log a \leq -235:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (log a) -235.0) (/ (/ x a) y) (/ x (* a (fma b y y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (log(a) <= -235.0) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (a * fma(b, y, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (log(a) <= -235.0)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(a * fma(b, y, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[Log[a], $MachinePrecision], -235.0], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(b * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 a) < -235

   1. Initial program 99.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 77.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 66.2

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
   10. Step-by-step derivation

   11. Simplified 48.6%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]

   if -235 < (log.f64 a)

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 63.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 63.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. accelerator-lowering-fma.f64 39.9

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 39.9%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq -6.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{0.5}{b}, y \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y \cdot \mathsf{fma}\left(b, t\_1 \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right)}{\mathsf{fma}\left(b, t\_1, -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b -1.3e-57)
     (fma
      b
      (*
       x
       (fma
        -0.16666666666666666
        (/ (* b b) (* y a))
        (/ (fma 0.5 b -1.0) (* y a))))
      (/ x (* y a)))
     (if (<= b -6.1e-96)
       (/
        x
        (* a (* (* b (* b b)) (fma y (/ 0.5 b) (* y 0.16666666666666666)))))
       (if (<= b 1.05e+103)
         (/
          x
          (*
           a
           (/
            (*
             y
             (fma
              b
              (* t_1 (fma b (* b (fma b 0.16666666666666666 0.5)) b))
              -1.0))
            (fma b t_1 -1.0))))
         (/ x (* a (* y (* (* b b) (fma 0.16666666666666666 b 0.5))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (b <= -1.3e-57) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else if (b <= -6.1e-96) {
		tmp = x / (a * ((b * (b * b)) * fma(y, (0.5 / b), (y * 0.16666666666666666))));
	} else if (b <= 1.05e+103) {
		tmp = x / (a * ((y * fma(b, (t_1 * fma(b, (b * fma(b, 0.16666666666666666, 0.5)), b)), -1.0)) / fma(b, t_1, -1.0)));
	} else {
		tmp = x / (a * (y * ((b * b) * fma(0.16666666666666666, b, 0.5))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (b <= -1.3e-57)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	elseif (b <= -6.1e-96)
		tmp = Float64(x / Float64(a * Float64(Float64(b * Float64(b * b)) * fma(y, Float64(0.5 / b), Float64(y * 0.16666666666666666)))));
	elseif (b <= 1.05e+103)
		tmp = Float64(x / Float64(a * Float64(Float64(y * fma(b, Float64(t_1 * fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), b)), -1.0)) / fma(b, t_1, -1.0))));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(Float64(b * b) * fma(0.16666666666666666, b, 0.5)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, -1.3e-57], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.1e-96], N[(x / N[(a * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(y * N[(0.5 / b), $MachinePrecision] + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(x / N[(a * N[(N[(y * N[(b * N[(t$95$1 * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(N[(b * b), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.29999999999999993e-57

   1. Initial program 99.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 66.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 76.8

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 69.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 73.2

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 73.2%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -1.29999999999999993e-57 < b < -6.1000000000000001e-96

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 3.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 3.6

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 3.6%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left({b}^{3} \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left({b}^{3} \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)}} \]

       2. cube-mult N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot \left(b \cdot b\right)\right)} \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \color{blue}{{b}^{2}}\right) \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(b \cdot {b}^{2}\right)} \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot \left(\frac{1}{6} \cdot y + \frac{1}{2} \cdot \frac{y}{b}\right)\right)} \]

       7. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{y}{b} + \frac{1}{6} \cdot y\right)}\right)} \]

       8. associate-*r/ N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{\frac{\frac{1}{2} \cdot y}{b}} + \frac{1}{6} \cdot y\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\frac{\color{blue}{y \cdot \frac{1}{2}}}{b} + \frac{1}{6} \cdot y\right)\right)} \]

       10. associate-/l* N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(\color{blue}{y \cdot \frac{\frac{1}{2}}{b}} + \frac{1}{6} \cdot y\right)\right)} \]

       11. metadata-eval N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(y \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b} + \frac{1}{6} \cdot y\right)\right)} \]

       12. associate-*r/ N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b}\right)} + \frac{1}{6} \cdot y\right)\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot \frac{1}{b}, \frac{1}{6} \cdot y\right)}\right)} \]

       14. associate-*r/ N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b}}, \frac{1}{6} \cdot y\right)\right)} \]

