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

Percentage Accurate: 98.4% → 98.4%

Time: 14.7s

Alternatives: 24

Speedup: 1.0×

• 46.9% 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

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * 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 = (exp((((log(a) * (t - 1.0d0)) + (log(z) * y)) - b)) * x) / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Final simplification 97.4%

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

Alternative 2: 44.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (- (+ (* (log a) (- t 1.0)) (* (log z) y)) b)) x) y)))
   (if (<= t_1 -5e+305)
     (/ (/ x a) y)
     (if (<= t_1 1e-205)
       (/ (/ x (fma (fma (* (fma 0.16666666666666666 b 0.5) y) b y) b y)) a)
       (* (/ (* (/ 1.0 a) 1.0) (* 1.0 y)) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	double tmp;
	if (t_1 <= -5e+305) {
		tmp = (x / a) / y;
	} else if (t_1 <= 1e-205) {
		tmp = (x / fma(fma((fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a;
	} else {
		tmp = (((1.0 / a) * 1.0) / (1.0 * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(Float64(Float64(log(a) * Float64(t - 1.0)) + Float64(log(z) * y)) - b)) * x) / y)
	tmp = 0.0
	if (t_1 <= -5e+305)
		tmp = Float64(Float64(x / a) / y);
	elseif (t_1 <= 1e-205)
		tmp = Float64(Float64(x / fma(fma(Float64(fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / a) * 1.0) / Float64(1.0 * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+305], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e-205], N[(N[(x / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * y), $MachinePrecision] * b + y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(1.0 / a), $MachinePrecision] * 1.0), $MachinePrecision] / N[(1.0 * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5.00000000000000009e305

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 21.1%

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

   if -5.00000000000000009e305 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e-205

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 64.7%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 56.5%

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

   if 1e-205 < (/.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 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 76.4%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 40.2

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

   13. Applied rewrites 40.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{a}} \cdot 1}{y \cdot 1} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot y, b, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (- (+ (* (log a) (- t 1.0)) (* (log z) y)) b)) x) y)))
   (if (<= t_1 -5e+305)
     (/ (/ x a) y)
     (if (<= t_1 1e-205)
       (/ (/ x (fma (* (fma 0.5 b 1.0) y) b y)) a)
       (* (/ (* (/ 1.0 a) 1.0) (* 1.0 y)) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	double tmp;
	if (t_1 <= -5e+305) {
		tmp = (x / a) / y;
	} else if (t_1 <= 1e-205) {
		tmp = (x / fma((fma(0.5, b, 1.0) * y), b, y)) / a;
	} else {
		tmp = (((1.0 / a) * 1.0) / (1.0 * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(Float64(Float64(log(a) * Float64(t - 1.0)) + Float64(log(z) * y)) - b)) * x) / y)
	tmp = 0.0
	if (t_1 <= -5e+305)
		tmp = Float64(Float64(x / a) / y);
	elseif (t_1 <= 1e-205)
		tmp = Float64(Float64(x / fma(Float64(fma(0.5, b, 1.0) * y), b, y)) / a);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / a) * 1.0) / Float64(1.0 * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+305], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e-205], N[(N[(x / N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(1.0 / a), $MachinePrecision] * 1.0), $MachinePrecision] / N[(1.0 * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5.00000000000000009e305

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 21.1%

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

   if -5.00000000000000009e305 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e-205

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 64.7%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 53.3%

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

   if 1e-205 < (/.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 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 76.4%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 40.2

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

   13. Applied rewrites 40.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{a}} \cdot 1}{y \cdot 1} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot y, b, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (exp (- (+ (* (log a) (- t 1.0)) (* (log z) y)) b)) x) y)))
   (if (<= t_1 -5e+305)
     (/ (/ x a) y)
     (if (<= t_1 1e-205)
       (/ (/ x (fma b y y)) a)
       (* (/ (* (/ 1.0 a) 1.0) (* 1.0 y)) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp((((log(a) * (t - 1.0)) + (log(z) * y)) - b)) * x) / y;
	double tmp;
	if (t_1 <= -5e+305) {
		tmp = (x / a) / y;
	} else if (t_1 <= 1e-205) {
		tmp = (x / fma(b, y, y)) / a;
	} else {
		tmp = (((1.0 / a) * 1.0) / (1.0 * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(Float64(Float64(log(a) * Float64(t - 1.0)) + Float64(log(z) * y)) - b)) * x) / y)
	tmp = 0.0
	if (t_1 <= -5e+305)
		tmp = Float64(Float64(x / a) / y);
	elseif (t_1 <= 1e-205)
		tmp = Float64(Float64(x / fma(b, y, y)) / a);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / a) * 1.0) / Float64(1.0 * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[N[(N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+305], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e-205], N[(N[(x / N[(b * y + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(1.0 / a), $MachinePrecision] * 1.0), $MachinePrecision] / N[(1.0 * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -5.00000000000000009e305

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 21.1%

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

   if -5.00000000000000009e305 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e-205

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 64.7%

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

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y + b \cdot y}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 46.0%

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

   if 1e-205 < (/.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 99.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 76.4%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 40.2

