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

Percentage Accurate: 98.4% → 98.4%

Time: 16.5s

Alternatives: 19

Speedup: 1.0×

• 46.8% 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: 98.4% 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: 98.4% 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 2: 56.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ t_2 := x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (/ (fma b (fma b 0.5 -1.0) 1.0) y))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 2e+300)
       (/
        x
        (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))
       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 * (fma(b, fma(b, 0.5, -1.0), 1.0) / y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+300) {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	} 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(x * Float64(fma(b, fma(b, 0.5, -1.0), 1.0) / y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+300)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	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[(x * N[(N[(b * N[(b * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+300], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 2.0000000000000001e300 < (/.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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 81.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 81.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 40.4

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 40.4%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

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

   11. Simplified 32.9%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 41.4

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

   13. Applied egg-rr 41.4%

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

   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) < 2.0000000000000001e300

   1. Initial program 96.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 64.5

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

   8. Simplified 64.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 71.3

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

   11. Simplified 71.3%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ t_2 := x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* x (/ (fma b (fma b 0.5 -1.0) 1.0) y))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 2e+300) (/ x (* a (* y (fma b (fma b 0.5 1.0) 1.0)))) 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 * (fma(b, fma(b, 0.5, -1.0), 1.0) / y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+300) {
		tmp = x / (a * (y * fma(b, fma(b, 0.5, 1.0), 1.0)));
	} 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(x * Float64(fma(b, fma(b, 0.5, -1.0), 1.0) / y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+300)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, 0.5, 1.0), 1.0))));
	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[(x * N[(N[(b * N[(b * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+300], N[(x / N[(a * N[(y * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $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 2.0000000000000001e300 < (/.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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 81.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 81.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 40.4

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 40.4%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

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

   11. Simplified 32.9%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 41.4

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

   13. Applied egg-rr 41.4%

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

   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) < 2.0000000000000001e300

   1. Initial program 96.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 64.5

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

   8. Simplified 64.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 64.9

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

   11. Simplified 64.9%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+203}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (log a) (+ t -1.0))) b))) y)))
   (if (<= t_1 (- INFINITY))
     (* x (/ (- 1.0 b) y))
     (if (<= t_1 1e+203) (/ x (* a (fma y b y))) (/ (/ x a) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x * ((1.0 - b) / y);
	} else if (t_1 <= 1e+203) {
		tmp = x / (a * fma(y, b, y));
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(log(a) * Float64(t + -1.0))) - b))) / y)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x * Float64(Float64(1.0 - b) / y));
	elseif (t_1 <= 1e+203)
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+203], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 81.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 81.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 43.6

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 43.6%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   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. *-lft-identity N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. neg-mul-1 N/A

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

       13. unsub-neg N/A

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

       14. --lowering--.f64 16.7

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

   11. Simplified 16.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. --lowering--.f64 21.8

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

   13. Applied egg-rr 21.8%

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

   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) < 9.9999999999999999e202

   1. Initial program 97.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 64.4

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

   8. Simplified 64.4%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 48.3

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

   11. Simplified 48.3%

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

   if 9.9999999999999999e202 < (/.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 99.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. exp-prod N/A

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

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

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. exp-lowering-exp.f64 68.1

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

   5. Simplified 68.1%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 61.2%

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

   9. Taylor expanded in t around 0

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

       1. /-lowering-/.f64 29.8

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

   11. Simplified 29.8%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{elif}\;\frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \leq 10^{+203}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot {a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+145)
     t_2
     (if (<= t_1 5e+130) (/ (* x (exp (- (fma y (log z) 0.0) b))) 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 <= -5e+145) {
		tmp = t_2;
	} else if (t_1 <= 5e+130) {
		tmp = (x * exp((fma(y, log(z), 0.0) - b))) / 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 <= -5e+145)
		tmp = t_2;
	elseif (t_1 <= 5e+130)
		tmp = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / 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, -5e+145], t$95$2, If[LessEqual[t$95$1, 5e+130], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.99999999999999967e145 or 4.9999999999999996e130 < (*.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. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 98.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 98.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   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 92.5

