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

Percentage Accurate: 98.5% → 98.5%

Time: 6.8s

Alternatives: 16

Speedup: 1.0×

• 47.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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]

Derivation

1. Initial program 98.6%

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

Alternative 2: 47.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+251}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)))
   (if (<= t_1 -5e+25)
     (/ (* x (/ 1.0 a)) y)
     (if (<= t_1 1e+251)
       (/
        x
        (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) y) a))
       (/ x (* y a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -5e+25) {
		tmp = (x * (1.0 / a)) / y;
	} else if (t_1 <= 1e+251) {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e+25)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	elseif (t_1 <= 1e+251)
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * y) * a));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+25], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e+251], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $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) < -5.00000000000000024e25

   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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 30.5%

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

   if -5.00000000000000024e25 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e251

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 62.3

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 60.3

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

   11. Applied rewrites 60.3%

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

   if 1e251 < (/.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 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 37.3%

       \[\leadsto \frac{x}{y \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 45.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+251}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)))
   (if (<= t_1 -5e+25)
     (/ (* x (/ 1.0 a)) y)
     (if (<= t_1 1e+251)
       (/ x (* (* (fma (fma 0.5 b 1.0) b 1.0) y) a))
       (/ x (* y a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -5e+25) {
		tmp = (x * (1.0 / a)) / y;
	} else if (t_1 <= 1e+251) {
		tmp = x / ((fma(fma(0.5, b, 1.0), b, 1.0) * y) * a);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e+25)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	elseif (t_1 <= 1e+251)
		tmp = Float64(x / Float64(Float64(fma(fma(0.5, b, 1.0), b, 1.0) * y) * a));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+25], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e+251], N[(x / N[(N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $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) < -5.00000000000000024e25

   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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 30.5%

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

   if -5.00000000000000024e25 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e251

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 62.3

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 57.2

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

   11. Applied rewrites 57.2%

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

   if 1e251 < (/.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 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 37.3%

       \[\leadsto \frac{x}{y \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 43.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+251}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot y, 0.5, y\right), b, y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)))
   (if (<= t_1 -5e+25)
     (/ (* x (/ 1.0 a)) y)
     (if (<= t_1 1e+251)
       (/ x (* (fma (fma (* b y) 0.5 y) b y) a))
       (/ x (* y a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -5e+25) {
		tmp = (x * (1.0 / a)) / y;
	} else if (t_1 <= 1e+251) {
		tmp = x / (fma(fma((b * y), 0.5, y), b, y) * a);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -5e+25)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	elseif (t_1 <= 1e+251)
		tmp = Float64(x / Float64(fma(fma(Float64(b * y), 0.5, y), b, y) * a));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+25], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 1e+251], N[(x / N[(N[(N[(N[(b * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * b + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $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) < -5.00000000000000024e25

   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. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 30.5%

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

   if -5.00000000000000024e25 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 1e251

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 62.3

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 54.1

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

   11. Applied rewrites 54.1%

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

   if 1e251 < (/.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 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 37.3%

       \[\leadsto \frac{x}{y \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := \frac{x}{a \cdot y}\\ t_3 := x \cdot \frac{{a}^{t}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -70:\\ \;\;\;\;\mathsf{fma}\left(\left(-b\right) \cdot \left(t\_2 \cdot -0.5\right) - t\_2, b, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a)))
        (t_2 (/ x (* a y)))
        (t_3 (* x (/ (pow a t) y))))
   (if (<= t_1 -2e+16)
     t_3
     (if (<= t_1 -70.0)
       (fma (- (* (- b) (* t_2 -0.5)) t_2) b t_2)
       (if (<= t_1 1000.0) (* x (/ (exp (- b)) y)) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double t_2 = x / (a * y);
	double t_3 = x * (pow(a, t) / y);
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_3;
	} else if (t_1 <= -70.0) {
		tmp = fma(((-b * (t_2 * -0.5)) - t_2), b, t_2);
	} else if (t_1 <= 1000.0) {
		tmp = x * (exp(-b) / y);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	t_2 = Float64(x / Float64(a * y))
	t_3 = Float64(x * Float64((a ^ t) / y))
	tmp = 0.0
	if (t_1 <= -2e+16)
		tmp = t_3;
	elseif (t_1 <= -70.0)
		tmp = fma(Float64(Float64(Float64(-b) * Float64(t_2 * -0.5)) - t_2), b, t_2);
	elseif (t_1 <= 1000.0)
		tmp = Float64(x * Float64(exp(Float64(-b)) / y));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+16], t$95$3, If[LessEqual[t$95$1, -70.0], N[(N[(N[((-b) * N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] * b + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(x * N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e16 or 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 78.8