       15. metadata-eval N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{\color{blue}{\frac{1}{2}}}{b}, \frac{1}{6} \cdot y\right)\right)} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\frac{1}{2}}{b}}, \frac{1}{6} \cdot y\right)\right)} \]

       17. *-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{\frac{1}{2}}{b}, \color{blue}{y \cdot \frac{1}{6}}\right)\right)} \]

       18. *-lowering-*.f64 64.0

           \[\leadsto \frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{0.5}{b}, \color{blue}{y \cdot 0.16666666666666666}\right)\right)} \]

   14. Simplified 64.0%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{0.5}{b}, y \cdot 0.16666666666666666\right)\right)}} \]

   if -6.1000000000000001e-96 < b < 1.0500000000000001e103

   1. Initial program 97.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 74.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 50.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 50.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 43.2

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 43.2%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + 1\right) \cdot y\right)}} \]

       2. flip-+ N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\frac{\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}} \cdot y\right)} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1\right) \cdot y}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\left(\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 1 \cdot 1\right) \cdot y}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) - 1}}} \]

   13. Applied egg-rr 48.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right) \cdot y}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -1\right)}}} \]

   if 1.0500000000000001e103 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 78.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 78.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 82.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 82.6

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left({b}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot {b}^{3}\right)}\right)} \]

       2. cube-mult N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)\right)} \]

       4. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right) \cdot b\right) \cdot {b}^{2}\right)}\right)} \]

       5. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)} \cdot {b}^{2}\right)\right)} \]

       6. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot b + \left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b\right)} \cdot {b}^{2}\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right)}\right) \cdot {b}^{2}\right)\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \frac{1}{2} \cdot \color{blue}{1}\right) \cdot {b}^{2}\right)\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\left(\frac{1}{6} \cdot b + \color{blue}{\frac{1}{2}}\right) \cdot {b}^{2}\right)\right)} \]

       10. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)} \cdot {b}^{2}\right)\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{6} \cdot b\right) \cdot {b}^{2}\right)}\right)} \]

       12. +-commutative N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot b + \frac{1}{2}\right)} \cdot {b}^{2}\right)\right)} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6}, b, \frac{1}{2}\right)} \cdot {b}^{2}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\mathsf{fma}\left(\frac{1}{6}, b, \frac{1}{2}\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

       15. *-lowering-*.f64 82.6

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)} \]

   14. Simplified 82.6%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \left(b \cdot b\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{elif}\;b \leq -6.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(y, \frac{0.5}{b}, y \cdot 0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{a \cdot \frac{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -9e-46)
   (fma
    b
    (*
     x
     (fma
      -0.16666666666666666
      (/ (* b b) (* y a))
      (/ (fma 0.5 b -1.0) (* y a))))
    (/ x (* y a)))
   (/ (/ x a) (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -9e-46) {
		tmp = fma(b, (x * fma(-0.16666666666666666, ((b * b) / (y * a)), (fma(0.5, b, -1.0) / (y * a)))), (x / (y * a)));
	} else {
		tmp = (x / a) / (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -9e-46)
		tmp = fma(b, Float64(x * fma(-0.16666666666666666, Float64(Float64(b * b) / Float64(y * a)), Float64(fma(0.5, b, -1.0) / Float64(y * a)))), Float64(x / Float64(y * a)));
	else
		tmp = Float64(Float64(x / a) / Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -9e-46], N[(b * N[(x * N[(-0.16666666666666666 * N[(N[(b * b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + -1.0), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.00000000000000001e-46

   1. Initial program 99.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 65.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 78.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \left(\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \frac{1}{2} \cdot \frac{b}{a \cdot y}\right) - \frac{1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       2. associate--l+ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\frac{1}{2} \cdot \frac{b}{a \cdot y} - \frac{1}{a \cdot y}\right)\right)}, \frac{x}{a \cdot y}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \left(\color{blue}{\frac{\frac{1}{2} \cdot b}{a \cdot y}} - \frac{1}{a \cdot y}\right)\right), \frac{x}{a \cdot y}\right) \]

       4. div-sub N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \left(\frac{-1}{6} \cdot \frac{{b}^{2}}{a \cdot y} + \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{b}^{2}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{b}^{2}}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{b \cdot b}}{a \cdot y}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{\color{blue}{a \cdot y}}, \frac{\frac{1}{2} \cdot b - 1}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \color{blue}{\frac{\frac{1}{2} \cdot b - 1}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\frac{1}{2} \cdot b + \color{blue}{-1}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{b \cdot b}{a \cdot y}, \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}}{a \cdot y}\right), \frac{x}{a \cdot y}\right) \]

