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

Percentage Accurate: 98.4% → 98.4%

Time: 10.7s

Alternatives: 22

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 98.3%

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

Alternative 2: 45.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot y, a, \left(-a\right) \cdot \frac{y}{b}\right), b, a \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \end{array} \end{array} \]

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 -2e-188)
     (/ (/ x a) y)
     (if (<= t_1 2e+14)
       (/ x (fma (* (- b) (fma (* -0.5 y) a (* (- a) (/ y b)))) b (* a y)))
       (/ (fma (- x) b x) (* a y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -2e-188) {
		tmp = (x / a) / y;
	} else if (t_1 <= 2e+14) {
		tmp = x / fma((-b * fma((-0.5 * y), a, (-a * (y / b)))), b, (a * y));
	} else {
		tmp = fma(-x, b, x) / (a * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -2e-188)
		tmp = Float64(Float64(x / a) / y);
	elseif (t_1 <= 2e+14)
		tmp = Float64(x / fma(Float64(Float64(-b) * fma(Float64(-0.5 * y), a, Float64(Float64(-a) * Float64(y / b)))), b, Float64(a * y)));
	else
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-188], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 2e+14], N[(x / N[(N[((-b) * N[(N[(-0.5 * y), $MachinePrecision] * a + N[((-a) * N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $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) < -1.9999999999999999e-188

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 38.1%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 42.5%

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

   if -1.9999999999999999e-188 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 2e14

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

   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

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 44.3%

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

   12. Taylor expanded in b around -inf

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

   14. Applied rewrites 59.5%

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

   if 2e14 < (/.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 99.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

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

   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

   8. Applied rewrites 51.9%

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

   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

   11. Applied rewrites 38.5%

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

Alternative 3: 43.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(\left(b \cdot y\right) \cdot 0.5\right) \cdot a, b, a \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \end{array} \end{array} \]

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 -4e+133)
     (/ (* x (/ 1.0 a)) y)
     (if (<= t_1 2e+143)
       (/ x (fma (* (* (* b y) 0.5) a) b (* a y)))
       (/ (fma (- x) b x) (* a y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -4e+133) {
		tmp = (x * (1.0 / a)) / y;
	} else if (t_1 <= 2e+143) {
		tmp = x / fma((((b * y) * 0.5) * a), b, (a * y));
	} else {
		tmp = fma(-x, b, x) / (a * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -4e+133)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	elseif (t_1 <= 2e+143)
		tmp = Float64(x / fma(Float64(Float64(Float64(b * y) * 0.5) * a), b, Float64(a * y)));
	else
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+133], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 2e+143], N[(x / N[(N[(N[(N[(b * y), $MachinePrecision] * 0.5), $MachinePrecision] * a), $MachinePrecision] * b + N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $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) < -4.0000000000000001e133

   1. Initial program 99.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 39.3%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 39.3%

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

   if -4.0000000000000001e133 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 2e143

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 60.8

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

   5. Applied rewrites 60.8%

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

   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

   8. Applied rewrites 53.9%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 47.3%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 52.9%

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

   if 2e143 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 52.9

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

   5. Applied rewrites 52.9%

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

   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

   8. Applied rewrites 50.5%

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

   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

   11. Applied rewrites 36.3%

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

Alternative 4: 40.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \end{array} \end{array} \]

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 -4e+133)
     (/ (* x (/ 1.0 a)) y)
     (if (<= t_1 0.0) (/ x (* (fma b y y) a)) (/ (fma (- x) b x) (* a y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -4e+133) {
		tmp = (x * (1.0 / a)) / y;
	} else if (t_1 <= 0.0) {
		tmp = x / (fma(b, y, y) * a);
	} else {
		tmp = fma(-x, b, x) / (a * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -4e+133)
		tmp = Float64(Float64(x * Float64(1.0 / a)) / y);
	elseif (t_1 <= 0.0)
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	else
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+133], N[(N[(x * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $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) < -4.0000000000000001e133

   1. Initial program 99.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 39.3%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 39.3%

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

   if -4.0000000000000001e133 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 0.0

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

   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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 43.9%

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

   if 0.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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

   8. Applied rewrites 54.3%

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

   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

   11. Applied rewrites 41.6%

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

Alternative 5: 40.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \end{array} \end{array} \]