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

   13. Applied rewrites 40.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{a}} \cdot 1}{y \cdot 1} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq -5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;\frac{e^{\left(\log a \cdot \left(t - 1\right) + \log z \cdot y\right) - b} \cdot x}{y} \leq 10^{-205}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 1}{1 \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{if}\;t \leq -55000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+82}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{e^{b} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (pow a (- t 1.0)) x) y)))
   (if (<= t -55000000000000.0)
     t_1
     (if (<= t 1.9e+82) (* (/ (pow z y) a) (/ x (* (exp b) y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -55000000000000.0) {
		tmp = t_1;
	} else if (t <= 1.9e+82) {
		tmp = (pow(z, y) / a) * (x / (exp(b) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((a ** (t - 1.0d0)) * x) / y
    if (t <= (-55000000000000.0d0)) then
        tmp = t_1
    else if (t <= 1.9d+82) then
        tmp = ((z ** y) / a) * (x / (exp(b) * y))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -55000000000000.0) {
		tmp = t_1;
	} else if (t <= 1.9e+82) {
		tmp = (Math.pow(z, y) / a) * (x / (Math.exp(b) * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.pow(a, (t - 1.0)) * x) / y
	tmp = 0
	if t <= -55000000000000.0:
		tmp = t_1
	elif t <= 1.9e+82:
		tmp = (math.pow(z, y) / a) * (x / (math.exp(b) * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y)
	tmp = 0.0
	if (t <= -55000000000000.0)
		tmp = t_1;
	elseif (t <= 1.9e+82)
		tmp = Float64(Float64((z ^ y) / a) * Float64(x / Float64(exp(b) * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a ^ (t - 1.0)) * x) / y;
	tmp = 0.0;
	if (t <= -55000000000000.0)
		tmp = t_1;
	elseif (t <= 1.9e+82)
		tmp = ((z ^ y) / a) * (x / (exp(b) * y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t, -55000000000000.0], t$95$1, If[LessEqual[t, 1.9e+82], N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5e13 or 1.90000000000000017e82 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.4%

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

   if -5.5e13 < t < 1.90000000000000017e82

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 91.3

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

   7. Applied rewrites 91.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -55000000000000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+82}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{e^{b} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 \cdot {a}^{\left(t - 1\right)}}{1 \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.5e+51)
   (/ (/ (* (pow z y) x) a) y)
   (if (<= y -1.42e-260)
     (/ (/ x (* (exp b) y)) a)
     (if (<= y 6.5e+35)
       (* (/ (* 1.0 (pow a (- t 1.0))) (* 1.0 y)) x)
       (/ (* (exp (* (log z) y)) x) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e+51) {
		tmp = ((pow(z, y) * x) / a) / y;
	} else if (y <= -1.42e-260) {
		tmp = (x / (exp(b) * y)) / a;
	} else if (y <= 6.5e+35) {
		tmp = ((1.0 * pow(a, (t - 1.0))) / (1.0 * y)) * x;
	} else {
		tmp = (exp((log(z) * y)) * x) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.5d+51)) then
        tmp = (((z ** y) * x) / a) / y
    else if (y <= (-1.42d-260)) then
        tmp = (x / (exp(b) * y)) / a
    else if (y <= 6.5d+35) then
        tmp = ((1.0d0 * (a ** (t - 1.0d0))) / (1.0d0 * y)) * x
    else
        tmp = (exp((log(z) * y)) * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.5e+51) {
		tmp = ((Math.pow(z, y) * x) / a) / y;
	} else if (y <= -1.42e-260) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else if (y <= 6.5e+35) {
		tmp = ((1.0 * Math.pow(a, (t - 1.0))) / (1.0 * y)) * x;
	} else {
		tmp = (Math.exp((Math.log(z) * y)) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.5e+51:
		tmp = ((math.pow(z, y) * x) / a) / y
	elif y <= -1.42e-260:
		tmp = (x / (math.exp(b) * y)) / a
	elif y <= 6.5e+35:
		tmp = ((1.0 * math.pow(a, (t - 1.0))) / (1.0 * y)) * x
	else:
		tmp = (math.exp((math.log(z) * y)) * x) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.5e+51)
		tmp = Float64(Float64(Float64((z ^ y) * x) / a) / y);
	elseif (y <= -1.42e-260)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	elseif (y <= 6.5e+35)
		tmp = Float64(Float64(Float64(1.0 * (a ^ Float64(t - 1.0))) / Float64(1.0 * y)) * x);
	else
		tmp = Float64(Float64(exp(Float64(log(z) * y)) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.5e+51)
		tmp = (((z ^ y) * x) / a) / y;
	elseif (y <= -1.42e-260)
		tmp = (x / (exp(b) * y)) / a;
	elseif (y <= 6.5e+35)
		tmp = ((1.0 * (a ^ (t - 1.0))) / (1.0 * y)) * x;
	else
		tmp = (exp((log(z) * y)) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.5e+51], N[(N[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, -1.42e-260], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 6.5e+35], N[(N[(N[(1.0 * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.5e51