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 78.8%

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

   if -2e16 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -70

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 81.6

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 58.8%

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

   if -70 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 99.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 61.8

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

   5. Applied rewrites 61.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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. lower-*.f64 N/A

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

      5. lower-/.f64 61.8

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

   7. Applied rewrites 61.8%

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

Alternative 6: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+50} \lor \neg \left(b \leq 8.5 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y} \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -6.2e+50) (not (<= b 8.5e-93)))
   (/ (* x (exp (- (* (log a) (- t 1.0)) b))) y)
   (* x (/ (* (pow z y) (pow a (- t 1.0))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.2e+50) || !(b <= 8.5e-93)) {
		tmp = (x * exp(((log(a) * (t - 1.0)) - b))) / y;
	} else {
		tmp = x * ((pow(z, y) * pow(a, (t - 1.0))) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-6.2d+50)) .or. (.not. (b <= 8.5d-93))) then
        tmp = (x * exp(((log(a) * (t - 1.0d0)) - b))) / y
    else
        tmp = x * (((z ** y) * (a ** (t - 1.0d0))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -6.2e+50) || !(b <= 8.5e-93)) {
		tmp = (x * Math.exp(((Math.log(a) * (t - 1.0)) - b))) / y;
	} else {
		tmp = x * ((Math.pow(z, y) * Math.pow(a, (t - 1.0))) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -6.2e+50) or not (b <= 8.5e-93):
		tmp = (x * math.exp(((math.log(a) * (t - 1.0)) - b))) / y
	else:
		tmp = x * ((math.pow(z, y) * math.pow(a, (t - 1.0))) / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -6.2e+50) || !(b <= 8.5e-93))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * Float64(t - 1.0)) - b))) / y);
	else
		tmp = Float64(x * Float64(Float64((z ^ y) * (a ^ Float64(t - 1.0))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -6.2e+50) || ~((b <= 8.5e-93)))
		tmp = (x * exp(((log(a) * (t - 1.0)) - b))) / y;
	else
		tmp = x * (((z ^ y) * (a ^ (t - 1.0))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -6.2e+50], N[Not[LessEqual[b, 8.5e-93]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * N[(t - 1.0), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.20000000000000006e50 or 8.5000000000000007e-93 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -6.20000000000000006e50 < b < 8.5000000000000007e-93

   1. Initial program 97.2%

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

   3. Taylor expanded in 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. exp-sum N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 88.7