       14. *-lowering-*.f64 74.4

           \[\leadsto \mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{\color{blue}{a \cdot y}}\right), \frac{x}{a \cdot y}\right) \]

   14. Simplified 74.4%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{a \cdot y}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{a \cdot y}\right)}, \frac{x}{a \cdot y}\right) \]

   if -9.00000000000000001e-46 < b

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 58.1

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 58.1%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)} \]

       7. accelerator-lowering-fma.f64 54.0

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)} \]

   11. Simplified 54.0%

       \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(b, x \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{b \cdot b}{y \cdot a}, \frac{\mathsf{fma}\left(0.5, b, -1\right)}{y \cdot a}\right), \frac{x}{y \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.75e-45)
   (/ (* -0.16666666666666666 (* x (* b (* b b)))) (* y a))
   (/ (/ x a) (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.75e-45) {
		tmp = (-0.16666666666666666 * (x * (b * (b * b)))) / (y * a);
	} else {
		tmp = (x / a) / (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.75e-45)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * Float64(b * Float64(b * b)))) / Float64(y * a));
	else
		tmp = Float64(Float64(x / a) / Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.75e-45], N[(N[(-0.16666666666666666 * N[(x * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.75e-45

   1. Initial program 99.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 65.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 78.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{a \cdot y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}}{a \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       6. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)}{a \cdot y} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)}{a \cdot y} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot {b}^{2}\right)}\right)}{a \cdot y} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       11. *-lowering-*.f64 71.8

           \[\leadsto \frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\color{blue}{a \cdot y}} \]

   14. Simplified 71.8%

       \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot y}} \]

   if -1.75e-45 < b

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 58.1

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 58.1%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)} \]

       7. accelerator-lowering-fma.f64 54.0

          \[\leadsto \frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)} \]

   11. Simplified 54.0%

       \[\leadsto \frac{\frac{x}{a}}{y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot t\_1\right)}{y \cdot a}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(b, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (* b b))))
   (if (<= b -1.6e-45)
     (/ (* -0.16666666666666666 (* x t_1)) (* y a))
     (if (<= b 4.1e+31)
       (/ (/ x a) (fma b y y))
       (/ x (* a (* y (* 0.16666666666666666 t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (b * b);
	double tmp;
	if (b <= -1.6e-45) {
		tmp = (-0.16666666666666666 * (x * t_1)) / (y * a);
	} else if (b <= 4.1e+31) {
		tmp = (x / a) / fma(b, y, y);
	} else {
		tmp = x / (a * (y * (0.16666666666666666 * t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= -1.6e-45)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * t_1)) / Float64(y * a));
	elseif (b <= 4.1e+31)
		tmp = Float64(Float64(x / a) / fma(b, y, y));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(0.16666666666666666 * t_1))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.6e-45], N[(N[(-0.16666666666666666 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e+31], N[(N[(x / a), $MachinePrecision] / N[(b * y + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.60000000000000004e-45

   1. Initial program 99.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 65.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 78.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{a \cdot y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}}{a \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       6. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)}{a \cdot y} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)}{a \cdot y} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot {b}^{2}\right)}\right)}{a \cdot y} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       11. *-lowering-*.f64 71.8

           \[\leadsto \frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\color{blue}{a \cdot y}} \]

   14. Simplified 71.8%

       \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot y}} \]

   if -1.60000000000000004e-45 < b < 4.1000000000000002e31

   1. Initial program 97.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 77.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 46.9

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 46.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y + b \cdot y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{\color{blue}{b \cdot y + y}} \]

       2. accelerator-lowering-fma.f64 46.6

          \[\leadsto \frac{\frac{x}{a}}{\color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 46.6%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   if 4.1000000000000002e31 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 80.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 69.4

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 69.4%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot \left(a \cdot \left({b}^{3} \cdot y\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot \left({b}^{3} \cdot y\right)\right) \cdot \frac{1}{6}}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(\left({b}^{3} \cdot y\right) \cdot \frac{1}{6}\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left({b}^{3} \cdot \left(y \cdot \frac{1}{6}\right)\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left({b}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left({b}^{3} \cdot \left(\frac{1}{6} \cdot y\right)\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left({b}^{3} \cdot \frac{1}{6}\right) \cdot y\right)}} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {b}^{3}\right)} \cdot y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {b}^{3}\right)\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {b}^{3}\right)\right)}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot {b}^{3}\right)}\right)} \]

