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

Percentage Accurate: 98.3% → 98.3%

Time: 17.3s

Alternatives: 25

Speedup: 1.0×

• 47.1% 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: 98.3% 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: 98.3% 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 98.8%

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

3. Final simplification 98.8%

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

Alternative 2: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ t_2 := \frac{b \cdot \left(b \cdot \mathsf{fma}\left(x, 0.5, \frac{x}{b \cdot b} - \frac{x}{b}\right)\right)}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+219}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
        (t_2 (/ (* b (* b (fma x 0.5 (- (/ x (* b b)) (/ x b))))) y)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+219) (/ x (* a (* b (+ y (/ y b))))) t_2))))

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 t_2 = (b * (b * fma(x, 0.5, ((x / (b * b)) - (x / b))))) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+219) {
		tmp = x / (a * (b * (y + (y / b))));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(Float64(b * Float64(b * fma(x, 0.5, Float64(Float64(x / Float64(b * b)) - Float64(x / b))))) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+219)
		tmp = Float64(x / Float64(a * Float64(b * Float64(y + Float64(y / b)))));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(b * N[(b * N[(x * 0.5 + N[(N[(x / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(x / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+219], N[(x / N[(a * N[(b * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 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) < -inf.0 or 9.99999999999999965e218 < (/.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 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 71.3

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

   5. Simplified 71.3%

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

   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 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. metadata-eval N/A

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

       12. sub-neg N/A

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

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

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

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

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

       15. sub-neg N/A

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

       16. *-commutative N/A

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

       17. metadata-eval N/A

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

       18. accelerator-lowering-fma.f64 34.8

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

   11. Simplified 34.8%

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

   12. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. +-commutative N/A

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

       6. associate-+l+ N/A

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

       7. *-commutative N/A

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

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

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

       9. mul-1-neg N/A

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

       10. unsub-neg N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

       15. /-lowering-/.f64 52.4

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

   14. Simplified 52.4%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 9.99999999999999965e218

   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 64.9

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

   5. Simplified 64.9%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 65.3

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

   8. Simplified 65.3%

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

   9. Taylor expanded in t around 0

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

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

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

       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 47.5

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

   11. Simplified 47.5%

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

   12. Taylor expanded in b around inf

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

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

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

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

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

       3. /-lowering-/.f64 59.0

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

   14. Simplified 59.0%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{b \cdot \left(b \cdot \mathsf{fma}\left(x, 0.5, \frac{x}{b \cdot b} - \frac{x}{b}\right)\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 10^{+219}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \left(b \cdot \mathsf{fma}\left(x, 0.5, \frac{x}{b \cdot b} - \frac{x}{b}\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ t_2 := \frac{x \cdot \frac{1 - b}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
        (t_2 (/ (* x (/ (- 1.0 b) a)) y)))
   (if (<= t_1 -2e+68) t_2 (if (<= t_1 2e+59) (/ x (* y (fma b a a))) t_2))))

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 t_2 = (x * ((1.0 - b) / a)) / y;
	double tmp;
	if (t_1 <= -2e+68) {
		tmp = t_2;
	} else if (t_1 <= 2e+59) {
		tmp = x / (y * fma(b, a, a));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(Float64(x * Float64(Float64(1.0 - b) / a)) / y)
	tmp = 0.0
	if (t_1 <= -2e+68)
		tmp = t_2;
	elseif (t_1 <= 2e+59)
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x * N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+68], t$95$2, If[LessEqual[t$95$1, 2e+59], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 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) < -1.99999999999999991e68 or 1.99999999999999994e59 < (/.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 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 b around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt1-in N/A

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

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

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

      8. exp-sum N/A

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

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

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

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

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

      13. exp-prod N/A

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

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

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

      15. rem-exp-log N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 74.1

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

   5. Simplified 74.1%

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

   6. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

      8. neg-lowering-neg.f64 74.1

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

   8. Simplified 74.1%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 68.1

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

   11. Simplified 68.1%

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

   12. Taylor expanded in y around 0

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

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

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

       2. --lowering--.f64 46.9

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

   14. Simplified 46.9%

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

   if -1.99999999999999991e68 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.99999999999999994e59

   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 61.6

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

   5. Simplified 61.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in t around 0

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

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

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

       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 45.3

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

   11. Simplified 45.3%

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

   12. Taylor expanded in x around 0

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

       1. distribute-rgt1-in N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. distribute-lft-in N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 48.0

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

   14. Simplified 48.0%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x \cdot \frac{1 - b}{a}}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1 - b}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, x, x\right)}{y \cdot a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

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 (- INFINITY))
     (/ (fma (- b) x x) (* y a))
     (if (<= t_1 2e+59) (/ x (* y (fma b a a))) (/ (/ 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 <= -((double) INFINITY)) {
		tmp = fma(-b, x, x) / (y * a);
	} else if (t_1 <= 2e+59) {
		tmp = x / (y * fma(b, a, a));
	} 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 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-b), x, x) / Float64(y * a));
	elseif (t_1 <= 2e+59)
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	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, (-Infinity)], N[(N[((-b) * x + x), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+59], N[(x / N[(y * N[(b * a + a), $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) < -inf.0

   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 b around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt1-in N/A

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

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

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

      8. exp-sum N/A

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

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

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

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

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

      13. exp-prod N/A

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

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

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

      15. rem-exp-log N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

      8. neg-lowering-neg.f64 60.9

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

   8. Simplified 60.9%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 54.8

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

   11. Simplified 54.8%

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

   12. Taylor expanded in y around 0

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

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

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

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

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

       8. mul-1-neg N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 41.0

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

   14. Simplified 41.0%

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

   if -inf.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.99999999999999994e59

   1. Initial program 98.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 63.8

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

   8. Simplified 63.8%

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

   9. Taylor expanded in t around 0

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

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

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

       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 46.0

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

   11. Simplified 46.0%

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

   12. Taylor expanded in x around 0

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

       1. distribute-rgt1-in N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. distribute-lft-in N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 48.6

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

   14. Simplified 48.6%

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

   if 1.99999999999999994e59 < (/.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 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 71.4

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

   5. Simplified 71.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 67.1%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 40.4

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

   11. Simplified 40.4%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, x, x\right)}{y \cdot a}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ t_2 := \frac{\frac{x}{a}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))
        (t_2 (/ (/ x a) y)))
   (if (<= t_1 -2e+68) t_2 (if (<= t_1 2e+59) (/ x (* y (fma b a a))) t_2))))

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 t_2 = (x / a) / y;
	double tmp;
	if (t_1 <= -2e+68) {
		tmp = t_2;
	} else if (t_1 <= 2e+59) {
		tmp = x / (y * fma(b, a, a));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(Float64(x / a) / y)
	tmp = 0.0
	if (t_1 <= -2e+68)
		tmp = t_2;
	elseif (t_1 <= 2e+59)
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+68], t$95$2, If[LessEqual[t$95$1, 2e+59], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 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) < -1.99999999999999991e68 or 1.99999999999999994e59 < (/.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 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 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 75.9