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

   5. Applied rewrites 88.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+50} \lor \neg \left(b \leq 8.5 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{z}^{y} \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{elif}\;y \leq 150000:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\log z \cdot y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.1e+23)
   (/ (* x (/ (pow z y) a)) y)
   (if (<= y 7.6e-255)
     (* x (/ (pow a (- t 1.0)) y))
     (if (<= y 150000.0)
       (/ x (* (* (exp b) y) a))
       (/ (* x (exp (* (log z) y))) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.1e+23) {
		tmp = (x * (pow(z, y) / a)) / y;
	} else if (y <= 7.6e-255) {
		tmp = x * (pow(a, (t - 1.0)) / y);
	} else if (y <= 150000.0) {
		tmp = x / ((exp(b) * y) * a);
	} else {
		tmp = (x * exp((log(z) * y))) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-1.1d+23)) then
        tmp = (x * ((z ** y) / a)) / y
    else if (y <= 7.6d-255) then
        tmp = x * ((a ** (t - 1.0d0)) / y)
    else if (y <= 150000.0d0) then
        tmp = x / ((exp(b) * y) * a)
    else
        tmp = (x * exp((log(z) * y))) / 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 <= -1.1e+23) {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	} else if (y <= 7.6e-255) {
		tmp = x * (Math.pow(a, (t - 1.0)) / y);
	} else if (y <= 150000.0) {
		tmp = x / ((Math.exp(b) * y) * a);
	} else {
		tmp = (x * Math.exp((Math.log(z) * y))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.1e+23:
		tmp = (x * (math.pow(z, y) / a)) / y
	elif y <= 7.6e-255:
		tmp = x * (math.pow(a, (t - 1.0)) / y)
	elif y <= 150000.0:
		tmp = x / ((math.exp(b) * y) * a)
	else:
		tmp = (x * math.exp((math.log(z) * y))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.1e+23)
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	elseif (y <= 7.6e-255)
		tmp = Float64(x * Float64((a ^ Float64(t - 1.0)) / y));
	elseif (y <= 150000.0)
		tmp = Float64(x / Float64(Float64(exp(b) * y) * a));
	else
		tmp = Float64(Float64(x * exp(Float64(log(z) * y))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.1e+23)
		tmp = (x * ((z ^ y) / a)) / y;
	elseif (y <= 7.6e-255)
		tmp = x * ((a ^ (t - 1.0)) / y);
	elseif (y <= 150000.0)
		tmp = x / ((exp(b) * y) * a);
	else
		tmp = (x * exp((log(z) * y))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.1e+23], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 7.6e-255], N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150000.0], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.10000000000000004e23

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if -1.10000000000000004e23 < y < 7.6e-255

   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 y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 79.7

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

   8. Applied rewrites 79.7%

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

   if 7.6e-255 < y < 1.5e5

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 87.9

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

   8. Applied rewrites 87.9%

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

   if 1.5e5 < 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

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 80.3

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

   5. Applied rewrites 80.3%

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

Alternative 8: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+83}:\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{{z}^{y} \cdot {a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.7e+83)
   (* x (/ (exp (- b)) y))
   (if (<= b 1.8e+18)
     (* x (/ (* (pow z y) (pow a (- t 1.0))) y))
     (/ x (* (* (exp b) y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.7e+83) {
		tmp = x * (exp(-b) / y);
	} else if (b <= 1.8e+18) {
		tmp = x * ((pow(z, y) * pow(a, (t - 1.0))) / y);
	} else {
		tmp = x / ((exp(b) * y) * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-3.7d+83)) then
        tmp = x * (exp(-b) / y)
    else if (b <= 1.8d+18) then
        tmp = x * (((z ** y) * (a ** (t - 1.0d0))) / y)
    else
        tmp = x / ((exp(b) * y) * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.7e+83) {
		tmp = x * (Math.exp(-b) / y);
	} else if (b <= 1.8e+18) {
		tmp = x * ((Math.pow(z, y) * Math.pow(a, (t - 1.0))) / y);
	} else {
		tmp = x / ((Math.exp(b) * y) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.7e+83:
		tmp = x * (math.exp(-b) / y)
	elif b <= 1.8e+18:
		tmp = x * ((math.pow(z, y) * math.pow(a, (t - 1.0))) / y)
	else:
		tmp = x / ((math.exp(b) * y) * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.7e+83)
		tmp = Float64(x * Float64(exp(Float64(-b)) / y));
	elseif (b <= 1.8e+18)
		tmp = Float64(x * Float64(Float64((z ^ y) * (a ^ Float64(t - 1.0))) / y));
	else
		tmp = Float64(x / Float64(Float64(exp(b) * y) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.7e+83)
		tmp = x * (exp(-b) / y);
	elseif (b <= 1.8e+18)
		tmp = x * (((z ^ y) * (a ^ (t - 1.0))) / y);
	else
		tmp = x / ((exp(b) * y) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.7e+83], N[(x * N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e+18], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7000000000000002e83