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.6%

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

   if -6.5e51 < y < -1.42000000000000004e-260

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 83.0%

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

   if -1.42000000000000004e-260 < y < 6.5000000000000003e35

   1. Initial program 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 75.5%

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

   if 6.5000000000000003e35 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 \cdot {a}^{\left(t - 1\right)}}{1 \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log z \cdot y} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{if}\;t \leq -55000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+82}:\\ \;\;\;\;\frac{{z}^{y} \cdot x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (pow a (- t 1.0)) x) y)))
   (if (<= t -55000000000000.0)
     t_1
     (if (<= t 8e+82) (/ (* (pow z y) x) (* (* (exp b) y) a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -55000000000000.0) {
		tmp = t_1;
	} else if (t <= 8e+82) {
		tmp = (pow(z, y) * x) / ((exp(b) * y) * a);
	} 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 = ((a ** (t - 1.0d0)) * x) / y
    if (t <= (-55000000000000.0d0)) then
        tmp = t_1
    else if (t <= 8d+82) then
        tmp = ((z ** y) * x) / ((exp(b) * y) * a)
    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 = (Math.pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -55000000000000.0) {
		tmp = t_1;
	} else if (t <= 8e+82) {
		tmp = (Math.pow(z, y) * x) / ((Math.exp(b) * y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.pow(a, (t - 1.0)) * x) / y
	tmp = 0
	if t <= -55000000000000.0:
		tmp = t_1
	elif t <= 8e+82:
		tmp = (math.pow(z, y) * x) / ((math.exp(b) * y) * a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y)
	tmp = 0.0
	if (t <= -55000000000000.0)
		tmp = t_1;
	elseif (t <= 8e+82)
		tmp = Float64(Float64((z ^ y) * x) / Float64(Float64(exp(b) * y) * a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a ^ (t - 1.0)) * x) / y;
	tmp = 0.0;
	if (t <= -55000000000000.0)
		tmp = t_1;
	elseif (t <= 8e+82)
		tmp = ((z ^ y) * x) / ((exp(b) * y) * a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t, -55000000000000.0], t$95$1, If[LessEqual[t, 8e+82], N[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5e13 or 7.9999999999999997e82 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.4%

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

   if -5.5e13 < t < 7.9999999999999997e82

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 91.3

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 90.6%

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

Alternative 8: 82.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;b \leq 6500000000:\\ \;\;\;\;\frac{\left({z}^{y} \cdot x\right) \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.6e-10)
   (/ (/ x (* (exp b) y)) a)
   (if (<= b 6500000000.0)
     (/ (* (* (pow z y) x) (pow a (- t 1.0))) y)
     (* (/ (exp (- b)) y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.6e-10) {
		tmp = (x / (exp(b) * y)) / a;
	} else if (b <= 6500000000.0) {
		tmp = ((pow(z, y) * x) * pow(a, (t - 1.0))) / y;
	} else {
		tmp = (exp(-b) / y) * x;
	}
	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 <= (-3.6d-10)) then
        tmp = (x / (exp(b) * y)) / a
    else if (b <= 6500000000.0d0) then
        tmp = (((z ** y) * x) * (a ** (t - 1.0d0))) / y
    else
        tmp = (exp(-b) / y) * x
    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 <= -3.6e-10) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else if (b <= 6500000000.0) {
		tmp = ((Math.pow(z, y) * x) * Math.pow(a, (t - 1.0))) / y;
	} else {
		tmp = (Math.exp(-b) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.6e-10:
		tmp = (x / (math.exp(b) * y)) / a
	elif b <= 6500000000.0:
		tmp = ((math.pow(z, y) * x) * math.pow(a, (t - 1.0))) / y
	else:
		tmp = (math.exp(-b) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.6e-10)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	elseif (b <= 6500000000.0)
		tmp = Float64(Float64(Float64((z ^ y) * x) * (a ^ Float64(t - 1.0))) / y);
	else
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.6e-10)
		tmp = (x / (exp(b) * y)) / a;
	elseif (b <= 6500000000.0)
		tmp = (((z ^ y) * x) * (a ^ (t - 1.0))) / y;
	else
		tmp = (exp(-b) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.6e-10], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6500000000.0], N[(N[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6e-10

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 78.5%

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

   if -3.6e-10 < b < 6.5e9

   1. Initial program 96.6%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 6.5e9 < 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 b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.8

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

   5. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.8

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

   7. Applied rewrites 81.8%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;b \leq 6500000000:\\ \;\;\;\;\frac{\left({z}^{y} \cdot x\right) \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6500000000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot \left({z}^{y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -880000000.0)
     t_1
     (if (<= b 6500000000.0)
       (* (/ (pow a (- t 1.0)) y) (* (pow z y) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_1;
	} else if (b <= 6500000000.0) {
		tmp = (pow(a, (t - 1.0)) / y) * (pow(z, y) * x);
	} 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 = (exp(-b) / y) * x
    if (b <= (-880000000.0d0)) then
        tmp = t_1
    else if (b <= 6500000000.0d0) then
        tmp = ((a ** (t - 1.0d0)) / y) * ((z ** y) * x)
    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 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -880000000.0) {
		tmp = t_1;
	} else if (b <= 6500000000.0) {
		tmp = (Math.pow(a, (t - 1.0)) / y) * (Math.pow(z, y) * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -880000000.0:
		tmp = t_1
	elif b <= 6500000000.0:
		tmp = (math.pow(a, (t - 1.0)) / y) * (math.pow(z, y) * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -880000000.0)
		tmp = t_1;
	elseif (b <= 6500000000.0)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * Float64((z ^ y) * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -880000000.0)
		tmp = t_1;
	elseif (b <= 6500000000.0)
		tmp = ((a ^ (t - 1.0)) / y) * ((z ^ y) * x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -880000000.0], t$95$1, If[LessEqual[b, 6500000000.0], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e8 or 6.5e9 < 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 b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.5

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

   5. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if -8.8e8 < b < 6.5e9

   1. Initial program 95.3%

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

   3. Taylor expanded in 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. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. exp-prod N/A