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

   5. Simplified 75.9%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 66.2%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 40.1

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

   11. Simplified 40.1%

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

   if -1.99999999999999991e68 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1.99999999999999994e59

   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 61.6

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

   5. Simplified 61.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in t around 0

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

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

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

       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 45.3

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

   11. Simplified 45.3%

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

   12. Taylor expanded in x around 0

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

       1. distribute-rgt1-in N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. distribute-lft-in N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 48.0

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

   14. Simplified 48.0%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot {a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -230:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{elif}\;t\_1 \leq -14:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;t\_1 \leq 10^{+67}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))) (t_2 (/ (* x (pow a t)) y)))
   (if (<= t_1 -2e+27)
     t_2
     (if (<= t_1 -230.0)
       (/ x (* y (fma b a a)))
       (if (<= t_1 -14.0)
         (/ x (* y (exp b)))
         (if (<= t_1 1e+67) (/ (* x (pow z y)) 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 = (x * pow(a, t)) / y;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= -230.0) {
		tmp = x / (y * fma(b, a, a));
	} else if (t_1 <= -14.0) {
		tmp = x / (y * exp(b));
	} else if (t_1 <= 1e+67) {
		tmp = (x * pow(z, y)) / 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(Float64(x * (a ^ t)) / y)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= -230.0)
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	elseif (t_1 <= -14.0)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (t_1 <= 1e+67)
		tmp = Float64(Float64(x * (z ^ y)) / 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[(x * N[Power[a, t], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$2, If[LessEqual[t$95$1, -230.0], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -14.0], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+67], N[(N[(x * N[Power[z, y], $MachinePrecision]), $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)) < -2e27 or 9.99999999999999983e66 < (*.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 91.4

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

   5. Simplified 91.4%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 82.9

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

   8. Simplified 82.9%

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

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -230

   1. Initial program 95.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 55.3

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

   8. Simplified 55.3%

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

   9. Taylor expanded in t around 0

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

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

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

       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 61.8

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

   11. Simplified 61.8%

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

   12. Taylor expanded in x around 0

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

       1. distribute-rgt1-in N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. distribute-lft-in N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 70.1

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

   14. Simplified 70.1%

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

   if -230 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -14

   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 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 81.7

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

   5. Simplified 81.7%

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

   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 76.8

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

   8. Simplified 76.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{b}}} \cdot \frac{x}{y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{e^{b} \cdot y}} \]

      5. *-lft-identity N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 76.8

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

   10. Applied egg-rr 76.8%

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

   if -14 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 9.99999999999999983e66

   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 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. pow-lowering-pow.f64 59.1

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

   8. Simplified 59.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -230:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -14:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 10^{+67}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ \mathbf{if}\;t\_1 \leq -705:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;t\_1 \leq -14:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 10^{+67}:\\ \;\;\;\;\frac{x \cdot \left(\mathsf{fma}\left(b, \frac{1}{b}, -b\right) \cdot \left({z}^{y} \cdot \frac{1}{a}\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))))
   (if (<= t_1 -705.0)
     (/ (* x (pow a (+ t -1.0))) y)
     (if (<= t_1 -14.0)
       (/ x (* y (* a (exp b))))
       (if (<= t_1 1e+67)
         (/ (* x (* (fma b (/ 1.0 b) (- b)) (* (pow z y) (/ 1.0 a)))) y)
         (/ (* x (pow a t)) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double tmp;
	if (t_1 <= -705.0) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else if (t_1 <= -14.0) {
		tmp = x / (y * (a * exp(b)));
	} else if (t_1 <= 1e+67) {
		tmp = (x * (fma(b, (1.0 / b), -b) * (pow(z, y) * (1.0 / a)))) / y;
	} else {
		tmp = (x * pow(a, t)) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	tmp = 0.0
	if (t_1 <= -705.0)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	elseif (t_1 <= -14.0)
		tmp = Float64(x / Float64(y * Float64(a * exp(b))));
	elseif (t_1 <= 1e+67)
		tmp = Float64(Float64(x * Float64(fma(b, Float64(1.0 / b), Float64(-b)) * Float64((z ^ y) * Float64(1.0 / a)))) / y);
	else
		tmp = Float64(Float64(x * (a ^ t)) / y);
	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]}, If[LessEqual[t$95$1, -705.0], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, -14.0], N[(x / N[(y * N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+67], N[(N[(x * N[(N[(b * N[(1.0 / b), $MachinePrecision] + (-b)), $MachinePrecision] * N[(N[Power[z, y], $MachinePrecision] * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Power[a, t], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -705

   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 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 83.6%

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

   if -705 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -14

   1. Initial program 96.5%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

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

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

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

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

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

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

      14. rem-exp-log N/A

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

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

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

      16. exp-lowering-exp.f64 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in y 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. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 92.0

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

   8. Simplified 92.0%

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

   if -14 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 9.99999999999999983e66

   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 b around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt1-in N/A

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

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

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

      5. +-commutative N/A

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

      6. unsub-neg N/A

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

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

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

      8. exp-sum N/A

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

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

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

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

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

      13. exp-prod N/A

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

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

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

      15. rem-exp-log N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-lowering-+.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in b around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

      8. neg-lowering-neg.f64 64.4

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

   8. Simplified 64.4%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 75.1

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

   11. Simplified 75.1%

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

   if 9.99999999999999983e66 < (*.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 92.6

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

   5. Simplified 92.6%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 79.6

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

   8. Simplified 79.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -705:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -14:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 10^{+67}:\\ \;\;\;\;\frac{x \cdot \left(\mathsf{fma}\left(b, \frac{1}{b}, -b\right) \cdot \left({z}^{y} \cdot \frac{1}{a}\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* t (log a)) b))) y)))
   (if (<= t -1.3e+32)
     t_1
     (if (<= t 1.45e+64) (* x (/ (/ (pow z y) a) (* 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(((t * log(a)) - b))) / y;
	double tmp;
	if (t <= -1.3e+32) {
		tmp = t_1;
	} else if (t <= 1.45e+64) {
		tmp = x * ((pow(z, y) / a) / (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(((t * log(a)) - b))) / y
    if (t <= (-1.3d+32)) then
        tmp = t_1
    else if (t <= 1.45d+64) then
        tmp = x * (((z ** y) / a) / (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(((t * Math.log(a)) - b))) / y;
	double tmp;
	if (t <= -1.3e+32) {
		tmp = t_1;
	} else if (t <= 1.45e+64) {
		tmp = x * ((Math.pow(z, y) / a) / (y * Math.exp(b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((t * math.log(a)) - b))) / y
	tmp = 0
	if t <= -1.3e+32:
		tmp = t_1
	elif t <= 1.45e+64:
		tmp = x * ((math.pow(z, y) / a) / (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(t * log(a)) - b))) / y)
	tmp = 0.0
	if (t <= -1.3e+32)
		tmp = t_1;
	elseif (t <= 1.45e+64)
		tmp = Float64(x * Float64(Float64((z ^ y) / a) / 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(((t * log(a)) - b))) / y;
	tmp = 0.0;
	if (t <= -1.3e+32)
		tmp = t_1;
	elseif (t <= 1.45e+64)
		tmp = x * (((z ^ y) / a) / (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[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t, -1.3e+32], t$95$1, If[LessEqual[t, 1.45e+64], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3000000000000001e32 or 1.44999999999999997e64 < 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 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 91.6