       11. cube-mult N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)\right)} \]

       12. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot {b}^{2}\right)}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

       15. *-lowering-*.f64 69.4

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

   14. Simplified 69.4%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{y \cdot a}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(b, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5.4e-15)
   (/ (* -0.16666666666666666 (* x (* b (* b b)))) (* y a))
   (/ x (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5.4e-15) {
		tmp = (-0.16666666666666666 * (x * (b * (b * b)))) / (y * a);
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5.4e-15)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x * Float64(b * Float64(b * b)))) / Float64(y * a));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5.4e-15], N[(N[(-0.16666666666666666 * N[(x * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.40000000000000018e-15

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 64.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 78.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right) + \left(\frac{-1}{2} \cdot \frac{x}{a \cdot y} + \frac{1}{6} \cdot \frac{x}{a \cdot y}\right)\right)\right) - \left(-1 \cdot \frac{x}{a \cdot y} + \frac{1}{2} \cdot \frac{x}{a \cdot y}\right)\right) - \frac{x}{a \cdot y}\right) + \frac{x}{a \cdot y}} \]

   11. Simplified 70.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{x \cdot 0.16666666666666666}{a \cdot y} \cdot \left(-b\right), \frac{x}{a \cdot y} \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right), \frac{x}{a \cdot y}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{a \cdot y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{a \cdot y}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}}{a \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{a \cdot y} \]

       6. cube-mult N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)}{a \cdot y} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)}{a \cdot y} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{\left(b \cdot {b}^{2}\right)}\right)}{a \cdot y} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{-1}{6} \cdot \left(x \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)}{a \cdot y} \]

       11. *-lowering-*.f64 73.0

           \[\leadsto \frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{\color{blue}{a \cdot y}} \]

   14. Simplified 73.0%

       \[\leadsto \color{blue}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot y}} \]

   if -5.40000000000000018e-15 < b

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 75.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 57.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 57.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 53.1

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 53.1%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.1e-127)
   (* (/ x (* y a)) (- 1.0 b))
   (/ x (* a (* y (* 0.16666666666666666 (* b (* b b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.1e-127) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = x / (a * (y * (0.16666666666666666 * (b * (b * b)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.1d-127) then
        tmp = (x / (y * a)) * (1.0d0 - b)
    else
        tmp = x / (a * (y * (0.16666666666666666d0 * (b * (b * b)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.1e-127) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = x / (a * (y * (0.16666666666666666 * (b * (b * b)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 1.1e-127:
		tmp = (x / (y * a)) * (1.0 - b)
	else:
		tmp = x / (a * (y * (0.16666666666666666 * (b * (b * b)))))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.1e-127)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	else
		tmp = Float64(x / Float64(a * Float64(y * Float64(0.16666666666666666 * Float64(b * Float64(b * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 1.1e-127)
		tmp = (x / (y * a)) * (1.0 - b);
	else
		tmp = x / (a * (y * (0.16666666666666666 * (b * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.1e-127], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(0.16666666666666666 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001e-127

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 72.8

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.8%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 62.4

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 62.4%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{a \cdot y}\right)\right)} + \frac{x}{a \cdot y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{a \cdot y}}\right)\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot \frac{x}{a \cdot y} \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       9. *-lowering-*.f64 51.0

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 51.0%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if 1.1000000000000001e-127 < b

   1. Initial program 99.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 72.7

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 64.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 1\right)\right)} \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 1\right)\right)} \]

       5. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 1\right)\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right)\right)} \]

       7. accelerator-lowering-fma.f64 55.5

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 1\right)\right)} \]

   11. Simplified 55.5%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{\color{blue}{\frac{1}{6} \cdot \left(a \cdot \left({b}^{3} \cdot y\right)\right)}} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot \left({b}^{3} \cdot y\right)\right) \cdot \frac{1}{6}}} \]

       2. associate-*l* N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(\left({b}^{3} \cdot y\right) \cdot \frac{1}{6}\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left({b}^{3} \cdot \left(y \cdot \frac{1}{6}\right)\right)}} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left({b}^{3} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left({b}^{3} \cdot \left(\frac{1}{6} \cdot y\right)\right)}} \]

       6. associate-*r* N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left({b}^{3} \cdot \frac{1}{6}\right) \cdot y\right)}} \]