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 -4e+133)
     (/ (/ x a) y)
     (if (<= t_1 0.0) (/ x (* (fma b y y) a)) (/ (fma (- x) b x) (* a y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if (t_1 <= -4e+133) {
		tmp = (x / a) / y;
	} else if (t_1 <= 0.0) {
		tmp = x / (fma(b, y, y) * a);
	} else {
		tmp = fma(-x, b, x) / (a * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
	tmp = 0.0
	if (t_1 <= -4e+133)
		tmp = Float64(Float64(x / a) / y);
	elseif (t_1 <= 0.0)
		tmp = Float64(x / Float64(fma(b, y, y) * a));
	else
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+133], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $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) < -4.0000000000000001e133

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 56.8

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

   5. Applied rewrites 56.8%

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

   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

   8. Applied rewrites 60.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 34.6%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 39.3%

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

   if -4.0000000000000001e133 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 0.0

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

   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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 43.9%

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

   if 0.0 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y)

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

   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

   8. Applied rewrites 54.3%

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

   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

   11. Applied rewrites 41.6%

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

Alternative 6: 39.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+133} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y)))
   (if (or (<= t_1 -4e+133) (not (<= t_1 5e-54)))
     (/ (/ x a) y)
     (/ x (* (fma b y y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
	double tmp;
	if ((t_1 <= -4e+133) || !(t_1 <= 5e-54)) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (fma(b, y, 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 <= -4e+133) || !(t_1 <= 5e-54))
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(fma(b, y, 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[Or[LessEqual[t$95$1, -4e+133], N[Not[LessEqual[t$95$1, 5e-54]], $MachinePrecision]], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(N[(b * y + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < -4.0000000000000001e133 or 5.00000000000000015e-54 < (/.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 99.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 35.2%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 40.1%

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

   if -4.0000000000000001e133 < (/.f64 (*.f64 x (exp.f64 (-.f64 (+.f64 (*.f64 y (log.f64 z)) (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))) b))) y) < 5.00000000000000015e-54

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 59.9

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

   5. Applied rewrites 59.9%

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

   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

   8. Applied rewrites 52.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 45.5%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \leq -4 \cdot 10^{+133} \lor \neg \left(\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, y, y\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+72} \lor \neg \left(t\_1 \leq 1000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, -\log a\right) - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -1e+72) (not (<= t_1 1000.0)))
     (/ (* x (exp (- (* (log a) t) b))) y)
     (/ (* x (exp (- (fma (log z) y (- (log a))) b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -1e+72) || !(t_1 <= 1000.0)) {
		tmp = (x * exp(((log(a) * t) - b))) / y;
	} else {
		tmp = (x * exp((fma(log(z), y, -log(a)) - b))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -1e+72) || !(t_1 <= 1000.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(fma(log(z), y, Float64(-log(a))) - b))) / y);
	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]}, If[Or[LessEqual[t$95$1, -1e+72], N[Not[LessEqual[t$95$1, 1000.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y + (-N[Log[a], $MachinePrecision])), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -9.99999999999999944e71 or 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -9.99999999999999944e71 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 96.9%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-log.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -1 \cdot 10^{+72} \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 1000\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log z, y, -\log a\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := e^{b} \cdot y\\ t_3 := \frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -195:\\ \;\;\;\;\frac{x}{t\_2 \cdot a}\\ \mathbf{elif}\;t\_1 \leq 10^{+89}:\\ \;\;\;\;\frac{\frac{x}{t\_2}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a)))
        (t_2 (* (exp b) y))
        (t_3 (* (/ (pow a (- t 1.0)) y) x)))
   (if (<= t_1 -2e+74)
     t_3
     (if (<= t_1 -195.0)
       (/ x (* t_2 a))
       (if (<= t_1 1e+89) (/ (/ x t_2) a) 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 = exp(b) * y;
	double t_3 = (pow(a, (t - 1.0)) / y) * x;
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = t_3;
	} else if (t_1 <= -195.0) {
		tmp = x / (t_2 * a);
	} else if (t_1 <= 1e+89) {
		tmp = (x / t_2) / a;
	} else {
		tmp = t_3;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    t_2 = exp(b) * y
    t_3 = ((a ** (t - 1.0d0)) / y) * x
    if (t_1 <= (-2d+74)) then
        tmp = t_3
    else if (t_1 <= (-195.0d0)) then
        tmp = x / (t_2 * a)
    else if (t_1 <= 1d+89) then
        tmp = (x / t_2) / a
    else
        tmp = t_3
    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 = (t - 1.0) * Math.log(a);
	double t_2 = Math.exp(b) * y;
	double t_3 = (Math.pow(a, (t - 1.0)) / y) * x;
	double tmp;
	if (t_1 <= -2e+74) {
		tmp = t_3;
	} else if (t_1 <= -195.0) {
		tmp = x / (t_2 * a);
	} else if (t_1 <= 1e+89) {
		tmp = (x / t_2) / a;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	t_2 = math.exp(b) * y
	t_3 = (math.pow(a, (t - 1.0)) / y) * x
	tmp = 0
	if t_1 <= -2e+74:
		tmp = t_3
	elif t_1 <= -195.0:
		tmp = x / (t_2 * a)
	elif t_1 <= 1e+89:
		tmp = (x / t_2) / a
	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(exp(b) * y)
	t_3 = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x)
	tmp = 0.0
	if (t_1 <= -2e+74)
		tmp = t_3;
	elseif (t_1 <= -195.0)
		tmp = Float64(x / Float64(t_2 * a));
	elseif (t_1 <= 1e+89)
		tmp = Float64(Float64(x / t_2) / a);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	t_2 = exp(b) * y;
	t_3 = ((a ^ (t - 1.0)) / y) * x;
	tmp = 0.0;
	if (t_1 <= -2e+74)
		tmp = t_3;
	elseif (t_1 <= -195.0)
		tmp = x / (t_2 * a);
	elseif (t_1 <= 1e+89)
		tmp = (x / t_2) / a;
	else
		tmp = t_3;
	end
	tmp_2 = 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[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+74], t$95$3, If[LessEqual[t$95$1, -195.0], N[(x / N[(t$95$2 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+89], N[(N[(x / t$95$2), $MachinePrecision] / a), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1.9999999999999999e74 or 9.99999999999999995e88 < (*.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 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.6