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

      11. lower-pow.f64 N/A

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

      12. rem-exp-log N/A

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

      13. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -880000000:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{elif}\;b \leq 6500000000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot \left({z}^{y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 \cdot {a}^{\left(t - 1\right)}}{1 \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ (* (pow z y) x) a) y)))
   (if (<= y -6.5e+51)
     t_1
     (if (<= y -1.42e-260)
       (/ (/ x (* (exp b) y)) a)
       (if (<= y 6.5e+35)
         (* (/ (* 1.0 (pow a (- t 1.0))) (* 1.0 y)) x)
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((pow(z, y) * x) / a) / y;
	double tmp;
	if (y <= -6.5e+51) {
		tmp = t_1;
	} else if (y <= -1.42e-260) {
		tmp = (x / (exp(b) * y)) / a;
	} else if (y <= 6.5e+35) {
		tmp = ((1.0 * pow(a, (t - 1.0))) / (1.0 * y)) * x;
	} 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 = (((z ** y) * x) / a) / y
    if (y <= (-6.5d+51)) then
        tmp = t_1
    else if (y <= (-1.42d-260)) then
        tmp = (x / (exp(b) * y)) / a
    else if (y <= 6.5d+35) then
        tmp = ((1.0d0 * (a ** (t - 1.0d0))) / (1.0d0 * y)) * x
    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 = ((Math.pow(z, y) * x) / a) / y;
	double tmp;
	if (y <= -6.5e+51) {
		tmp = t_1;
	} else if (y <= -1.42e-260) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else if (y <= 6.5e+35) {
		tmp = ((1.0 * Math.pow(a, (t - 1.0))) / (1.0 * y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((math.pow(z, y) * x) / a) / y
	tmp = 0
	if y <= -6.5e+51:
		tmp = t_1
	elif y <= -1.42e-260:
		tmp = (x / (math.exp(b) * y)) / a
	elif y <= 6.5e+35:
		tmp = ((1.0 * math.pow(a, (t - 1.0))) / (1.0 * y)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64((z ^ y) * x) / a) / y)
	tmp = 0.0
	if (y <= -6.5e+51)
		tmp = t_1;
	elseif (y <= -1.42e-260)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	elseif (y <= 6.5e+35)
		tmp = Float64(Float64(Float64(1.0 * (a ^ Float64(t - 1.0))) / Float64(1.0 * y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z ^ y) * x) / a) / y;
	tmp = 0.0;
	if (y <= -6.5e+51)
		tmp = t_1;
	elseif (y <= -1.42e-260)
		tmp = (x / (exp(b) * y)) / a;
	elseif (y <= 6.5e+35)
		tmp = ((1.0 * (a ^ (t - 1.0))) / (1.0 * y)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6.5e+51], t$95$1, If[LessEqual[y, -1.42e-260], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 6.5e+35], N[(N[(N[(1.0 * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e51 or 6.5000000000000003e35 < 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 b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.1%

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

   if -6.5e51 < y < -1.42000000000000004e-260

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 83.0%

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

   if -1.42000000000000004e-260 < y < 6.5000000000000003e35

   1. Initial program 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 75.5%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 \cdot {a}^{\left(t - 1\right)}}{1 \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{z}^{y} \cdot x}{a \cdot y}\\ t_2 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 6200000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (pow z y) x) (* a y))) (t_2 (* (/ (exp (- b)) y) x)))
   (if (<= b -6.6e-7)
     t_2
     (if (<= b -1.6e-308)
       t_1
       (if (<= b 4e-55)
         (/ (* (pow a (- t 1.0)) x) y)
         (if (<= b 6200000000.0) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (pow(z, y) * x) / (a * y);
	double t_2 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_2;
	} else if (b <= -1.6e-308) {
		tmp = t_1;
	} else if (b <= 4e-55) {
		tmp = (pow(a, (t - 1.0)) * x) / y;
	} else if (b <= 6200000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((z ** y) * x) / (a * y)
    t_2 = (exp(-b) / y) * x
    if (b <= (-6.6d-7)) then
        tmp = t_2
    else if (b <= (-1.6d-308)) then
        tmp = t_1
    else if (b <= 4d-55) then
        tmp = ((a ** (t - 1.0d0)) * x) / y
    else if (b <= 6200000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (Math.pow(z, y) * x) / (a * y);
	double t_2 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_2;
	} else if (b <= -1.6e-308) {
		tmp = t_1;
	} else if (b <= 4e-55) {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	} else if (b <= 6200000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.pow(z, y) * x) / (a * y)
	t_2 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -6.6e-7:
		tmp = t_2
	elif b <= -1.6e-308:
		tmp = t_1
	elif b <= 4e-55:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	elif b <= 6200000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64((z ^ y) * x) / Float64(a * y))
	t_2 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -6.6e-7)
		tmp = t_2;
	elseif (b <= -1.6e-308)
		tmp = t_1;
	elseif (b <= 4e-55)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	elseif (b <= 6200000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((z ^ y) * x) / (a * y);
	t_2 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -6.6e-7)
		tmp = t_2;
	elseif (b <= -1.6e-308)
		tmp = t_1;
	elseif (b <= 4e-55)
		tmp = ((a ^ (t - 1.0)) * x) / y;
	elseif (b <= 6200000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] / N[(a * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -6.6e-7], t$95$2, If[LessEqual[b, -1.6e-308], t$95$1, If[LessEqual[b, 4e-55], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 6200000000.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.6000000000000003e-7 or 6.2e9 < 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 b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.0

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

   5. Applied rewrites 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.0

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

   7. Applied rewrites 80.0%

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

   if -6.6000000000000003e-7 < b < -1.6000000000000001e-308 or 3.99999999999999998e-55 < b < 6.2e9