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

   8. Simplified 92.5%

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

   if -4.99999999999999967e145 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 4.9999999999999996e130

   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 inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 81.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 81.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -5 \cdot 10^{+145}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot {a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -620:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -620.0)
     t_2
     (if (<= t_1 1000.0)
       (/
        x
        (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))
       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 <= -620.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	} 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 <= -620.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	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, -620.0], t$95$2, If[LessEqual[t$95$1, 1000.0], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -620 or 1e3 < (*.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. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 89.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 89.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   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 79.0

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

   8. Simplified 79.0%

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

   if -620 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 96.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 64.1

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

   5. Simplified 64.1%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 68.6

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

   8. Simplified 68.6%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 54.5

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

   11. Simplified 54.5%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -620:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 1000:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-237}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (fma y (log z) 0.0) b))) y)))
   (if (<= y -3.6e+43)
     t_1
     (if (<= y 2.95e-237)
       (/ (* x (exp (- (fma (log a) t 0.0) b))) y)
       (if (<= y 2.1e-37) (* (pow a (+ t -1.0)) (/ x y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	double tmp;
	if (y <= -3.6e+43) {
		tmp = t_1;
	} else if (y <= 2.95e-237) {
		tmp = (x * exp((fma(log(a), t, 0.0) - b))) / y;
	} else if (y <= 2.1e-37) {
		tmp = pow(a, (t + -1.0)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y)
	tmp = 0.0
	if (y <= -3.6e+43)
		tmp = t_1;
	elseif (y <= 2.95e-237)
		tmp = Float64(Float64(x * exp(Float64(fma(log(a), t, 0.0) - b))) / y);
	elseif (y <= 2.1e-37)
		tmp = Float64((a ^ Float64(t + -1.0)) * Float64(x / y));
	else
		tmp = t_1;
	end
	return 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] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.6e+43], t$95$1, If[LessEqual[y, 2.95e-237], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.1e-37], N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6000000000000001e43 or 2.1000000000000001e-37 < 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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 92.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 92.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   if -3.6000000000000001e43 < y < 2.95000000000000018e-237

   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. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 85.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 85.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   if 2.95000000000000018e-237 < y < 2.1000000000000001e-37

   1. Initial program 93.4%

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

   3. Taylor expanded in b around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y 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 80.6

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

   8. Simplified 80.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-237}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{if}\;b \leq -6.1 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.3:\\ \;\;\;\;\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (fma (log a) t 0.0) b))) y)))
   (if (<= b -6.1e+27)
     t_1
     (if (<= b 2.3) (fma x (* (pow a (+ t -1.0)) (/ (pow z y) y)) 0.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((fma(log(a), t, 0.0) - b))) / y;
	double tmp;
	if (b <= -6.1e+27) {
		tmp = t_1;
	} else if (b <= 2.3) {
		tmp = fma(x, (pow(a, (t + -1.0)) * (pow(z, y) / y)), 0.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(fma(log(a), t, 0.0) - b))) / y)
	tmp = 0.0
	if (b <= -6.1e+27)
		tmp = t_1;
	elseif (b <= 2.3)
		tmp = fma(x, Float64((a ^ Float64(t + -1.0)) * Float64((z ^ y) / y)), 0.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[b, -6.1e+27], t$95$1, If[LessEqual[b, 2.3], N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.0999999999999997e27 or 2.2999999999999998 < 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 t around inf

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 90.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 90.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   if -6.0999999999999997e27 < b < 2.2999999999999998