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

   10. Applied rewrites 82.6%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -195

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 47.5

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

   5. Applied rewrites 47.5%

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

   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

   8. Applied rewrites 67.4%

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

   if -195 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 9.99999999999999995e88

   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. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 64.5

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

   5. Applied rewrites 64.5%

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

   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

   8. Applied rewrites 65.2%

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

   10. Applied rewrites 70.3%

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

Alternative 9: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+36} \lor \neg \left(t\_1 \leq 560\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -4e+36) (not (<= t_1 560.0)))
     (/ (* x (exp (- (* (log a) t) b))) y)
     (/ (* x (/ (pow z y) a)) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -4e+36) || !(t_1 <= 560.0)) {
		tmp = (x * exp(((log(a) * t) - b))) / y;
	} else {
		tmp = (x * (pow(z, y) / a)) / 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-4d+36)) .or. (.not. (t_1 <= 560.0d0))) then
        tmp = (x * exp(((log(a) * t) - b))) / y
    else
        tmp = (x * ((z ** y) / a)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -4e+36) || !(t_1 <= 560.0)) {
		tmp = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	} else {
		tmp = (x * (Math.pow(z, y) / a)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -4e+36) or not (t_1 <= 560.0):
		tmp = (x * math.exp(((math.log(a) * t) - b))) / y
	else:
		tmp = (x * (math.pow(z, y) / a)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -4e+36) || !(t_1 <= 560.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	else
		tmp = Float64(Float64(x * Float64((z ^ y) / a)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -4e+36) || ~((t_1 <= 560.0)))
		tmp = (x * exp(((log(a) * t) - b))) / y;
	else
		tmp = (x * ((z ^ y) / a)) / y;
	end
	tmp_2 = 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]}, If[Or[LessEqual[t$95$1, -4e+36], N[Not[LessEqual[t$95$1, 560.0]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -4.00000000000000017e36 or 560 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -4.00000000000000017e36 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 560