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

   5. Simplified 91.6%

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

   if -1.3000000000000001e32 < t < 1.44999999999999997e64

   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 t around 0

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

      1. associate-/l* N/A

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

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

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

      3. exp-diff N/A

         \[\leadsto x \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

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

         \[\leadsto x \cdot \color{blue}{\frac{e^{-1 \cdot \log a + y \cdot \log z}}{y \cdot e^{b}}} \]

      6. +-commutative N/A

         \[\leadsto x \cdot \frac{e^{\color{blue}{y \cdot \log z + -1 \cdot \log a}}}{y \cdot e^{b}} \]

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. exp-diff N/A

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

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

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

      11. *-commutative N/A

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

      12. exp-to-pow N/A

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

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

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

      14. rem-exp-log N/A

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

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

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

      16. exp-lowering-exp.f64 89.3

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

   5. Simplified 89.3%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+64}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{a}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -4.5e+46)
     t_1
     (if (<= y 1.45e-111)
       (/ (* x (exp (- (* t (log a)) b))) y)
       (if (<= y 1.56e+19) (/ (* x (pow a (+ t -1.0))) 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 <= -4.5e+46) {
		tmp = t_1;
	} else if (y <= 1.45e-111) {
		tmp = (x * exp(((t * log(a)) - b))) / y;
	} else if (y <= 1.56e+19) {
		tmp = (x * pow(a, (t + -1.0))) / 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 <= (-4.5d+46)) then
        tmp = t_1
    else if (y <= 1.45d-111) then
        tmp = (x * exp(((t * log(a)) - b))) / y
    else if (y <= 1.56d+19) then
        tmp = (x * (a ** (t + (-1.0d0)))) / 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 <= -4.5e+46) {
		tmp = t_1;
	} else if (y <= 1.45e-111) {
		tmp = (x * Math.exp(((t * Math.log(a)) - b))) / y;
	} else if (y <= 1.56e+19) {
		tmp = (x * Math.pow(a, (t + -1.0))) / 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 <= -4.5e+46:
		tmp = t_1
	elif y <= 1.45e-111:
		tmp = (x * math.exp(((t * math.log(a)) - b))) / y
	elif y <= 1.56e+19:
		tmp = (x * math.pow(a, (t + -1.0))) / 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 <= -4.5e+46)
		tmp = t_1;
	elseif (y <= 1.45e-111)
		tmp = Float64(Float64(x * exp(Float64(Float64(t * log(a)) - b))) / y);
	elseif (y <= 1.56e+19)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / 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 <= -4.5e+46)
		tmp = t_1;
	elseif (y <= 1.45e-111)
		tmp = (x * exp(((t * log(a)) - b))) / y;
	elseif (y <= 1.56e+19)
		tmp = (x * (a ^ (t + -1.0))) / 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, -4.5e+46], t$95$1, If[LessEqual[y, 1.45e-111], N[(N[(x * N[Exp[N[(N[(t * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.56e+19], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5000000000000001e46 or 1.56e19 < 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 95.7

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

   5. Simplified 95.7%

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

   if -4.5000000000000001e46 < y < 1.45000000000000001e-111

   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 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 82.7

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

   5. Simplified 82.7%

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

   if 1.45000000000000001e-111 < y < 1.56e19

   1. Initial program 96.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 84.5

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

   5. Simplified 84.5%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 83.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot e^{t \cdot \log a - b}}{y}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -1.7e+45)
     t_1
     (if (<= y 2.7e+35) (/ (* 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 <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 2.7e+35) {
		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 <= (-1.7d+45)) then
        tmp = t_1
    else if (y <= 2.7d+35) 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 <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 2.7e+35) {
		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 <= -1.7e+45:
		tmp = t_1
	elif y <= 2.7e+35:
		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 <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 2.7e+35)
		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 <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 2.7e+35)
		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, -1.7e+45], t$95$1, If[LessEqual[y, 2.7e+35], 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 < -1.7e45 or 2.70000000000000003e35 < 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 96.4

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

   5. Simplified 96.4%

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

   if -1.7e45 < y < 2.70000000000000003e35

   1. Initial program 98.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* y (log z)) b))) y)))
   (if (<= y -1.7e+45)
     t_1
     (if (<= y 1.35e+19) (/ (* x (pow a (+ t -1.0))) 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 <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 1.35e+19) {
		tmp = (x * pow(a, (t + -1.0))) / 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 <= (-1.7d+45)) then
        tmp = t_1
    else if (y <= 1.35d+19) then
        tmp = (x * (a ** (t + (-1.0d0)))) / 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 <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 1.35e+19) {
		tmp = (x * Math.pow(a, (t + -1.0))) / 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 <= -1.7e+45:
		tmp = t_1
	elif y <= 1.35e+19:
		tmp = (x * math.pow(a, (t + -1.0))) / 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 <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 1.35e+19)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / 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 <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 1.35e+19)
		tmp = (x * (a ^ (t + -1.0))) / 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, -1.7e+45], t$95$1, If[LessEqual[y, 1.35e+19], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e45 or 1.35e19 < 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 95.7