   1. Initial program 94.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 77.6

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

   7. Applied rewrites 77.6%

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

   9. Applied rewrites 77.6%

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

   10. Taylor expanded in b around 0

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

   12. Applied rewrites 77.6%

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

   if -1.6000000000000001e-308 < b < 3.99999999999999998e-55

   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 b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{{z}^{y} \cdot x}{a \cdot y}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-55}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 6200000000:\\ \;\;\;\;\frac{{z}^{y} \cdot x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{{z}^{y} \cdot x}{a}}{y}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ (* (pow z y) x) a) y)))
   (if (<= y -6.5e+51)
     t_1
     (if (<= y -2.35e-260)
       (/ (/ x (* (exp b) y)) a)
       (if (<= y 5.2e+35) (/ (* (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 = ((pow(z, y) * x) / a) / y;
	double tmp;
	if (y <= -6.5e+51) {
		tmp = t_1;
	} else if (y <= -2.35e-260) {
		tmp = (x / (exp(b) * y)) / a;
	} else if (y <= 5.2e+35) {
		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 = (((z ** y) * x) / a) / y
    if (y <= (-6.5d+51)) then
        tmp = t_1
    else if (y <= (-2.35d-260)) then
        tmp = (x / (exp(b) * y)) / a
    else if (y <= 5.2d+35) 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 = ((Math.pow(z, y) * x) / a) / y;
	double tmp;
	if (y <= -6.5e+51) {
		tmp = t_1;
	} else if (y <= -2.35e-260) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else if (y <= 5.2e+35) {
		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 = ((math.pow(z, y) * x) / a) / y
	tmp = 0
	if y <= -6.5e+51:
		tmp = t_1
	elif y <= -2.35e-260:
		tmp = (x / (math.exp(b) * y)) / a
	elif y <= 5.2e+35:
		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(Float64((z ^ y) * x) / a) / y)
	tmp = 0.0
	if (y <= -6.5e+51)
		tmp = t_1;
	elseif (y <= -2.35e-260)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	elseif (y <= 5.2e+35)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z ^ y) * x) / a) / y;
	tmp = 0.0;
	if (y <= -6.5e+51)
		tmp = t_1;
	elseif (y <= -2.35e-260)
		tmp = (x / (exp(b) * y)) / a;
	elseif (y <= 5.2e+35)
		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[(N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -6.5e+51], t$95$1, If[LessEqual[y, -2.35e-260], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[y, 5.2e+35], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5e51 or 5.20000000000000013e35 < 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 b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.1%

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

   if -6.5e51 < y < -2.35e-260

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 83.0%

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

   if -2.35e-260 < y < 5.20000000000000013e35

   1. Initial program 95.5%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.5%

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

Alternative 13: 74.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{if}\;t \leq -105000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{e^{b} \cdot y}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* (pow a (- t 1.0)) x) y)))
   (if (<= t -105000.0)
     t_1
     (if (<= t 3.9e+74) (/ (/ x (* (exp b) y)) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -105000.0) {
		tmp = t_1;
	} else if (t <= 3.9e+74) {
		tmp = (x / (exp(b) * y)) / a;
	} 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 = ((a ** (t - 1.0d0)) * x) / y
    if (t <= (-105000.0d0)) then
        tmp = t_1
    else if (t <= 3.9d+74) then
        tmp = (x / (exp(b) * y)) / a
    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 = (Math.pow(a, (t - 1.0)) * x) / y;
	double tmp;
	if (t <= -105000.0) {
		tmp = t_1;
	} else if (t <= 3.9e+74) {
		tmp = (x / (Math.exp(b) * y)) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.pow(a, (t - 1.0)) * x) / y
	tmp = 0
	if t <= -105000.0:
		tmp = t_1
	elif t <= 3.9e+74:
		tmp = (x / (math.exp(b) * y)) / a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y)
	tmp = 0.0
	if (t <= -105000.0)
		tmp = t_1;
	elseif (t <= 3.9e+74)
		tmp = Float64(Float64(x / Float64(exp(b) * y)) / a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a ^ (t - 1.0)) * x) / y;
	tmp = 0.0;
	if (t <= -105000.0)
		tmp = t_1;
	elseif (t <= 3.9e+74)
		tmp = (x / (exp(b) * y)) / a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t, -105000.0], t$95$1, If[LessEqual[t, 3.9e+74], N[(N[(x / N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -105000 or 3.90000000000000008e74 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.2%

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

   if -105000 < t < 3.90000000000000008e74

   1. Initial program 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 91.7

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

   7. Applied rewrites 91.7%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 75.7%

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

Alternative 14: 75.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8000000000:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -1.3e+21)
     t_1
     (if (<= b 8000000000.0) (/ (* (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 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -1.3e+21) {
		tmp = t_1;
	} else if (b <= 8000000000.0) {
		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 = (exp(-b) / y) * x
    if (b <= (-1.3d+21)) then
        tmp = t_1
    else if (b <= 8000000000.0d0) 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 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -1.3e+21) {
		tmp = t_1;
	} else if (b <= 8000000000.0) {
		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 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -1.3e+21:
		tmp = t_1
	elif b <= 8000000000.0:
		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(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -1.3e+21)
		tmp = t_1;
	elseif (b <= 8000000000.0)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -1.3e+21)
		tmp = t_1;
	elseif (b <= 8000000000.0)
		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[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -1.3e+21], t$95$1, If[LessEqual[b, 8000000000.0], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3e21 or 8e9 < 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 b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.0