   1. Initial program 96.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -4 \cdot 10^{+36} \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 560\right):\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+74} \lor \neg \left(t\_1 \leq 200000000000\right):\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- t 1.0) (log a))))
   (if (or (<= t_1 -2e+74) (not (<= t_1 200000000000.0)))
     (* (/ (pow a (- t 1.0)) y) x)
     (/ x (* (* (exp b) y) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - 1.0) * log(a);
	double tmp;
	if ((t_1 <= -2e+74) || !(t_1 <= 200000000000.0)) {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t - 1.0d0) * log(a)
    if ((t_1 <= (-2d+74)) .or. (.not. (t_1 <= 200000000000.0d0))) then
        tmp = ((a ** (t - 1.0d0)) / y) * x
    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 t_1 = (t - 1.0) * Math.log(a);
	double tmp;
	if ((t_1 <= -2e+74) || !(t_1 <= 200000000000.0)) {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	} else {
		tmp = x / ((Math.exp(b) * y) * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - 1.0) * math.log(a)
	tmp = 0
	if (t_1 <= -2e+74) or not (t_1 <= 200000000000.0):
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	else:
		tmp = x / ((math.exp(b) * y) * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - 1.0) * log(a))
	tmp = 0.0
	if ((t_1 <= -2e+74) || !(t_1 <= 200000000000.0))
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	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)
	t_1 = (t - 1.0) * log(a);
	tmp = 0.0;
	if ((t_1 <= -2e+74) || ~((t_1 <= 200000000000.0)))
		tmp = ((a ^ (t - 1.0)) / y) * x;
	else
		tmp = x / ((exp(b) * y) * a);
	end
	tmp_2 = 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]}, If[Or[LessEqual[t$95$1, -2e+74], N[Not[LessEqual[t$95$1, 200000000000.0]], $MachinePrecision]], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -1.9999999999999999e74 or 2e11 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.2

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

   10. Applied rewrites 81.2%

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

   if -1.9999999999999999e74 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2e11

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   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

   8. Applied rewrites 66.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -2 \cdot 10^{+74} \lor \neg \left(\left(t - 1\right) \cdot \log a \leq 200000000000\right):\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+107} \lor \neg \left(y \leq 1.2 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.55e+107) (not (<= y 1.2e+136)))
   (* (/ (/ (pow z y) a) y) x)
   (/ (* x (exp (- (* (- t 1.0) (log a)) b))) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.55e+107) || !(y <= 1.2e+136)) {
		tmp = ((pow(z, y) / a) / y) * x;
	} else {
		tmp = (x * exp((((t - 1.0) * log(a)) - b))) / 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 <= (-2.55d+107)) .or. (.not. (y <= 1.2d+136))) then
        tmp = (((z ** y) / a) / y) * x
    else
        tmp = (x * exp((((t - 1.0d0) * log(a)) - b))) / 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 <= -2.55e+107) || !(y <= 1.2e+136)) {
		tmp = ((Math.pow(z, y) / a) / y) * x;
	} else {
		tmp = (x * Math.exp((((t - 1.0) * Math.log(a)) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.55e+107) or not (y <= 1.2e+136):
		tmp = ((math.pow(z, y) / a) / y) * x
	else:
		tmp = (x * math.exp((((t - 1.0) * math.log(a)) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.55e+107) || !(y <= 1.2e+136))
		tmp = Float64(Float64(Float64((z ^ y) / a) / y) * x);
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(t - 1.0) * log(a)) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.55e+107) || ~((y <= 1.2e+136)))
		tmp = (((z ^ y) / a) / y) * x;
	else
		tmp = (x * exp((((t - 1.0) * log(a)) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.55e+107], N[Not[LessEqual[y, 1.2e+136]], $MachinePrecision]], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5500000000000001e107 or 1.2e136 < 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.4

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

   10. Applied rewrites 94.4%

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

   if -2.5500000000000001e107 < y < 1.2e136

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+107} \lor \neg \left(y \leq 1.2 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29000 \lor \neg \left(y \leq 7 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -29000.0) (not (<= y 7e+134)))
   (* (/ (/ (pow z y) a) y) x)
   (/ (* x (/ (pow a t) a)) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -29000.0) || !(y <= 7e+134)) {
		tmp = ((pow(z, y) / a) / y) * x;
	} else {
		tmp = (x * (pow(a, t) / a)) / 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 <= (-29000.0d0)) .or. (.not. (y <= 7d+134))) then
        tmp = (((z ** y) / a) / y) * x
    else
        tmp = (x * ((a ** t) / a)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -29000.0) || !(y <= 7e+134)) {
		tmp = ((Math.pow(z, y) / a) / y) * x;
	} else {
		tmp = (x * (Math.pow(a, t) / a)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -29000.0) or not (y <= 7e+134):
		tmp = ((math.pow(z, y) / a) / y) * x
	else:
		tmp = (x * (math.pow(a, t) / a)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -29000.0) || !(y <= 7e+134))
		tmp = Float64(Float64(Float64((z ^ y) / a) / y) * x);
	else
		tmp = Float64(Float64(x * Float64((a ^ t) / a)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -29000.0) || ~((y <= 7e+134)))
		tmp = (((z ^ y) / a) / y) * x;
	else
		tmp = (x * ((a ^ t) / a)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -29000.0], N[Not[LessEqual[y, 7e+134]], $MachinePrecision]], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[Power[a, t], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -29000 or 7.00000000000000006e134 < 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.4