   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

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

      2. lower-neg.f64 86.2

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

   5. Applied rewrites 86.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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. lower-*.f64 N/A

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

      5. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

   if -3.7000000000000002e83 < b < 1.8e18

   1. Initial program 97.6%

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

   3. Taylor expanded in 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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. exp-sum N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. pow-to-exp N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 1.8e18 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 9: 75.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-255}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{elif}\;y \leq 150000:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (/ (pow z y) a)) y)))
   (if (<= y -1.1e+23)
     t_1
     (if (<= y 7.6e-255)
       (* x (/ (pow a (- t 1.0)) y))
       (if (<= y 150000.0) (/ 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 = (x * (pow(z, y) / a)) / y;
	double tmp;
	if (y <= -1.1e+23) {
		tmp = t_1;
	} else if (y <= 7.6e-255) {
		tmp = x * (pow(a, (t - 1.0)) / y);
	} else if (y <= 150000.0) {
		tmp = x / ((exp(b) * y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * ((z ** y) / a)) / y
    if (y <= (-1.1d+23)) then
        tmp = t_1
    else if (y <= 7.6d-255) then
        tmp = x * ((a ** (t - 1.0d0)) / y)
    else if (y <= 150000.0d0) 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 = (x * (Math.pow(z, y) / a)) / y;
	double tmp;
	if (y <= -1.1e+23) {
		tmp = t_1;
	} else if (y <= 7.6e-255) {
		tmp = x * (Math.pow(a, (t - 1.0)) / y);
	} else if (y <= 150000.0) {
		tmp = x / ((Math.exp(b) * y) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * (math.pow(z, y) / a)) / y
	tmp = 0
	if y <= -1.1e+23:
		tmp = t_1
	elif y <= 7.6e-255:
		tmp = x * (math.pow(a, (t - 1.0)) / y)
	elif y <= 150000.0:
		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(x * Float64((z ^ y) / a)) / y)
	tmp = 0.0
	if (y <= -1.1e+23)
		tmp = t_1;
	elseif (y <= 7.6e-255)
		tmp = Float64(x * Float64((a ^ Float64(t - 1.0)) / y));
	elseif (y <= 150000.0)
		tmp = Float64(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 = (x * ((z ^ y) / a)) / y;
	tmp = 0.0;
	if (y <= -1.1e+23)
		tmp = t_1;
	elseif (y <= 7.6e-255)
		tmp = x * ((a ^ (t - 1.0)) / y);
	elseif (y <= 150000.0)
		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[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -1.1e+23], t$95$1, If[LessEqual[y, 7.6e-255], N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 150000.0], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000004e23 or 1.5e5 < 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

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

      1. exp-sum N/A

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

      2. *-commutative N/A

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

      3. pow-to-exp N/A

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

      4. pow-to-exp N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lift-pow.f64 79.8

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

   8. Applied rewrites 79.8%

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

   if -1.10000000000000004e23 < y < 7.6e-255

   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 y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 79.7