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

   5. Simplified 95.7%

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

   if -1.7e45 < y < 1.35e19

   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 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 74.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{y \cdot \log z - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ t_2 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow a (+ t -1.0)) (/ x y))) (t_2 (/ (* x (pow z y)) y)))
   (if (<= y -1.7e+45)
     t_2
     (if (<= y 7.6e-299)
       t_1
       (if (<= y 1.15e-200)
         (/ x (* y (exp b)))
         (if (<= y 1.26e+20) t_1 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)) * (x / y);
	double t_2 = (x * pow(z, y)) / y;
	double tmp;
	if (y <= -1.7e+45) {
		tmp = t_2;
	} else if (y <= 7.6e-299) {
		tmp = t_1;
	} else if (y <= 1.15e-200) {
		tmp = x / (y * exp(b));
	} else if (y <= 1.26e+20) {
		tmp = t_1;
	} 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))) * (x / y)
    t_2 = (x * (z ** y)) / y
    if (y <= (-1.7d+45)) then
        tmp = t_2
    else if (y <= 7.6d-299) then
        tmp = t_1
    else if (y <= 1.15d-200) then
        tmp = x / (y * exp(b))
    else if (y <= 1.26d+20) then
        tmp = t_1
    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)) * (x / y);
	double t_2 = (x * Math.pow(z, y)) / y;
	double tmp;
	if (y <= -1.7e+45) {
		tmp = t_2;
	} else if (y <= 7.6e-299) {
		tmp = t_1;
	} else if (y <= 1.15e-200) {
		tmp = x / (y * Math.exp(b));
	} else if (y <= 1.26e+20) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t + -1.0)) * (x / y)
	t_2 = (x * math.pow(z, y)) / y
	tmp = 0
	if y <= -1.7e+45:
		tmp = t_2
	elif y <= 7.6e-299:
		tmp = t_1
	elif y <= 1.15e-200:
		tmp = x / (y * math.exp(b))
	elif y <= 1.26e+20:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((a ^ Float64(t + -1.0)) * Float64(x / y))
	t_2 = Float64(Float64(x * (z ^ y)) / y)
	tmp = 0.0
	if (y <= -1.7e+45)
		tmp = t_2;
	elseif (y <= 7.6e-299)
		tmp = t_1;
	elseif (y <= 1.15e-200)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (y <= 1.26e+20)
		tmp = t_1;
	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)) * (x / y);
	t_2 = (x * (z ^ y)) / y;
	tmp = 0.0;
	if (y <= -1.7e+45)
		tmp = t_2;
	elseif (y <= 7.6e-299)
		tmp = t_1;
	elseif (y <= 1.15e-200)
		tmp = x / (y * exp(b));
	elseif (y <= 1.26e+20)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.7e+45], t$95$2, If[LessEqual[y, 7.6e-299], t$95$1, If[LessEqual[y, 1.15e-200], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.26e+20], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e45 or 1.26e20 < 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 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. pow-lowering-pow.f64 88.8

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

   8. Simplified 88.8%

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

   if -1.7e45 < y < 7.6000000000000005e-299 or 1.15000000000000004e-200 < y < 1.26e20

   1. Initial program 97.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 82.2

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

   5. Simplified 82.2%

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

   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. +-commutative N/A

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

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

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

      10. /-lowering-/.f64 68.0

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

   8. Simplified 68.0%

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

   if 7.6000000000000005e-299 < y < 1.15000000000000004e-200

   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 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 66.4

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

   5. Simplified 66.4%

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

   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 66.4

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

   8. Simplified 66.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{b}}} \cdot \frac{x}{y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{e^{b} \cdot y}} \]

      5. *-lft-identity N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 66.4

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

   10. Applied egg-rr 66.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-299}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {a}^{t}}{y}\\ t_2 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 5600:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow a t)) y)) (t_2 (/ x (* y (exp b)))))
   (if (<= b -750.0)
     t_2
     (if (<= b -1.3e-86)
       t_1
       (if (<= b 3.1e-201)
         (/ x (* a (* b (+ y (/ y b)))))
         (if (<= b 5600.0) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * pow(a, t)) / y;
	double t_2 = x / (y * exp(b));
	double tmp;
	if (b <= -750.0) {
		tmp = t_2;
	} else if (b <= -1.3e-86) {
		tmp = t_1;
	} else if (b <= 3.1e-201) {
		tmp = x / (a * (b * (y + (y / b))));
	} else if (b <= 5600.0) {
		tmp = t_1;
	} 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 = (x * (a ** t)) / y
    t_2 = x / (y * exp(b))
    if (b <= (-750.0d0)) then
        tmp = t_2
    else if (b <= (-1.3d-86)) then
        tmp = t_1
    else if (b <= 3.1d-201) then
        tmp = x / (a * (b * (y + (y / b))))
    else if (b <= 5600.0d0) then
        tmp = t_1
    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 = (x * Math.pow(a, t)) / y;
	double t_2 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -750.0) {
		tmp = t_2;
	} else if (b <= -1.3e-86) {
		tmp = t_1;
	} else if (b <= 3.1e-201) {
		tmp = x / (a * (b * (y + (y / b))));
	} else if (b <= 5600.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.pow(a, t)) / y
	t_2 = x / (y * math.exp(b))
	tmp = 0
	if b <= -750.0:
		tmp = t_2
	elif b <= -1.3e-86:
		tmp = t_1
	elif b <= 3.1e-201:
		tmp = x / (a * (b * (y + (y / b))))
	elif b <= 5600.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * (a ^ t)) / y)
	t_2 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -750.0)
		tmp = t_2;
	elseif (b <= -1.3e-86)
		tmp = t_1;
	elseif (b <= 3.1e-201)
		tmp = Float64(x / Float64(a * Float64(b * Float64(y + Float64(y / b)))));
	elseif (b <= 5600.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * (a ^ t)) / y;
	t_2 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -750.0)
		tmp = t_2;
	elseif (b <= -1.3e-86)
		tmp = t_1;
	elseif (b <= 3.1e-201)
		tmp = x / (a * (b * (y + (y / b))));
	elseif (b <= 5600.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Power[a, t], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -750.0], t$95$2, If[LessEqual[b, -1.3e-86], t$95$1, If[LessEqual[b, 3.1e-201], N[(x / N[(a * N[(b * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5600.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -750 or 5600 < 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 89.9

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

   5. Simplified 89.9%

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

   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 79.8

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

   8. Simplified 79.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{b}}} \cdot \frac{x}{y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{e^{b} \cdot y}} \]

      5. *-lft-identity N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 79.8

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

   10. Applied egg-rr 79.8%

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

   if -750 < b < -1.3000000000000001e-86 or 3.0999999999999999e-201 < b < 5600

   1. Initial program 97.4%

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

   3. Taylor expanded in 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 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 60.4

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

   8. Simplified 60.4%

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

   if -1.3000000000000001e-86 < b < 3.0999999999999999e-201

   1. Initial program 98.3%

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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 76.8