   1. Initial program 96.8%

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

   3. Taylor expanded in b around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow z y)) y)))
   (if (<= y -2.05e+36)
     t_1
     (if (<= y -1.1e-113)
       (/ (* x (pow a (+ t -1.0))) y)
       (if (<= y 3.1e+81) (/ x (* 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 * pow(z, y)) / y;
	double tmp;
	if (y <= -2.05e+36) {
		tmp = t_1;
	} else if (y <= -1.1e-113) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else if (y <= 3.1e+81) {
		tmp = x / (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 * (z ** y)) / y
    if (y <= (-2.05d+36)) then
        tmp = t_1
    else if (y <= (-1.1d-113)) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else if (y <= 3.1d+81) then
        tmp = x / (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.pow(z, y)) / y;
	double tmp;
	if (y <= -2.05e+36) {
		tmp = t_1;
	} else if (y <= -1.1e-113) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else if (y <= 3.1e+81) {
		tmp = x / (a * (y * Math.exp(b)));
	} 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 <= -2.05e+36:
		tmp = t_1
	elif y <= -1.1e-113:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	elif y <= 3.1e+81:
		tmp = x / (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 * (z ^ y)) / y)
	tmp = 0.0
	if (y <= -2.05e+36)
		tmp = t_1;
	elseif (y <= -1.1e-113)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	elseif (y <= 3.1e+81)
		tmp = Float64(x / Float64(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 * (z ^ y)) / y;
	tmp = 0.0;
	if (y <= -2.05e+36)
		tmp = t_1;
	elseif (y <= -1.1e-113)
		tmp = (x * (a ^ (t + -1.0))) / y;
	elseif (y <= 3.1e+81)
		tmp = x / (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[Power[z, y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -2.05e+36], t$95$1, If[LessEqual[y, -1.1e-113], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3.1e+81], N[(x / N[(a * N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.05000000000000006e36 or 3.1e81 < 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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 93.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 93.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 88.7

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

   8. Simplified 88.7%

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

   if -2.05000000000000006e36 < y < -1.10000000000000002e-113

   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 79.5

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

   5. Simplified 79.5%

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

   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 76.9%

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

   if -1.10000000000000002e-113 < y < 3.1e81

   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 y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. exp-prod N/A

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

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

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. exp-lowering-exp.f64 78.2

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

   5. Simplified 78.2%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 76.5

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

   8. Simplified 76.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+36}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.95 \cdot 10^{-212}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow z y)) y)))
   (if (<= y -7.5e+38)
     t_1
     (if (<= y 3.95e-212)
       (/ x (* y (exp b)))
       (if (<= y 1.52e+36)
         (/
          x
          (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))
         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 <= -7.5e+38) {
		tmp = t_1;
	} else if (y <= 3.95e-212) {
		tmp = x / (y * exp(b));
	} else if (y <= 1.52e+36) {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	} 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 <= -7.5e+38)
		tmp = t_1;
	elseif (y <= 3.95e-212)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (y <= 1.52e+36)
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	else
		tmp = t_1;
	end
	return 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, -7.5e+38], t$95$1, If[LessEqual[y, 3.95e-212], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.52e+36], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999999e38 or 1.52e36 < 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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 92.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 92.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 86.5

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

   8. Simplified 86.5%

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

   if -7.4999999999999999e38 < y < 3.9500000000000002e-212

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 59.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 59.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 58.0

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 58.0%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 58.0

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

   10. Applied egg-rr 58.0%

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

   if 3.9500000000000002e-212 < y < 1.52e36

   1. Initial program 94.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   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 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 77.1

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

   8. Simplified 77.1%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 62.6

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

   11. Simplified 62.6%

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

Alternative 11: 74.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow z y)) y)))
   (if (<= y -2e+36)
     t_1
     (if (<= y 2.05e+61) (/ (* 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 <= -2e+36) {
		tmp = t_1;
	} else if (y <= 2.05e+61) {
		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 <= (-2d+36)) then
        tmp = t_1
    else if (y <= 2.05d+61) 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 <= -2e+36) {
		tmp = t_1;
	} else if (y <= 2.05e+61) {
		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 <= -2e+36:
		tmp = t_1
	elif y <= 2.05e+61:
		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 <= -2e+36)
		tmp = t_1;
	elseif (y <= 2.05e+61)
		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 <= -2e+36)
		tmp = t_1;
	elseif (y <= 2.05e+61)
		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, -2e+36], t$95$1, If[LessEqual[y, 2.05e+61], 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 < -2.00000000000000008e36 or 2.04999999999999986e61 < 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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 93.4