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

   10. Applied rewrites 88.4%

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

   if -29000 < y < 7.00000000000000006e134

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.7%

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

   10. Applied rewrites 74.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29000 \lor \neg \left(y \leq 7 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29000 \lor \neg \left(y \leq 7 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -29000.0) (not (<= y 7e+134)))
   (* (/ (/ (pow z y) a) y) x)
   (/ (* 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 ((y <= -29000.0) || !(y <= 7e+134)) {
		tmp = ((pow(z, y) / a) / y) * x;
	} 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 ((y <= (-29000.0d0)) .or. (.not. (y <= 7d+134))) then
        tmp = (((z ** y) / a) / y) * x
    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 ((y <= -29000.0) || !(y <= 7e+134)) {
		tmp = ((Math.pow(z, y) / a) / y) * x;
	} else {
		tmp = (x * Math.pow(a, (t - 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -29000.0) or not (y <= 7e+134):
		tmp = ((math.pow(z, y) / a) / y) * x
	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 ((y <= -29000.0) || !(y <= 7e+134))
		tmp = Float64(Float64(Float64((z ^ y) / a) / y) * x);
	else
		tmp = Float64(Float64(x * (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 ((y <= -29000.0) || ~((y <= 7e+134)))
		tmp = (((z ^ y) / a) / y) * x;
	else
		tmp = (x * (a ^ (t - 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -29000.0], N[Not[LessEqual[y, 7e+134]], $MachinePrecision]], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -29000 or 7.00000000000000006e134 < 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.4

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

   10. Applied rewrites 88.4%

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

   if -29000 < y < 7.00000000000000006e134

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29000 \lor \neg \left(y \leq 7 \cdot 10^{+134}\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+62} \lor \neg \left(b \leq 5.1 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -7.2e+62) (not (<= b 5.1e+159)))
   (* (/ (exp (- b)) y) x)
   (/ (* 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 <= -7.2e+62) || !(b <= 5.1e+159)) {
		tmp = (exp(-b) / y) * x;
	} 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 <= (-7.2d+62)) .or. (.not. (b <= 5.1d+159))) then
        tmp = (exp(-b) / y) * x
    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 <= -7.2e+62) || !(b <= 5.1e+159)) {
		tmp = (Math.exp(-b) / y) * x;
	} 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 <= -7.2e+62) or not (b <= 5.1e+159):
		tmp = (math.exp(-b) / y) * x
	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 <= -7.2e+62) || !(b <= 5.1e+159))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64(x * (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 <= -7.2e+62) || ~((b <= 5.1e+159)))
		tmp = (exp(-b) / y) * x;
	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, -7.2e+62], N[Not[LessEqual[b, 5.1e+159]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.2e62 or 5.09999999999999967e159 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-log.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.5

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

   8. Applied rewrites 81.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.5

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

   10. Applied rewrites 81.5%

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

   if -7.2e62 < b < 5.09999999999999967e159

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+62} \lor \neg \left(b \leq 5.1 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t - 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+142} \lor \neg \left(b \leq 5.1 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -3.5e+142) (not (<= b 5.1e+159)))
   (* (/ (exp (- b)) y) x)
   (* (/ (pow a (- t 1.0)) y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -3.5e+142) || !(b <= 5.1e+159)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = (pow(a, (t - 1.0)) / y) * x;
	}
	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 <= (-3.5d+142)) .or. (.not. (b <= 5.1d+159))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = ((a ** (t - 1.0d0)) / y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -3.5e+142) || !(b <= 5.1e+159)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) / y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -3.5e+142) or not (b <= 5.1e+159):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = (math.pow(a, (t - 1.0)) / y) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -3.5e+142) || !(b <= 5.1e+159))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) / y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -3.5e+142) || ~((b <= 5.1e+159)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = ((a ^ (t - 1.0)) / y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -3.5e+142], N[Not[LessEqual[b, 5.1e+159]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.49999999999999997e142 or 5.09999999999999967e159 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-log.f64 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.5