       7. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(\frac{1}{6} \cdot {b}^{3}\right)} \cdot y\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {b}^{3}\right)\right)}} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {b}^{3}\right)\right)}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot {b}^{3}\right)}\right)} \]

       11. cube-mult N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)\right)} \]

       12. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{{b}^{2}}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(b \cdot {b}^{2}\right)}\right)\right)} \]

       14. unpow2 N/A

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(\frac{1}{6} \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

       15. *-lowering-*.f64 56.5

           \[\leadsto \frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)} \]

   14. Simplified 56.5%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.0% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.15e+56)
   (* (/ x (* y a)) (- 1.0 b))
   (/ x (* a (* y (fma b (fma b 0.5 1.0) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.15e+56) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = x / (a * (y * fma(b, fma(b, 0.5, 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.15e+56)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, 0.5, 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.15e+56], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.15000000000000007e56

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 63.1

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 87.2

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{a \cdot y}\right)\right)} + \frac{x}{a \cdot y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{a \cdot y}}\right)\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot \frac{x}{a \cdot y} \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       9. *-lowering-*.f64 56.2

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 56.2%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if -1.15000000000000007e56 < b

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 74.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 74.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 57.9

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right) + 1\right)}\right)} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, 1 + \frac{1}{2} \cdot b, 1\right)}\right)} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot b + 1}, 1\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{2}} + 1, 1\right)\right)} \]

       5. accelerator-lowering-fma.f64 51.5

          \[\leadsto \frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 1\right)\right)} \]

   11. Simplified 51.5%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-196}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1e-196)
   (* (/ x (* y a)) (- 1.0 b))
   (if (<= b 2.2e+118) (/ (/ x a) y) (/ x (* a (fma b y y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1e-196) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else if (b <= 2.2e+118) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (a * fma(b, y, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1e-196)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	elseif (b <= 2.2e+118)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(a * fma(b, y, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1e-196], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e+118], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(b * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1e-196

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 72.2

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 62.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{a \cdot y}\right)\right)} + \frac{x}{a \cdot y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{a \cdot y}}\right)\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot \frac{x}{a \cdot y} \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       9. *-lowering-*.f64 49.8

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 49.8%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if 1e-196 < b < 2.19999999999999986e118

   1. Initial program 99.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 68.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 54.0

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 54.0%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]
   10. Step-by-step derivation

   11. Simplified 40.5%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y}} \]

   if 2.19999999999999986e118 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 82.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. accelerator-lowering-fma.f64 47.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 47.1%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-196}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 37.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-201}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(b, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1e-201) (* (/ x (* y a)) (- 1.0 b)) (/ (/ x a) (fma b y y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1e-201) {
		tmp = (x / (y * a)) * (1.0 - b);
	} else {
		tmp = (x / a) / fma(b, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1e-201)
		tmp = Float64(Float64(x / Float64(y * a)) * Float64(1.0 - b));
	else
		tmp = Float64(Float64(x / a) / fma(b, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1e-201], N[(N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / N[(b * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999946e-202

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 72.2

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 62.3

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{a \cdot y}\right)\right)} + \frac{x}{a \cdot y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{a \cdot y}}\right)\right) + \frac{x}{a \cdot y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{a \cdot y}} + \frac{x}{a \cdot y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot \frac{x}{a \cdot y} \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + 1\right) \cdot \frac{x}{a \cdot y} \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \color{blue}{\frac{x}{a \cdot y}} \]

       9. *-lowering-*.f64 49.8

          \[\leadsto \left(\left(-b\right) + 1\right) \cdot \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 49.8%

       \[\leadsto \color{blue}{\left(\left(-b\right) + 1\right) \cdot \frac{x}{a \cdot y}} \]

   if 9.99999999999999946e-202 < b

   1. Initial program 99.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 65.9

         \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   8. Simplified 65.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y \cdot e^{b}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{x}{a}}{\color{blue}{y + b \cdot y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{x}{a}}{\color{blue}{b \cdot y + y}} \]

       2. accelerator-lowering-fma.f64 40.7

          \[\leadsto \frac{\frac{x}{a}}{\color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 40.7%

       \[\leadsto \frac{\frac{x}{a}}{\color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-201}:\\ \;\;\;\;\frac{x}{y \cdot a} \cdot \left(1 - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(b, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 34.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.4e+118) (* x (/ 1.0 (* y a))) (/ x (* a (fma b y y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.4e+118) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / (a * fma(b, y, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.4e+118)
		tmp = Float64(x * Float64(1.0 / Float64(y * a)));
	else
		tmp = Float64(x / Float64(a * fma(b, y, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.4e+118], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(b * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.39999999999999993e118