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

   5. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.0

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

   7. Applied rewrites 81.0%

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

   if -1.3e21 < b < 8e9

   1. Initial program 95.4%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.6%

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

Alternative 15: 58.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{e^{-b}}{y} \cdot x\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (exp (- b)) y) x)))
   (if (<= b -6.6e-7) t_1 (if (<= b 4.3e-26) (/ 1.0 (/ y (/ x a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_1;
	} else if (b <= 4.3e-26) {
		tmp = 1.0 / (y / (x / a));
	} 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 = (exp(-b) / y) * x
    if (b <= (-6.6d-7)) then
        tmp = t_1
    else if (b <= 4.3d-26) then
        tmp = 1.0d0 / (y / (x / a))
    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 = (Math.exp(-b) / y) * x;
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_1;
	} else if (b <= 4.3e-26) {
		tmp = 1.0 / (y / (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (math.exp(-b) / y) * x
	tmp = 0
	if b <= -6.6e-7:
		tmp = t_1
	elif b <= 4.3e-26:
		tmp = 1.0 / (y / (x / a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(exp(Float64(-b)) / y) * x)
	tmp = 0.0
	if (b <= -6.6e-7)
		tmp = t_1;
	elseif (b <= 4.3e-26)
		tmp = Float64(1.0 / Float64(y / Float64(x / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (exp(-b) / y) * x;
	tmp = 0.0;
	if (b <= -6.6e-7)
		tmp = t_1;
	elseif (b <= 4.3e-26)
		tmp = 1.0 / (y / (x / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[b, -6.6e-7], t$95$1, If[LessEqual[b, 4.3e-26], N[(1.0 / N[(y / N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.6000000000000003e-7 or 4.29999999999999988e-26 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.8

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

   7. Applied rewrites 76.8%

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

   if -6.6000000000000003e-7 < b < 4.29999999999999988e-26

   1. Initial program 95.6%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.4%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 44.4

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

   13. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{a}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 54.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} \cdot e^{-b}\\ \mathbf{if}\;b \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ x y) (exp (- b)))))
   (if (<= b -6.6e-7) t_1 (if (<= b 4.3e-26) (/ 1.0 (/ y (/ x a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x / y) * exp(-b);
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_1;
	} else if (b <= 4.3e-26) {
		tmp = 1.0 / (y / (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) * exp(-b)
    if (b <= (-6.6d-7)) then
        tmp = t_1
    else if (b <= 4.3d-26) then
        tmp = 1.0d0 / (y / (x / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x / y) * Math.exp(-b);
	double tmp;
	if (b <= -6.6e-7) {
		tmp = t_1;
	} else if (b <= 4.3e-26) {
		tmp = 1.0 / (y / (x / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x / y) * math.exp(-b)
	tmp = 0
	if b <= -6.6e-7:
		tmp = t_1
	elif b <= 4.3e-26:
		tmp = 1.0 / (y / (x / a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x / y) * exp(Float64(-b)))
	tmp = 0.0
	if (b <= -6.6e-7)
		tmp = t_1;
	elseif (b <= 4.3e-26)
		tmp = Float64(1.0 / Float64(y / Float64(x / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x / y) * exp(-b);
	tmp = 0.0;
	if (b <= -6.6e-7)
		tmp = t_1;
	elseif (b <= 4.3e-26)
		tmp = 1.0 / (y / (x / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] * N[Exp[(-b)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.6e-7], t$95$1, If[LessEqual[b, 4.3e-26], N[(1.0 / N[(y / N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -6.6000000000000003e-7 or 4.29999999999999988e-26 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

   if -6.6000000000000003e-7 < b < 4.29999999999999988e-26

   1. Initial program 95.6%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

      10. rem-exp-log N/A

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

      11. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 44.4%

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

       1. lift-/.f64 N/A

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

       2. clear-num N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 44.4

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

   13. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{a}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y} \cdot e^{-b}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot e^{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 45.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(-0.5 \cdot t\_1, b, t\_1\right), b, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -3.5e+178)
     (fma (- (fma (* -0.5 t_1) b t_1)) b t_1)
     (/ (/ x (fma (fma (* (fma 0.16666666666666666 b 0.5) y) b y) b y)) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -3.5e+178) {
		tmp = fma(-fma((-0.5 * t_1), b, t_1), b, t_1);
	} else {
		tmp = (x / fma(fma((fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -3.5e+178)
		tmp = fma(Float64(-fma(Float64(-0.5 * t_1), b, t_1)), b, t_1);
	else
		tmp = Float64(Float64(x / fma(fma(Float64(fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e+178], N[((-N[(N[(-0.5 * t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision]) * b + t$95$1), $MachinePrecision], N[(N[(x / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * y), $MachinePrecision] * b + y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e178

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 90.2%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 77.5%

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

   if -3.5e178 < b

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 55.9%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 44.7%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(-\mathsf{fma}\left(-0.5 \cdot \frac{x}{a \cdot y}, b, \frac{x}{a \cdot y}\right), b, \frac{x}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\mathsf{fma}\left(-0.5 \cdot \frac{x}{y}, b, \frac{x}{y}\right), b, \frac{x}{y}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.5e+178)
   (/ (fma (- (fma (* -0.5 (/ x y)) b (/ x y))) b (/ x y)) a)
   (/ (/ x (fma (fma (* (fma 0.16666666666666666 b 0.5) y) b y) b y)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.5e+178) {
		tmp = fma(-fma((-0.5 * (x / y)), b, (x / y)), b, (x / y)) / a;
	} else {
		tmp = (x / fma(fma((fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.5e+178)
		tmp = Float64(fma(Float64(-fma(Float64(-0.5 * Float64(x / y)), b, Float64(x / y))), b, Float64(x / y)) / a);
	else
		tmp = Float64(Float64(x / fma(fma(Float64(fma(0.16666666666666666, b, 0.5) * y), b, y), b, y)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.5e+178], N[(N[((-N[(N[(-0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] * b + N[(x / y), $MachinePrecision]), $MachinePrecision]) * b + N[(x / y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * y), $MachinePrecision] * b + y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e178