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

   8. Simplified 76.8%

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

   9. Taylor expanded in t around 0

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

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

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

       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 47.7

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

   11. Simplified 47.7%

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

   12. Taylor expanded in b around inf

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

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

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

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

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

       3. /-lowering-/.f64 66.7

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

   14. Simplified 66.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -750:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{elif}\;b \leq 5600:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow z y)) y)))
   (if (<= y -1.7e+45)
     t_1
     (if (<= y 1.1e+20) (/ (* x (pow a (+ t -1.0))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * pow(z, y)) / y;
	double tmp;
	if (y <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 1.1e+20) {
		tmp = (x * pow(a, (t + -1.0))) / 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 * (z ** y)) / y
    if (y <= (-1.7d+45)) then
        tmp = t_1
    else if (y <= 1.1d+20) then
        tmp = (x * (a ** (t + (-1.0d0)))) / 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.pow(z, y)) / y;
	double tmp;
	if (y <= -1.7e+45) {
		tmp = t_1;
	} else if (y <= 1.1e+20) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.pow(z, y)) / y
	tmp = 0
	if y <= -1.7e+45:
		tmp = t_1
	elif y <= 1.1e+20:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * (z ^ y)) / y)
	tmp = 0.0
	if (y <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 1.1e+20)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * (z ^ y)) / y;
	tmp = 0.0;
	if (y <= -1.7e+45)
		tmp = t_1;
	elseif (y <= 1.1e+20)
		tmp = (x * (a ^ (t + -1.0))) / 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[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.7e+45], t$95$1, If[LessEqual[y, 1.1e+20], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e45 or 1.1e20 < 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 95.7

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

   5. Simplified 95.7%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. pow-lowering-pow.f64 88.8

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

   8. Simplified 88.8%

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

   if -1.7e45 < y < 1.1e20

   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 81.5

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

   5. Simplified 81.5%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 73.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+20}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -0.0056:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 0.0043:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -0.0056)
     t_1
     (if (<= b 0.0043) (/ x (* a (* b (+ y (/ y b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -0.0056) {
		tmp = t_1;
	} else if (b <= 0.0043) {
		tmp = x / (a * (b * (y + (y / 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 / (y * exp(b))
    if (b <= (-0.0056d0)) then
        tmp = t_1
    else if (b <= 0.0043d0) then
        tmp = x / (a * (b * (y + (y / 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 / (y * Math.exp(b));
	double tmp;
	if (b <= -0.0056) {
		tmp = t_1;
	} else if (b <= 0.0043) {
		tmp = x / (a * (b * (y + (y / b))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -0.0056:
		tmp = t_1
	elif b <= 0.0043:
		tmp = x / (a * (b * (y + (y / b))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -0.0056)
		tmp = t_1;
	elseif (b <= 0.0043)
		tmp = Float64(x / Float64(a * Float64(b * Float64(y + Float64(y / b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -0.0056)
		tmp = t_1;
	elseif (b <= 0.0043)
		tmp = x / (a * (b * (y + (y / b))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0056], t$95$1, If[LessEqual[b, 0.0043], N[(x / N[(a * N[(b * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00559999999999999994 or 0.0043 < 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 87.7

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

   5. Simplified 87.7%

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

   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 77.9

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

   8. Simplified 77.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. exp-neg N/A

         \[\leadsto \color{blue}{\frac{1}{e^{b}}} \cdot \frac{x}{y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{e^{b} \cdot y}} \]

      5. *-lft-identity N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 77.9

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

   10. Applied egg-rr 77.9%

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

   if -0.00559999999999999994 < b < 0.0043

   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 76.1

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

   5. Simplified 76.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 76.2

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

   8. Simplified 76.2%

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

   9. Taylor expanded in t around 0

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

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

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

       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 42.9

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

   11. Simplified 42.9%

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

   12. Taylor expanded in b around inf

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

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

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

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

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

       3. /-lowering-/.f64 54.9

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

   14. Simplified 54.9%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0056:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq 0.0043:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(1 + b\right)} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+121)
   (/ (* x (* (* b (* b b)) -0.16666666666666666)) y)
   (if (<= b 4.3e-34)
     (/ x (* a (* b (+ y (/ y b)))))
     (* (/ x (* a (+ 1.0 b))) (/ 1.0 y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+121) {
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	} else if (b <= 4.3e-34) {
		tmp = x / (a * (b * (y + (y / b))));
	} else {
		tmp = (x / (a * (1.0 + b))) * (1.0 / 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 (b <= (-6d+121)) then
        tmp = (x * ((b * (b * b)) * (-0.16666666666666666d0))) / y
    else if (b <= 4.3d-34) then
        tmp = x / (a * (b * (y + (y / b))))
    else
        tmp = (x / (a * (1.0d0 + b))) * (1.0d0 / 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 (b <= -6e+121) {
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	} else if (b <= 4.3e-34) {
		tmp = x / (a * (b * (y + (y / b))));
	} else {
		tmp = (x / (a * (1.0 + b))) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6e+121:
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y
	elif b <= 4.3e-34:
		tmp = x / (a * (b * (y + (y / b))))
	else:
		tmp = (x / (a * (1.0 + b))) * (1.0 / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+121)
		tmp = Float64(Float64(x * Float64(Float64(b * Float64(b * b)) * -0.16666666666666666)) / y);
	elseif (b <= 4.3e-34)
		tmp = Float64(x / Float64(a * Float64(b * Float64(y + Float64(y / b)))));
	else
		tmp = Float64(Float64(x / Float64(a * Float64(1.0 + b))) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6e+121)
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	elseif (b <= 4.3e-34)
		tmp = x / (a * (b * (y + (y / b))));
	else
		tmp = (x / (a * (1.0 + b))) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+121], N[(N[(x * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 4.3e-34], N[(x / N[(a * N[(b * N[(y + N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(a * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.0000000000000005e121

   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 90.8

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

   5. Simplified 90.8%

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

   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 78.5

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

   8. Simplified 78.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. +-commutative N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 69.5

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

   11. Simplified 69.5%

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

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}}{y} \]

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

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}{y}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{-1}{6}\right)} \cdot {b}^{3}}{y} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

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

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

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

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

       10. cube-mult N/A

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

       11. unpow2 N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 78.5

           \[\leadsto \frac{x \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.16666666666666666\right)}{y} \]

   14. Simplified 78.5%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}} \]

   if -6.0000000000000005e121 < b < 4.3e-34

   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 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 72.1

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

   5. Simplified 72.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 68.8

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

   8. Simplified 68.8%

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

   9. Taylor expanded in t around 0

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

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

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

       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 38.5

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

   11. Simplified 38.5%

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

   12. Taylor expanded in b around inf

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

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

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

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

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

       3. /-lowering-/.f64 49.3

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

   14. Simplified 49.3%

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

   if 4.3e-34 < 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 64.0

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

   5. Simplified 64.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 51.0

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

   8. Simplified 51.0%

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

   9. Taylor expanded in t around 0

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

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

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

       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 44.4

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

   11. Simplified 44.4%

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

       1. div-inv N/A

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

       2. associate-*r/ N/A

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

       3. distribute-lft1-in N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

          \[\leadsto \frac{x \cdot 1}{\left(a \cdot \left(\color{blue}{\left(1 - 0\right)} + b\right)\right) \cdot y} \]