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

   8. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.5

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

   10. Applied rewrites 85.5%

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

   if -3.49999999999999997e142 < b < 5.09999999999999967e159

   1. Initial program 97.7%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 65.1

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

   10. Applied rewrites 65.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+142} \lor \neg \left(b \leq 5.1 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)}}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -160 \lor \neg \left(b \leq 16000000000000\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot y, a, \left(-a\right) \cdot \frac{y}{b}\right), b, a \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -160.0) (not (<= b 16000000000000.0)))
   (* (/ (exp (- b)) y) x)
   (/ x (fma (* (- b) (fma (* -0.5 y) a (* (- a) (/ y b)))) b (* a y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -160.0) || !(b <= 16000000000000.0)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = x / fma((-b * fma((-0.5 * y), a, (-a * (y / b)))), b, (a * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -160.0) || !(b <= 16000000000000.0))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(x / fma(Float64(Float64(-b) * fma(Float64(-0.5 * y), a, Float64(Float64(-a) * Float64(y / b)))), b, Float64(a * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -160.0], N[Not[LessEqual[b, 16000000000000.0]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(x / N[(N[((-b) * N[(N[(-0.5 * y), $MachinePrecision] * a + N[((-a) * N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -160 or 1.6e13 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. distribute-lft-neg-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-in N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-log.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.1

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

   8. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.1

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

   10. Applied rewrites 70.1%

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

   if -160 < b < 1.6e13

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 61.9

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

   5. Applied rewrites 61.9%

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

   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

   8. Applied rewrites 40.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 39.3%

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

   12. Taylor expanded in b around -inf

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

   14. Applied rewrites 48.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -160 \lor \neg \left(b \leq 16000000000000\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot y, a, \left(-a\right) \cdot \frac{y}{b}\right), b, a \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{a}}{y}\\ \mathbf{if}\;b \leq -0.14:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_1 \cdot 0.5\right) \cdot b - t\_1, b, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot y, a, \left(-a\right) \cdot \frac{y}{b}\right), b, a \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ x a) y)))
   (if (<= b -0.14)
     (fma (- (* (* t_1 0.5) b) t_1) b t_1)
     (/ x (fma (* (- b) (fma (* -0.5 y) a (* (- a) (/ y b)))) b (* a y))))))

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 <= -0.14) {
		tmp = fma((((t_1 * 0.5) * b) - t_1), b, t_1);
	} else {
		tmp = x / fma((-b * fma((-0.5 * y), a, (-a * (y / b)))), b, (a * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x / a) / y)
	tmp = 0.0
	if (b <= -0.14)
		tmp = fma(Float64(Float64(Float64(t_1 * 0.5) * b) - t_1), b, t_1);
	else
		tmp = Float64(x / fma(Float64(Float64(-b) * fma(Float64(-0.5 * y), a, Float64(Float64(-a) * Float64(y / b)))), b, Float64(a * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[b, -0.14], N[(N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] * b), $MachinePrecision] - t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision], N[(x / N[(N[((-b) * N[(N[(-0.5 * y), $MachinePrecision] * a + N[((-a) * N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.14000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 35.4%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 60.8%

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

   if -0.14000000000000001 < b

   1. Initial program 97.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 58.3

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

   5. Applied rewrites 58.3%

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

   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

   8. Applied rewrites 47.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 39.9%

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

   12. Taylor expanded in b around -inf

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

   14. Applied rewrites 46.9%

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

Alternative 18: 47.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a \cdot y}\\ \mathbf{if}\;b \leq -1.24 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\left(t\_1 \cdot 0.5\right) \cdot b - t\_1, b, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(-b\right) \cdot \mathsf{fma}\left(-0.5 \cdot y, a, \left(-a\right) \cdot \frac{y}{b}\right), b, a \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* a y))))
   (if (<= b -1.24e-120)
     (fma (- (* (* t_1 0.5) b) t_1) b t_1)
     (/ x (fma (* (- b) (fma (* -0.5 y) a (* (- a) (/ y b)))) b (* a y))))))