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

   8. Applied rewrites 79.7%

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

   if 7.6e-255 < y < 1.5e5

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 10: 75.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t - 1 \leq -1 \cdot 10^{+23} \lor \neg \left(t - 1 \leq 20000000000\right):\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (- t 1.0) -1e+23) (not (<= (- t 1.0) 20000000000.0)))
   (* x (/ (pow a t) y))
   (/ x (* (* (exp b) y) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t - 1.0) <= -1e+23) || !((t - 1.0) <= 20000000000.0)) {
		tmp = x * (pow(a, t) / y);
	} else {
		tmp = x / ((exp(b) * y) * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((t - 1.0d0) <= (-1d+23)) .or. (.not. ((t - 1.0d0) <= 20000000000.0d0))) then
        tmp = x * ((a ** t) / y)
    else
        tmp = x / ((exp(b) * y) * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((t - 1.0) <= -1e+23) || !((t - 1.0) <= 20000000000.0)) {
		tmp = x * (Math.pow(a, t) / y);
	} else {
		tmp = x / ((Math.exp(b) * y) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((t - 1.0) <= -1e+23) or not ((t - 1.0) <= 20000000000.0):
		tmp = x * (math.pow(a, t) / y)
	else:
		tmp = x / ((math.exp(b) * y) * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(t - 1.0) <= -1e+23) || !(Float64(t - 1.0) <= 20000000000.0))
		tmp = Float64(x * Float64((a ^ t) / y));
	else
		tmp = Float64(x / Float64(Float64(exp(b) * y) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((t - 1.0) <= -1e+23) || ~(((t - 1.0) <= 20000000000.0)))
		tmp = x * ((a ^ t) / y);
	else
		tmp = x / ((exp(b) * y) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(t - 1.0), $MachinePrecision], -1e+23], N[Not[LessEqual[N[(t - 1.0), $MachinePrecision], 20000000000.0]], $MachinePrecision]], N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 t #s(literal 1 binary64)) < -9.9999999999999992e22 or 2e10 < (-.f64 t #s(literal 1 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 80.0

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 80.0%

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

   if -9.9999999999999992e22 < (-.f64 t #s(literal 1 binary64)) < 2e10

   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 y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 76.3

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

   8. Applied rewrites 76.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t - 1 \leq -1 \cdot 10^{+23} \lor \neg \left(t - 1 \leq 20000000000\right):\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+82} \lor \neg \left(b \leq 2.2 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -8.2e+82) (not (<= b 2.2e+17)))
   (* x (/ (exp (- b)) y))
   (* x (/ (pow a (- t 1.0)) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -8.2e+82) || !(b <= 2.2e+17)) {
		tmp = x * (exp(-b) / y);
	} else {
		tmp = x * (pow(a, (t - 1.0)) / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    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 <= (-8.2d+82)) .or. (.not. (b <= 2.2d+17))) then
        tmp = x * (exp(-b) / y)
    else
        tmp = x * ((a ** (t - 1.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -8.2e+82) || !(b <= 2.2e+17)) {
		tmp = x * (Math.exp(-b) / y);
	} else {
		tmp = x * (Math.pow(a, (t - 1.0)) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -8.2e+82) or not (b <= 2.2e+17):
		tmp = x * (math.exp(-b) / y)
	else:
		tmp = x * (math.pow(a, (t - 1.0)) / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -8.2e+82) || !(b <= 2.2e+17))
		tmp = Float64(x * Float64(exp(Float64(-b)) / y));
	else
		tmp = Float64(x * Float64((a ^ Float64(t - 1.0)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -8.2e+82) || ~((b <= 2.2e+17)))
		tmp = x * (exp(-b) / y);
	else
		tmp = x * ((a ^ (t - 1.0)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -8.2e+82], N[Not[LessEqual[b, 2.2e+17]], $MachinePrecision]], N[(x * N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.1999999999999999e82 or 2.2e17 < 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

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

      2. lower-neg.f64 83.2

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

   5. Applied rewrites 83.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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. lower-*.f64 N/A

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

      5. lower-/.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -8.1999999999999999e82 < b < 2.2e17