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f64 N/A

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

      6. lift--.f64 N/A

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

      7. exp-diff N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 80.1

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

   7. Applied rewrites 80.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 90.2%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 77.3%

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

   if -3.5e178 < b

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 71.3

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.9%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y + b \cdot \left(y + b \cdot \left(\frac{1}{6} \cdot \left(b \cdot y\right) + \frac{1}{2} \cdot y\right)\right)}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 44.7%

       \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, y\right), b, y\right)}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\mathsf{fma}\left(-0.5 \cdot \frac{x}{y}, b, \frac{x}{y}\right), b, \frac{x}{y}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot y, b, y\right), b, y\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(-b, t\_1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot y, b, y\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -3.5e+178)
     (fma (- b) t_1 t_1)
     (/ (/ x (fma (* (fma 0.5 b 1.0) y) b y)) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if (b <= -3.5e+178) {
		tmp = fma(-b, t_1, t_1);
	} else {
		tmp = (x / fma((fma(0.5, b, 1.0) * y), b, y)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -3.5e+178)
		tmp = fma(Float64(-b), t_1, t_1);
	else
		tmp = Float64(Float64(x / fma(Float64(fma(0.5, b, 1.0) * y), b, y)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e+178], N[((-b) * t$95$1 + t$95$1), $MachinePrecision], N[(N[(x / N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e178

   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. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 80.1

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.2%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto -1 \cdot \frac{b \cdot x}{a \cdot y} + \frac{x}{\color{blue}{a \cdot y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 55.0%

       \[\leadsto \mathsf{fma}\left(-b, \frac{x}{\color{blue}{y \cdot a}}, \frac{x}{y \cdot a}\right) \]

   if -3.5e178 < b

   1. Initial program 97.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 71.3

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.9%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y + b \cdot \left(y + \frac{1}{2} \cdot \left(b \cdot y\right)\right)}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 43.4%

       \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot y, b, y\right)}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(-b, \frac{x}{a \cdot y}, \frac{x}{a \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot y, b, y\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.2% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-257}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1.2e-257) (/ 1.0 (/ y (/ x a))) (/ (/ x (fma b y y)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1.2e-257) {
		tmp = 1.0 / (y / (x / a));
	} else {
		tmp = (x / fma(b, y, y)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1.2e-257)
		tmp = Float64(1.0 / Float64(y / Float64(x / a)));
	else
		tmp = Float64(Float64(x / fma(b, y, y)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1.2e-257], N[(1.0 / N[(y / N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(b * y + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.20000000000000008e-257

   1. Initial program 97.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}}{y} \]
   4. Step-by-step derivation

      1. exp-sum N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(e^{y \cdot \log z} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right)} \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot e^{\color{blue}{\log z \cdot y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      6. exp-to-pow N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      8. exp-prod N/A

         \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {\color{blue}{a}}^{\left(t - 1\right)}}{y} \]

      11. lower--.f64 74.6

          \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {a}^{\color{blue}{\left(t - 1\right)}}}{y} \]

   5. Applied rewrites 74.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot {z}^{y}\right) \cdot {a}^{\left(t - 1\right)}}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{x \cdot {z}^{y}}{\color{blue}{a}}}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.5%

      \[\leadsto \frac{\frac{{z}^{y} \cdot x}{\color{blue}{a}}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{a}}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.1%

       \[\leadsto \frac{\frac{x}{a}}{y} \]
   12. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{a}}{y}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{a}}}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{a}}}} \]

       4. lower-/.f64 40.1

          \[\leadsto \frac{1}{\color{blue}{\frac{y}{\frac{x}{a}}}} \]

   13. Applied rewrites 40.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{a}}}} \]

   if 1.20000000000000008e-257 < b

   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. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 72.0

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.4%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y + b \cdot y}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 41.1%

       \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 35.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-265}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b 1e-265) (/ (/ x a) y) (/ (/ x (fma b y y)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 1e-265) {
		tmp = (x / a) / y;
	} else {
		tmp = (x / fma(b, y, y)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 1e-265)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(Float64(x / fma(b, y, y)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 1e-265], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(b * y + y), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999985e-266

   1. Initial program 97.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}}{y} \]
   4. Step-by-step derivation

      1. exp-sum N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(e^{y \cdot \log z} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right)} \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(x \cdot e^{\color{blue}{\log z \cdot y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      6. exp-to-pow N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

      8. exp-prod N/A

         \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

      9. lower-pow.f64 N/A

         \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

      10. rem-exp-log N/A

          \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {\color{blue}{a}}^{\left(t - 1\right)}}{y} \]

      11. lower--.f64 74.2

          \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {a}^{\color{blue}{\left(t - 1\right)}}}{y} \]

   5. Applied rewrites 74.2%

      \[\leadsto \frac{\color{blue}{\left(x \cdot {z}^{y}\right) \cdot {a}^{\left(t - 1\right)}}}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{x \cdot {z}^{y}}{\color{blue}{a}}}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.0%

      \[\leadsto \frac{\frac{{z}^{y} \cdot x}{\color{blue}{a}}}{y} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{a}}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.9%

       \[\leadsto \frac{\frac{x}{a}}{y} \]

   if 9.99999999999999985e-266 < b

   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. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 76.2%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 72.5