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 <= -1.24e-120) {
		tmp = fma((((t_1 * 0.5) * b) - t_1), b, t_1);
	} else {
		tmp = x / fma((-b * fma((-0.5 * y), a, (-a * (y / b)))), b, (a * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a * y))
	tmp = 0.0
	if (b <= -1.24e-120)
		tmp = fma(Float64(Float64(Float64(t_1 * 0.5) * b) - t_1), b, t_1);
	else
		tmp = Float64(x / fma(Float64(Float64(-b) * fma(Float64(-0.5 * y), a, Float64(Float64(-a) * Float64(y / b)))), b, Float64(a * y)));
	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, -1.24e-120], N[(N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] * b), $MachinePrecision] - t$95$1), $MachinePrecision] * b + t$95$1), $MachinePrecision], N[(x / N[(N[((-b) * N[(N[(-0.5 * y), $MachinePrecision] * a + N[((-a) * N[(y / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.2399999999999999e-120

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 55.6

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

   5. Applied rewrites 55.6%

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

   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

   8. Applied rewrites 67.5%

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

   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

   11. Applied rewrites 55.3%

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

   if -1.2399999999999999e-120 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

   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

   8. Applied rewrites 48.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 39.7%

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

   12. Taylor expanded in b around -inf

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

   14. Applied rewrites 47.5%

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

Alternative 19: 44.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, y\right), b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.6e-125)
   (/ (fma (- x) b x) (* a y))
   (/ (/ x a) (fma (fma (* y (fma 0.16666666666666666 b 0.5)) b y) b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.6e-125) {
		tmp = fma(-x, b, x) / (a * y);
	} else {
		tmp = (x / a) / fma(fma((y * fma(0.16666666666666666, b, 0.5)), b, y), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.6e-125)
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	else
		tmp = Float64(Float64(x / a) / fma(fma(Float64(y * fma(0.16666666666666666, b, 0.5)), b, y), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.6e-125], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / N[(N[(N[(y * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] * b + y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.5999999999999999e-125

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 54.4

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

   5. Applied rewrites 54.4%

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

   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

   8. Applied rewrites 66.0%

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

   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

   11. Applied rewrites 46.3%

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

   if -1.5999999999999999e-125 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

   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

   8. Applied rewrites 48.7%

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

   10. Applied rewrites 46.0%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 43.8%

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

Alternative 20: 41.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, b, x\right)}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot y, 0.5, y\right), b, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.05e+142)
   (/ (fma (- x) b x) (* a y))
   (/ (/ x a) (fma (fma (* b y) 0.5 y) b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.05e+142) {
		tmp = fma(-x, b, x) / (a * y);
	} else {
		tmp = (x / a) / fma(fma((b * y), 0.5, y), b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.05e+142)
		tmp = Float64(fma(Float64(-x), b, x) / Float64(a * y));
	else
		tmp = Float64(Float64(x / a) / fma(fma(Float64(b * y), 0.5, y), b, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.05e+142], N[(N[((-x) * b + x), $MachinePrecision] / N[(a * y), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] * 0.5 + y), $MachinePrecision] * b + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05e142

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

   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

   8. Applied rewrites 83.6%

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

   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

   11. Applied rewrites 53.8%

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

   if -1.05e142 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 57.6

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

   5. Applied rewrites 57.6%

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

   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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 46.5%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 42.2%

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

Alternative 21: 33.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a 2e-34) (/ (/ x a) y) (/ x (* a y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2e-34) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (a * 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 (a <= 2d-34) then
        tmp = (x / a) / y
    else
        tmp = x / (a * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= 2e-34) {
		tmp = (x / a) / y;
	} else {
		tmp = x / (a * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= 2e-34:
		tmp = (x / a) / y
	else:
		tmp = x / (a * y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= 2e-34)
		tmp = Float64(Float64(x / a) / y);
	else
		tmp = Float64(x / Float64(a * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= 2e-34)
		tmp = (x / a) / y;
	else
		tmp = x / (a * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, 2e-34], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision], N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.99999999999999986e-34

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 30.7%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 36.5%

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

   if 1.99999999999999986e-34 < a

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. exp-diff N/A

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

      5. lower-/.f64 N/A

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

      6. exp-to-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

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

   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

   8. Applied rewrites 58.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 37.0%

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

Alternative 22: 31.7% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{a \cdot y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* a y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (a * y);
}

Fortran

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 / (a * y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (a * y);
}

Python

def code(x, y, z, t, a, b):
	return x / (a * y)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(a * y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (a * y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. exp-diff N/A

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

   5. lower-/.f64 N/A

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

   6. exp-to-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-/.f64 57.5

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

5. Applied rewrites 57.5%

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

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

8. Applied rewrites 54.7%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 34.0%

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

Developer Target 1: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

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) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2025016 
(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))