       7. associate--r- N/A

          \[\leadsto \frac{x \cdot 1}{\left(a \cdot \color{blue}{\left(1 - \left(0 - b\right)\right)}\right) \cdot y} \]

       8. neg-sub0 N/A

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

       9. times-frac N/A

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

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

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

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

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

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

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

       13. neg-sub0 N/A

           \[\leadsto \frac{x}{a \cdot \left(1 - \color{blue}{\left(0 - b\right)}\right)} \cdot \frac{1}{y} \]

       14. associate--r- N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(1 - 0\right) + b\right)}} \cdot \frac{1}{y} \]

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

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

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

       18. /-lowering-/.f64 55.3

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

   13. Applied egg-rr 55.3%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{a \cdot \left(b \cdot \left(y + \frac{y}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(1 + b\right)} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.9% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(1 + b\right)} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+121)
   (/ (* x (* (* b (* b b)) -0.16666666666666666)) y)
   (* (/ x (* a (+ 1.0 b))) (/ 1.0 y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+121) {
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	} else {
		tmp = (x / (a * (1.0 + b))) * (1.0 / 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 (b <= (-6d+121)) then
        tmp = (x * ((b * (b * b)) * (-0.16666666666666666d0))) / y
    else
        tmp = (x / (a * (1.0d0 + b))) * (1.0d0 / 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 (b <= -6e+121) {
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	} else {
		tmp = (x / (a * (1.0 + b))) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6e+121:
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y
	else:
		tmp = (x / (a * (1.0 + b))) * (1.0 / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+121)
		tmp = Float64(Float64(x * Float64(Float64(b * Float64(b * b)) * -0.16666666666666666)) / y);
	else
		tmp = Float64(Float64(x / Float64(a * Float64(1.0 + b))) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6e+121)
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	else
		tmp = (x / (a * (1.0 + b))) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+121], N[(N[(x * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(a * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.0000000000000005e121

   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 90.8

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

   5. Simplified 90.8%

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

   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 78.5

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

   8. Simplified 78.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. +-commutative N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 69.5

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

   11. Simplified 69.5%

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

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}}{y} \]

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

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}{y}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{-1}{6}\right)} \cdot {b}^{3}}{y} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

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

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

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

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

       10. cube-mult N/A

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

       11. unpow2 N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 78.5

           \[\leadsto \frac{x \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.16666666666666666\right)}{y} \]

   14. Simplified 78.5%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}} \]

   if -6.0000000000000005e121 < 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 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 63.1

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

   8. Simplified 63.1%

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

   9. Taylor expanded in t around 0

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

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

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

       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 40.4

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

   11. Simplified 40.4%

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

       1. div-inv N/A

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

       2. associate-*r/ N/A

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

       3. distribute-lft1-in N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. metadata-eval N/A

          \[\leadsto \frac{x \cdot 1}{\left(a \cdot \left(\color{blue}{\left(1 - 0\right)} + b\right)\right) \cdot y} \]

       7. associate--r- N/A

          \[\leadsto \frac{x \cdot 1}{\left(a \cdot \color{blue}{\left(1 - \left(0 - b\right)\right)}\right) \cdot y} \]

       8. neg-sub0 N/A

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

       9. times-frac N/A

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

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

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

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

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

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

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

       13. neg-sub0 N/A

           \[\leadsto \frac{x}{a \cdot \left(1 - \color{blue}{\left(0 - b\right)}\right)} \cdot \frac{1}{y} \]

       14. associate--r- N/A

           \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(1 - 0\right) + b\right)}} \cdot \frac{1}{y} \]

       15. metadata-eval N/A

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

       16. +-commutative N/A

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

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

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

       18. /-lowering-/.f64 43.3

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

   13. Applied egg-rr 43.3%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(1 + b\right)} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+121)
   (/ (* x (* (* b (* b b)) -0.16666666666666666)) y)
   (/ x (* y (fma b a a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+121) {
		tmp = (x * ((b * (b * b)) * -0.16666666666666666)) / y;
	} else {
		tmp = x / (y * fma(b, a, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+121)
		tmp = Float64(Float64(x * Float64(Float64(b * Float64(b * b)) * -0.16666666666666666)) / y);
	else
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+121], N[(N[(x * N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.0000000000000005e121

   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 90.8

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

   5. Simplified 90.8%

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

   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 78.5

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

   8. Simplified 78.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out N/A

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

       7. +-commutative N/A

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

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

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

       9. +-commutative N/A

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

       10. *-commutative N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 69.5

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

   11. Simplified 69.5%

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

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{{b}^{3} \cdot x}{y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \left({b}^{3} \cdot x\right)}{y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot {b}^{3}\right)}}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}}{y} \]

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

          \[\leadsto \color{blue}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {b}^{3}}{y}} \]

       5. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{-1}{6}\right)} \cdot {b}^{3}}{y} \]

       6. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

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

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {b}^{3}\right)}}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

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

          \[\leadsto \frac{x \cdot \color{blue}{\left({b}^{3} \cdot \frac{-1}{6}\right)}}{y} \]

       10. cube-mult N/A

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

       11. unpow2 N/A

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 78.5

           \[\leadsto \frac{x \cdot \left(\left(b \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot -0.16666666666666666\right)}{y} \]

   14. Simplified 78.5%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot -0.16666666666666666\right)}{y}} \]

   if -6.0000000000000005e121 < 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 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 63.1

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

   8. Simplified 63.1%

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

   9. Taylor expanded in t around 0

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

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

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

       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 40.4

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

   11. Simplified 40.4%

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

   12. Taylor expanded in x around 0

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

       1. distribute-rgt1-in N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

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

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

       5. distribute-rgt-in N/A

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

       6. *-lft-identity N/A

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

       7. distribute-lft-in N/A

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

       8. +-commutative N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. *-commutative N/A

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

       13. accelerator-lowering-fma.f64 42.5

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

   14. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \left(b \cdot 0.5\right), x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.45e+122)
   (/ (fma x (* b (* b 0.5)) x) y)
   (/ x (* y (fma b a a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.45e+122) {
		tmp = fma(x, (b * (b * 0.5)), x) / y;
	} else {
		tmp = x / (y * fma(b, a, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.45e+122)
		tmp = Float64(fma(x, Float64(b * Float64(b * 0.5)), x) / y);
	else
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.45e+122], N[(N[(x * N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.45e122

   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 90.8

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

   5. Simplified 90.8%

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

   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 78.5

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

   8. Simplified 78.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-lft-in N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. distribute-rgt-out N/A