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.2%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y + b \cdot y}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 41.3%

       \[\leadsto \frac{\frac{x}{\mathsf{fma}\left(b, y, y\right)}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 30.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6.4e-171) (/ (/ x y) a) (/ x (* a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.4e-171) {
		tmp = (x / y) / a;
	} else {
		tmp = x / (a * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-6.4d-171)) then
        tmp = (x / y) / a
    else
        tmp = x / (a * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6.4e-171) {
		tmp = (x / y) / a;
	} else {
		tmp = x / (a * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6.4e-171:
		tmp = (x / y) / a
	else:
		tmp = x / (a * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6.4e-171)
		tmp = Float64(Float64(x / y) / a);
	else
		tmp = Float64(x / Float64(a * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6.4e-171)
		tmp = (x / y) / a;
	else
		tmp = x / (a * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6.4e-171], N[(N[(x / y), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.4000000000000003e-171

   1. Initial program 97.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 77.4

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.9%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\frac{x}{y}}{a} \]
   12. Step-by-step derivation

   13. Applied rewrites 28.7%

       \[\leadsto \frac{\frac{x}{y}}{a} \]

   if -6.4000000000000003e-171 < y

   1. Initial program 97.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

      5. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      6. lift--.f64 N/A

         \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

      7. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

      8. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

      9. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   4. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

      9. lower-pow.f64 69.8

         \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{x}{a \cdot \color{blue}{y}} \]
   12. Step-by-step derivation

   13. Applied rewrites 38.8%

       \[\leadsto \frac{x}{y \cdot \color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 30.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{a}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (/ x a) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x / a) / 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 / a) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x / a) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x / a) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x / a) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x / a) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 97.4%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \frac{\color{blue}{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}}{y} \]
4. Step-by-step derivation

   1. exp-sum N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{y \cdot \log z} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x \cdot e^{y \cdot \log z}\right)} \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

   5. *-commutative N/A

      \[\leadsto \frac{\left(x \cdot e^{\color{blue}{\log z \cdot y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

   6. exp-to-pow N/A

      \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\left(x \cdot \color{blue}{{z}^{y}}\right) \cdot e^{\log a \cdot \left(t - 1\right)}}{y} \]

   8. exp-prod N/A

      \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot \color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}}}{y} \]

   10. rem-exp-log N/A

       \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {\color{blue}{a}}^{\left(t - 1\right)}}{y} \]

   11. lower--.f64 71.8

       \[\leadsto \frac{\left(x \cdot {z}^{y}\right) \cdot {a}^{\color{blue}{\left(t - 1\right)}}}{y} \]

5. Applied rewrites 71.8%

   \[\leadsto \frac{\color{blue}{\left(x \cdot {z}^{y}\right) \cdot {a}^{\left(t - 1\right)}}}{y} \]

6. Taylor expanded in t around 0

   \[\leadsto \frac{\frac{x \cdot {z}^{y}}{\color{blue}{a}}}{y} \]
7. Step-by-step derivation

8. Applied rewrites 64.2%

   \[\leadsto \frac{\frac{{z}^{y} \cdot x}{\color{blue}{a}}}{y} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{\frac{x}{a}}{y} \]
10. Step-by-step derivation

11. Applied rewrites 34.5%

    \[\leadsto \frac{\frac{x}{a}}{y} \]
12. Add Preprocessing

Alternative 24: 29.9% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a \cdot y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* a y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (a * 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 / (a * y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (a * y);
}

Python

def code(x, y, z, t, a, b):
	return x / (a * y)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(a * y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (a * y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}}{y} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

   5. lift-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

   6. lift--.f64 N/A

      \[\leadsto e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \cdot \frac{x}{y} \]

   7. exp-diff N/A

      \[\leadsto \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}} \cdot \frac{x}{y} \]

   8. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}}} \cdot \frac{x}{y} \]

   9. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{e^{b}}{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}} \cdot y}} \]

4. Applied rewrites 76.6%

   \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{e^{b}}{{a}^{\left(t - 1\right)} \cdot {z}^{y}} \cdot y}} \]

5. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{a \cdot \left(y \cdot e^{b}\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{x \cdot {z}^{y}}{\color{blue}{\left(y \cdot e^{b}\right) \cdot a}} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}} \cdot \frac{{z}^{y}}{a}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \cdot \frac{{z}^{y}}{a} \]

   5. *-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{e^{b} \cdot y}} \cdot \frac{{z}^{y}}{a} \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{e^{b}} \cdot y} \cdot \frac{{z}^{y}}{a} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \color{blue}{\frac{{z}^{y}}{a}} \]

   9. lower-pow.f64 72.4

      \[\leadsto \frac{x}{e^{b} \cdot y} \cdot \frac{\color{blue}{{z}^{y}}}{a} \]

7. Applied rewrites 72.4%

   \[\leadsto \color{blue}{\frac{x}{e^{b} \cdot y} \cdot \frac{{z}^{y}}{a}} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{x}{\color{blue}{a \cdot \left(y \cdot e^{b}\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 59.9%

    \[\leadsto \frac{\frac{x}{e^{b} \cdot y}}{\color{blue}{a}} \]

11. Taylor expanded in b around 0

    \[\leadsto \frac{x}{a \cdot \color{blue}{y}} \]
12. Step-by-step derivation

13. Applied rewrites 33.1%

    \[\leadsto \frac{x}{y \cdot \color{blue}{a}} \]

14. Final simplification 33.1%

    \[\leadsto \frac{x}{a \cdot y} \]
15. Add Preprocessing

Developer Target 1: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024268 
(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))