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 70.9

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 62.7

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 39.5

          \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]

   11. Simplified 39.5%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. frac-times N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{a \cdot y}} \]

       2. div-inv N/A

          \[\leadsto \color{blue}{\left(1 \cdot x\right) \cdot \frac{1}{a \cdot y}} \]

       3. *-lft-identity N/A

          \[\leadsto \color{blue}{x} \cdot \frac{1}{a \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y} \cdot x} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y} \cdot x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y}} \cdot x \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{y \cdot a}} \cdot x \]

       8. *-lowering-*.f64 40.5

          \[\leadsto \frac{1}{\color{blue}{y \cdot a}} \cdot x \]

   13. Applied egg-rr 40.5%

       \[\leadsto \color{blue}{\frac{1}{y \cdot a} \cdot x} \]

   if 1.39999999999999993e118 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 82.0

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot e^{b}\right)}} \]

      4. exp-lowering-exp.f64 86.6

         \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{e^{b}}\right)} \]

   8. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot e^{b}\right)}} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y + b \cdot y\right)}} \]

       3. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y + y\right)}} \]

       4. accelerator-lowering-fma.f64 47.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 47.1%

       \[\leadsto \frac{x}{\color{blue}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 31.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 3.2e+48) (* x (/ 1.0 (* y a))) (/ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.2e+48) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 3.2d+48) then
        tmp = x * (1.0d0 / (y * a))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.2e+48) {
		tmp = x * (1.0 / (y * a));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 3.2e+48:
		tmp = x * (1.0 / (y * a))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 3.2e+48)
		tmp = Float64(x * Float64(1.0 / Float64(y * a)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 3.2e+48)
		tmp = x * (1.0 / (y * a));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 3.2e+48], N[(x * N[(1.0 / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2000000000000001e48

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 56.0

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 38.1

          \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{x}{y} \]
   12. Step-by-step derivation

       1. frac-times N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{a \cdot y}} \]

       2. div-inv N/A

          \[\leadsto \color{blue}{\left(1 \cdot x\right) \cdot \frac{1}{a \cdot y}} \]

       3. *-lft-identity N/A

          \[\leadsto \color{blue}{x} \cdot \frac{1}{a \cdot y} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y} \cdot x} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y} \cdot x} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{a \cdot y}} \cdot x \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{y \cdot a}} \cdot x \]

       8. *-lowering-*.f64 41.0

          \[\leadsto \frac{1}{\color{blue}{y \cdot a}} \cdot x \]

   13. Applied egg-rr 41.0%

       \[\leadsto \color{blue}{\frac{1}{y \cdot a} \cdot x} \]

   if 3.2000000000000001e48 < t

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 61.7

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 61.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 42.7

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 22.3

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Simplified 22.3%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 31.5% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 2.9e+47) (/ x (* y a)) (/ x y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.9e+47) {
		tmp = x / (y * a);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 2.9d+47) then
        tmp = x / (y * a)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 2.9e+47) {
		tmp = x / (y * a);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 2.9e+47:
		tmp = x / (y * a)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 2.9e+47)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 2.9e+47)
		tmp = x / (y * a);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 2.9e+47], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.8999999999999998e47

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 73.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot \frac{x}{y}} \]

      4. exp-to-pow N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t - 1\right)}} \cdot \frac{x}{y} \]

      6. sub-neg N/A

         \[\leadsto {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot \frac{x}{y} \]

      7. metadata-eval N/A

         \[\leadsto {a}^{\left(t + \color{blue}{-1}\right)} \cdot \frac{x}{y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      9. /-lowering-/.f64 56.0

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 40.9

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 40.9%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 2.8999999999999998e47 < t

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

      2. log-lowering-log.f64 61.7

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 61.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 42.7

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 22.3

          \[\leadsto \color{blue}{\frac{x}{y}} \]

   11. Simplified 22.3%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.9 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 15.7% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Python

def code(x, y, z, t, a, b):
	return x / y

Julia

function code(x, y, z, t, a, b)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   2. log-lowering-log.f64 71.8

      \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

5. Simplified 71.8%

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

   3. pow-lowering-pow.f64 44.9

      \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

8. Simplified 44.9%

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 17.6

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Simplified 17.6%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024199 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8845848504127471/10000000000000000) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 8520312288374073/10000000000) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))