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

       9. *-commutative N/A

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

       10. distribute-lft-in N/A

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

       11. metadata-eval N/A

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

       12. sub-neg N/A

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

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

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

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

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

       15. sub-neg N/A

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

       16. *-commutative N/A

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

       17. metadata-eval N/A

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

       18. accelerator-lowering-fma.f64 69.6

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

   11. Simplified 69.6%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 69.6

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

   14. Simplified 69.6%

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(b \cdot 0.5\right)}, x\right)}{y} \]

   if -1.45e122 < 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 69.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y + b \cdot y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{b \cdot y + y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot b} + y} \]

      3. accelerator-lowering-fma.f64 63.1

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   8. Simplified 63.1%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]

       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 40.4

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 40.4%

       \[\leadsto \color{blue}{\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b + 1\right) \cdot y\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(1 + b\right)} \cdot y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(1 + b\right)\right)}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}} \]

       5. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(1 \cdot y + b \cdot y\right)}} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y} + b \cdot y\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

       8. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(b \cdot y\right) + a \cdot y}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot b\right) \cdot y} + a \cdot y} \]

       10. distribute-rgt-out N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       12. *-commutative N/A

           \[\leadsto \frac{x}{y \cdot \left(\color{blue}{b \cdot a} + a\right)} \]

       13. accelerator-lowering-fma.f64 42.5

           \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(b, a, a\right)}} \]

   14. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 41.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+121}:\\ \;\;\;\;\frac{b \cdot \left(b \cdot \left(x \cdot 0.5\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+121) (/ (* b (* b (* x 0.5))) y) (/ x (* y (fma b a a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+121) {
		tmp = (b * (b * (x * 0.5))) / y;
	} else {
		tmp = x / (y * fma(b, a, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+121)
		tmp = Float64(Float64(b * Float64(b * Float64(x * 0.5))) / y);
	else
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+121], N[(N[(b * N[(b * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.0000000000000005e121

   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 90.8

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 90.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   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 78.5

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 78.5%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right) + x}}{y} \]

       2. +-commutative N/A

          \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot x\right) + -1 \cdot x\right)} + x}{y} \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right) + b \cdot \left(-1 \cdot x\right)\right)} + x}{y} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot x\right)} + b \cdot \left(-1 \cdot x\right)\right) + x}{y} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x} + b \cdot \left(-1 \cdot x\right)\right) + x}{y} \]

       6. associate-*r* N/A

          \[\leadsto \frac{\left(\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x + \color{blue}{\left(b \cdot -1\right) \cdot x}\right) + x}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{\left(\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x + \color{blue}{\left(-1 \cdot b\right)} \cdot x\right) + x}{y} \]

       8. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b\right) + -1 \cdot b\right)} + x}{y} \]

       9. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b \cdot -1}\right) + x}{y} \]

       10. distribute-lft-in N/A

           \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b + -1\right)\right)} + x}{y} \]

       11. metadata-eval N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) + x}{y} \]

       12. sub-neg N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}\right) + x}{y} \]

       13. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b - 1\right), x\right)}}{y} \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b - 1\right)}, x\right)}{y} \]

       15. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, x\right)}{y} \]

       16. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(\color{blue}{b \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)}{y} \]

       17. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(b \cdot \frac{1}{2} + \color{blue}{-1}\right), x\right)}{y} \]

       18. accelerator-lowering-fma.f64 69.6

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\mathsf{fma}\left(b, 0.5, -1\right)}, x\right)}{y} \]

   11. Simplified 69.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(b, 0.5, -1\right), x\right)}}{y} \]

   12. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{b}^{2} \cdot x}{y}} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left({b}^{2} \cdot x\right)}{y}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({b}^{2} \cdot x\right) \cdot \frac{1}{2}}}{y} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{{b}^{2} \cdot \left(x \cdot \frac{1}{2}\right)}}{y} \]

       4. *-commutative N/A

          \[\leadsto \frac{{b}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{y} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{{b}^{2} \cdot \left(\frac{1}{2} \cdot x\right)}{y}} \]

       6. unpow2 N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} \cdot x\right)}{y} \]

       7. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} \cdot x\right)\right)}}{y} \]

       8. *-commutative N/A

          \[\leadsto \frac{b \cdot \left(b \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right)}{y} \]

       9. associate-*r* N/A

          \[\leadsto \frac{b \cdot \color{blue}{\left(\left(b \cdot x\right) \cdot \frac{1}{2}\right)}}{y} \]

       10. *-commutative N/A

           \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]

       12. *-commutative N/A

           \[\leadsto \frac{b \cdot \color{blue}{\left(\left(b \cdot x\right) \cdot \frac{1}{2}\right)}}{y} \]

       13. associate-*r* N/A

           \[\leadsto \frac{b \cdot \color{blue}{\left(b \cdot \left(x \cdot \frac{1}{2}\right)\right)}}{y} \]

       14. *-commutative N/A

           \[\leadsto \frac{b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot x\right)}\right)}{y} \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \frac{b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot x\right)\right)}}{y} \]

       16. *-commutative N/A

           \[\leadsto \frac{b \cdot \left(b \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right)}{y} \]

       17. *-lowering-*.f64 60.6

           \[\leadsto \frac{b \cdot \left(b \cdot \color{blue}{\left(x \cdot 0.5\right)}\right)}{y} \]

   14. Simplified 60.6%

       \[\leadsto \color{blue}{\frac{b \cdot \left(b \cdot \left(x \cdot 0.5\right)\right)}{y}} \]

   if -6.0000000000000005e121 < 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 69.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y + b \cdot y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{b \cdot y + y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot b} + y} \]

      3. accelerator-lowering-fma.f64 63.1

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   8. Simplified 63.1%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]

       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 40.4

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 40.4%

       \[\leadsto \color{blue}{\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b + 1\right) \cdot y\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(1 + b\right)} \cdot y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(1 + b\right)\right)}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}} \]

       5. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(1 \cdot y + b \cdot y\right)}} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y} + b \cdot y\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

       8. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(b \cdot y\right) + a \cdot y}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot b\right) \cdot y} + a \cdot y} \]

       10. distribute-rgt-out N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       12. *-commutative N/A

           \[\leadsto \frac{x}{y \cdot \left(\color{blue}{b \cdot a} + a\right)} \]

       13. accelerator-lowering-fma.f64 42.5

           \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(b, a, a\right)}} \]

   14. Simplified 42.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.3e+192) (/ (- x (* x b)) y) (/ x (* y (fma b a a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.3e+192) {
		tmp = (x - (x * b)) / y;
	} else {
		tmp = x / (y * fma(b, a, a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.3e+192)
		tmp = Float64(Float64(x - Float64(x * b)) / y);
	else
		tmp = Float64(x / Float64(y * fma(b, a, a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.3e+192], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(y * N[(b * a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.30000000000000002e192

   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 95.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 95.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   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 90.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 90.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{y} + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot \frac{b \cdot x}{y}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{y}\right)\right)} \]