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 93.4%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 88.4

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

   8. Simplified 88.4%

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

   if -2.00000000000000008e36 < y < 2.04999999999999986e61

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. exp-prod N/A

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

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

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

      9. rem-exp-log N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. exp-lowering-exp.f64 78.2

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

   5. Simplified 78.2%

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

   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 69.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+36}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{y}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+61}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (pow z y)) y)))
   (if (<= y -2.6e+36)
     t_1
     (if (<= y 5.5e+61) (* (pow a (+ t -1.0)) (/ x 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 <= -2.6e+36) {
		tmp = t_1;
	} else if (y <= 5.5e+61) {
		tmp = pow(a, (t + -1.0)) * (x / 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 <= (-2.6d+36)) then
        tmp = t_1
    else if (y <= 5.5d+61) then
        tmp = (a ** (t + (-1.0d0))) * (x / 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 <= -2.6e+36) {
		tmp = t_1;
	} else if (y <= 5.5e+61) {
		tmp = Math.pow(a, (t + -1.0)) * (x / 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 <= -2.6e+36:
		tmp = t_1
	elif y <= 5.5e+61:
		tmp = math.pow(a, (t + -1.0)) * (x / 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 <= -2.6e+36)
		tmp = t_1;
	elseif (y <= 5.5e+61)
		tmp = Float64((a ^ Float64(t + -1.0)) * Float64(x / 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 <= -2.6e+36)
		tmp = t_1;
	elseif (y <= 5.5e+61)
		tmp = (a ^ (t + -1.0)) * (x / 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, -2.6e+36], t$95$1, If[LessEqual[y, 5.5e+61], N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6000000000000001e36 or 5.50000000000000036e61 < 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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 93.4

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 93.4%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

      3. pow-lowering-pow.f64 88.4

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

   8. Simplified 88.4%

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

   if -2.6000000000000001e36 < y < 5.50000000000000036e61

   1. Initial program 97.0%

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

   3. Taylor expanded in b around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y 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 64.7

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

   8. Simplified 64.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+61}:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -115:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -115.0)
   (/ x (* y (exp b)))
   (/ x (* a (* y (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -115.0) {
		tmp = x / (y * exp(b));
	} else {
		tmp = x / (a * (y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -115.0)
		tmp = Float64(x / Float64(y * exp(b)));
	else
		tmp = Float64(x / Float64(a * Float64(y * fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -115.0], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -115

   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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 85.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 85.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 74.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. exp-lowering-exp.f64 74.2

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

   10. Applied egg-rr 74.2%

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

   if -115 < b

   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 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 52.7

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

   8. Simplified 52.7%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

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

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

       3. +-commutative N/A

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

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

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. accelerator-lowering-fma.f64 50.8

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

   11. Simplified 50.8%

       \[\leadsto \frac{x}{a \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -180:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -180.0)
   (* x (/ (fma b (fma b 0.5 -1.0) 1.0) y))
   (/ x (* a (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -180.0) {
		tmp = x * (fma(b, fma(b, 0.5, -1.0), 1.0) / y);
	} else {
		tmp = x / (a * fma(y, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -180.0)
		tmp = Float64(x * Float64(fma(b, fma(b, 0.5, -1.0), 1.0) / y));
	else
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -180.0], N[(x * N[(N[(b * N[(b * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -180

   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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 85.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 85.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 74.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

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

   11. Simplified 57.5%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

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

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

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

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

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

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

       7. *-commutative N/A

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

       8. accelerator-lowering-fma.f64 66.7

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

   13. Applied egg-rr 66.7%

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

   if -180 < b

   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 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 52.7

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

   8. Simplified 52.7%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 36.8

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

   11. Simplified 36.8%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -180:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;\frac{x}{y} \cdot \left(b \cdot \mathsf{fma}\left(b, 0.5, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -11.0) (* (/ x y) (* b (fma b 0.5 -1.0))) (/ x (* a (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -11.0) {
		tmp = (x / y) * (b * fma(b, 0.5, -1.0));
	} else {
		tmp = x / (a * fma(y, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -11.0)
		tmp = Float64(Float64(x / y) * Float64(b * fma(b, 0.5, -1.0)));
	else
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -11.0], N[(N[(x / y), $MachinePrecision] * N[(b * N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -11