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 71.4

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

   8. Applied rewrites 71.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{+82} \lor \neg \left(b \leq 2.2 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \frac{e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;t \leq -1.85 \lor \neg \left(t \leq 20000000000\right):\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-b\right) \cdot \left(t\_1 \cdot -0.5\right) - t\_1, b, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (or (<= t -1.85) (not (<= t 20000000000.0)))
     (* x (/ (pow a t) y))
     (fma (- (* (- b) (* t_1 -0.5)) t_1) b t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a * y);
	double tmp;
	if ((t <= -1.85) || !(t <= 20000000000.0)) {
		tmp = x * (pow(a, t) / y);
	} else {
		tmp = fma(((-b * (t_1 * -0.5)) - t_1), b, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if ((t <= -1.85) || !(t <= 20000000000.0))
		tmp = Float64(x * Float64((a ^ t) / y));
	else
		tmp = fma(Float64(Float64(Float64(-b) * Float64(t_1 * -0.5)) - t_1), b, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.85], N[Not[LessEqual[t, 20000000000.0]], $MachinePrecision]], N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-b) * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001 or 2e10 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in b around 0

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

      1. pow-to-exp N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 79.4

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 79.4%

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

   if -1.8500000000000001 < t < 2e10

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 48.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \lor \neg \left(t \leq 20000000000\right):\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-b\right) \cdot \left(\frac{x}{a \cdot y} \cdot -0.5\right) - \frac{x}{a \cdot y}, b, \frac{x}{a \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -250:\\ \;\;\;\;\mathsf{fma}\left(\left(-b\right) \cdot \left(t\_1 \cdot -0.5\right) - t\_1, b, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -250.0)
     (fma (- (* (- b) (* t_1 -0.5)) t_1) b t_1)
     (/
      x
      (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) 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 <= -250.0) {
		tmp = fma(((-b * (t_1 * -0.5)) - t_1), b, t_1);
	} else {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * 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 <= -250.0)
		tmp = fma(Float64(Float64(Float64(-b) * Float64(t_1 * -0.5)) - t_1), b, t_1);
	else
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * 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, -250.0], N[(N[(N[((-b) * N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -250

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

   11. Applied rewrites 56.3%

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

   if -250 < b

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 55.2

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 49.9

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

   11. Applied rewrites 49.9%

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

Alternative 14: 46.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -250:\\ \;\;\;\;\mathsf{fma}\left(b \cdot t\_1, -1, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -250.0)
     (fma (* b t_1) -1.0 t_1)
     (/
      x
      (* (* (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) 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 <= -250.0) {
		tmp = fma((b * t_1), -1.0, t_1);
	} else {
		tmp = x / ((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * 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 <= -250.0)
		tmp = fma(Float64(b * t_1), -1.0, t_1);
	else
		tmp = Float64(x / Float64(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) * 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, -250.0], N[(N[(b * t$95$1), $MachinePrecision] * -1.0 + t$95$1), $MachinePrecision], N[(x / N[(N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -250

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 40.7

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

   11. Applied rewrites 40.7%

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

   if -250 < b

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 55.2

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 49.9

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

   11. Applied rewrites 49.9%

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

Alternative 15: 36.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -250:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -250.0) (/ x (* y 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 <= -250.0) {
		tmp = x / (y * 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 <= -250.0)
		tmp = Float64(x / Float64(y * a));
	else
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -250.0], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -250

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 81.1

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

   8. Applied rewrites 81.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 33.8%

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

   if -250 < b

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. div-exp N/A

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

      5. pow-to-exp N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-exp.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-exp.f64 55.2

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

   8. Applied rewrites 55.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 40.9

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

   11. Applied rewrites 40.9%

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

Alternative 16: 32.1% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* y a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / (y * a)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Python

def code(x, y, z, t, a, b):
	return x / (y * a)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(y * a))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (y * a);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. div-exp N/A

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

   5. pow-to-exp N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lower-exp.f64 69.0

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

5. Applied rewrites 69.0%

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

6. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 62.6

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

8. Applied rewrites 62.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2025085 
(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))