       3. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{y} - \frac{b \cdot x}{y}} \]

       4. div-sub N/A

          \[\leadsto \color{blue}{\frac{x - b \cdot x}{y}} \]

       5. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       6. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{-1 \cdot \left(b \cdot x\right)}}{y} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(b \cdot x\right)}{y}} \]

       8. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       9. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       10. --lowering--.f64 N/A

           \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       11. *-lowering-*.f64 47.0

           \[\leadsto \frac{x - \color{blue}{b \cdot x}}{y} \]

   11. Simplified 47.0%

       \[\leadsto \color{blue}{\frac{x - b \cdot x}{y}} \]

   if -1.30000000000000002e192 < 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 68.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y + b \cdot y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{b \cdot y + y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot b} + y} \]

      3. accelerator-lowering-fma.f64 62.1

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   8. Simplified 62.1%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]

       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 40.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 40.1%

       \[\leadsto \color{blue}{\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. distribute-rgt1-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(\left(b + 1\right) \cdot y\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{\left(1 + b\right)} \cdot y\right)} \]

       3. *-commutative N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(y \cdot \left(1 + b\right)\right)}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}} \]

       5. distribute-rgt-in N/A

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(1 \cdot y + b \cdot y\right)}} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{x}{a \cdot \left(\color{blue}{y} + b \cdot y\right)} \]

       7. distribute-lft-in N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(b \cdot y\right)}} \]

       8. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{a \cdot \left(b \cdot y\right) + a \cdot y}} \]

       9. associate-*r* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(a \cdot b\right) \cdot y} + a \cdot y} \]

       10. distribute-rgt-out N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a \cdot b + a\right)}} \]

       12. *-commutative N/A

           \[\leadsto \frac{x}{y \cdot \left(\color{blue}{b \cdot a} + a\right)} \]

       13. accelerator-lowering-fma.f64 41.7

           \[\leadsto \frac{x}{y \cdot \color{blue}{\mathsf{fma}\left(b, a, a\right)}} \]

   14. Simplified 41.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \mathsf{fma}\left(b, a, a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+192}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.15e+192) (/ (- x (* x b)) 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 <= -1.15e+192) {
		tmp = (x - (x * b)) / 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 <= -1.15e+192)
		tmp = Float64(Float64(x - Float64(x * b)) / 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, -1.15e+192], N[(N[(x - N[(x * b), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * N[(b * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.15e192

   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 95.1

         \[\leadsto \frac{x \cdot e^{y \cdot \color{blue}{\log z} - b}}{y} \]

   5. Simplified 95.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]

   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 90.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]

   8. Simplified 90.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{-b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{y} + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot \frac{b \cdot x}{y}} \]

       2. mul-1-neg N/A

          \[\leadsto \frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{y}\right)\right)} \]

       3. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{x}{y} - \frac{b \cdot x}{y}} \]

       4. div-sub N/A

          \[\leadsto \color{blue}{\frac{x - b \cdot x}{y}} \]

       5. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x + \left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       6. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{-1 \cdot \left(b \cdot x\right)}}{y} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x + -1 \cdot \left(b \cdot x\right)}{y}} \]

       8. mul-1-neg N/A

          \[\leadsto \frac{x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}}{y} \]

       9. unsub-neg N/A

          \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       10. --lowering--.f64 N/A

           \[\leadsto \frac{\color{blue}{x - b \cdot x}}{y} \]

       11. *-lowering-*.f64 47.0

           \[\leadsto \frac{x - \color{blue}{b \cdot x}}{y} \]

   11. Simplified 47.0%

       \[\leadsto \color{blue}{\frac{x - b \cdot x}{y}} \]

   if -1.15e192 < 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 68.5

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y + b \cdot y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{b \cdot y + y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot b} + y} \]

      3. accelerator-lowering-fma.f64 62.1

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   8. Simplified 62.1%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]

       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 40.1

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 40.1%

       \[\leadsto \color{blue}{\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{+192}:\\ \;\;\;\;\frac{x - x \cdot b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0048:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 0.0048) (/ x (* y a)) (/ x (* a (* y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 0.0048) {
		tmp = x / (y * a);
	} else {
		tmp = x / (a * (y * 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 <= 0.0048d0) then
        tmp = x / (y * a)
    else
        tmp = x / (a * (y * 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 <= 0.0048) {
		tmp = x / (y * a);
	} else {
		tmp = x / (a * (y * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 0.0048:
		tmp = x / (y * a)
	else:
		tmp = x / (a * (y * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 0.0048)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(a * Float64(y * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 0.0048)
		tmp = x / (y * a);
	else
		tmp = x / (a * (y * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 0.0048], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00479999999999999958

   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 70.4

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 70.4%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Simplified 68.6%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]

   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 35.8

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 35.8%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

   if 0.00479999999999999958 < 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 61.3

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 61.3%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y + b \cdot y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{b \cdot y + y}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot b} + y} \]

      3. accelerator-lowering-fma.f64 46.3

         \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   8. Simplified 46.3%

      \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{\mathsf{fma}\left(y, b, y\right)}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot \left(y + b \cdot y\right)}} \]

       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 44.8

          \[\leadsto \frac{x}{a \cdot \color{blue}{\mathsf{fma}\left(b, y, y\right)}} \]

   11. Simplified 44.8%

       \[\leadsto \color{blue}{\frac{x}{a \cdot \mathsf{fma}\left(b, y, y\right)}} \]

   12. Taylor expanded in b around inf

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y\right)}} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 44.8

          \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y\right)}} \]

   14. Simplified 44.8%

       \[\leadsto \frac{x}{a \cdot \color{blue}{\left(b \cdot y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0048:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 31.0% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* y a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

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 * a)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Python

def code(x, y, z, t, a, b):
	return x / (y * a)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(y * a))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (y * a);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]

Derivation

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 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 b around 0

   \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]
7. Step-by-step derivation

8. Simplified 61.7%

   \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y}} \]

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 30.8

       \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

11. Simplified 30.8%

    \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

12. Final simplification 30.8%

    \[\leadsto \frac{x}{y \cdot a} \]
13. Add Preprocessing

Alternative 25: 15.8% 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 98.8%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in 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 69.7

      \[\leadsto \frac{x \cdot e^{\log \color{blue}{a} \cdot t - b}}{y} \]

5. Simplified 69.7%

   \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot t} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot {a}^{t}}}{y} \]

   3. pow-lowering-pow.f64 45.7

      \[\leadsto \frac{x \cdot \color{blue}{{a}^{t}}}{y} \]

8. Simplified 45.7%

   \[\leadsto \color{blue}{\frac{x \cdot {a}^{t}}{y}} \]

9. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 15.1

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Simplified 15.1%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 72.0% 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 2024205 
(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))