   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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 85.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 85.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 74.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

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

   11. Simplified 57.5%

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

   12. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. sub-neg N/A

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

       4. distribute-rgt-in N/A

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

       5. distribute-lft-neg-out N/A

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

       6. lft-mult-inverse N/A

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

       7. sub-neg N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. *-commutative N/A

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

       12. accelerator-lowering-fma.f64 57.5

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

   14. Simplified 57.5%

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

   if -11 < b

   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 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 52.7

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

   8. Simplified 52.7%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 36.8

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

   11. Simplified 36.8%

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

Alternative 16: 39.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -35:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -35.0) (* x (/ (- 1.0 b) y)) (/ x (* a (fma y b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -35.0) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = x / (a * fma(y, b, y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -35.0)
		tmp = Float64(x * Float64(Float64(1.0 - b) / y));
	else
		tmp = Float64(x / Float64(a * fma(y, b, y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -35.0], N[(x * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(a * N[(y * b + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -35

   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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 85.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 85.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 74.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 74.2%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   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. *-lft-identity N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. neg-mul-1 N/A

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

       13. unsub-neg N/A

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

       14. --lowering--.f64 29.2

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

   11. Simplified 29.2%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. --lowering--.f64 32.4

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

   13. Applied egg-rr 32.4%

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

   if -35 < b

   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 64.7

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

   5. Simplified 64.7%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. exp-lowering-exp.f64 52.7

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

   8. Simplified 52.7%

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

   9. Taylor expanded in b around 0

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

       1. distribute-lft-out N/A

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

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

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. accelerator-lowering-fma.f64 36.8

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

   11. Simplified 36.8%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -35:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \mathsf{fma}\left(y, b, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.8e+24) (* x (/ (- 1.0 b) y)) (/ x (* y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.8e+24) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = x / (y * a);
	}
	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 <= (-4.8d+24)) then
        tmp = x * ((1.0d0 - b) / y)
    else
        tmp = x / (y * a)
    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 <= -4.8e+24) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -4.8e+24:
		tmp = x * ((1.0 - b) / y)
	else:
		tmp = x / (y * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.8e+24)
		tmp = Float64(x * Float64(Float64(1.0 - b) / y));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -4.8e+24)
		tmp = x * ((1.0 - b) / y);
	else
		tmp = x / (y * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.8e+24], N[(x * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.8000000000000001e24

   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. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

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

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 85.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 85.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   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-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      4. --lowering--.f64 73.8

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 73.8%

      \[\leadsto \frac{\color{blue}{x \cdot e^{0 - b}}}{y} \]

   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. *-lft-identity N/A

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

       9. associate-*r* N/A

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

       10. distribute-rgt-out N/A

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

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

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

       12. neg-mul-1 N/A

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

       13. unsub-neg N/A

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

       14. --lowering--.f64 29.7

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

   11. Simplified 29.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. --lowering--.f64 32.9

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

   13. Applied egg-rr 32.9%

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

   if -4.8000000000000001e24 < b

   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 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

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

   8. Simplified 60.2%

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

   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 31.5

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

   11. Simplified 31.5%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.6% 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 98.4%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
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 64.2

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

5. Simplified 64.2%

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

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 58.5%

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

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 29.3

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

11. Simplified 29.3%

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

12. Final simplification 29.3%

    \[\leadsto \frac{x}{y \cdot a} \]
13. Add Preprocessing

Alternative 19: 16.3% 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 98.4%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
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. +-rgt-identity N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   4. rem-exp-log N/A

      \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

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

      \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

   6. rem-exp-log 68.3

      \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

5. Simplified 68.3%

   \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

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 46.8

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

8. Simplified 46.8%

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

9. Taylor expanded in t around 0

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

    1. /-lowering-/.f64 12.2

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

11. Simplified 12.2%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 71.9% 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 2024196 
(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))