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

Percentage Accurate: 98.2% → 98.2%

Time: 10.0s

Alternatives: 10

Speedup: 1.0×

• 46.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 98.2% 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: 98.2% 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 98.5%

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

Alternative 2: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\log z \cdot y - b}}{y}\\ \mathbf{if}\;y \leq -72:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-219}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;y \leq 0.00092:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (* (log z) y) b))) y)))
   (if (<= y -72.0)
     t_1
     (if (<= y -3.1e-219)
       (/ (* (pow a (- t 1.0)) x) y)
       (if (<= y 0.00092) (/ (* x (exp (- (* (log a) t) b))) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp(((log(z) * y) - b))) / y;
	double tmp;
	if (y <= -72.0) {
		tmp = t_1;
	} else if (y <= -3.1e-219) {
		tmp = (pow(a, (t - 1.0)) * x) / y;
	} else if (y <= 0.00092) {
		tmp = (x * exp(((log(a) * t) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * exp(((log(z) * y) - b))) / y
    if (y <= (-72.0d0)) then
        tmp = t_1
    else if (y <= (-3.1d-219)) then
        tmp = ((a ** (t - 1.0d0)) * x) / y
    else if (y <= 0.00092d0) then
        tmp = (x * exp(((log(a) * t) - b))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * Math.exp(((Math.log(z) * y) - b))) / y;
	double tmp;
	if (y <= -72.0) {
		tmp = t_1;
	} else if (y <= -3.1e-219) {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	} else if (y <= 0.00092) {
		tmp = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x * math.exp(((math.log(z) * y) - b))) / y
	tmp = 0
	if y <= -72.0:
		tmp = t_1
	elif y <= -3.1e-219:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	elif y <= 0.00092:
		tmp = (x * math.exp(((math.log(a) * t) - b))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(Float64(log(z) * y) - b))) / y)
	tmp = 0.0
	if (y <= -72.0)
		tmp = t_1;
	elseif (y <= -3.1e-219)
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	elseif (y <= 0.00092)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x * exp(((log(z) * y) - b))) / y;
	tmp = 0.0;
	if (y <= -72.0)
		tmp = t_1;
	elseif (y <= -3.1e-219)
		tmp = ((a ^ (t - 1.0)) * x) / y;
	elseif (y <= 0.00092)
		tmp = (x * exp(((log(a) * t) - b))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -72.0], t$95$1, If[LessEqual[y, -3.1e-219], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 0.00092], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -72 or 9.2000000000000003e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.0%

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

   if -72 < y < -3.0999999999999997e-219

   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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.5%

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

   if -3.0999999999999997e-219 < y < 9.2000000000000003e-4

   1. Initial program 96.8%

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

   3. Taylor expanded in 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 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 3: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{{z}^{y}}{a} \cdot x}{y}\\ \mathbf{elif}\;y \leq 0.00092:\\ \;\;\;\;\frac{x \cdot e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\log z \cdot y - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.18e+91)
   (/ (* (/ (pow z y) a) x) y)
   (if (<= y 0.00092)
     (/ (* x (exp (- (* (- t 1.0) (log a)) b))) y)
     (/ (* x (exp (- (* (log z) y) b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.18e+91) {
		tmp = ((pow(z, y) / a) * x) / y;
	} else if (y <= 0.00092) {
		tmp = (x * exp((((t - 1.0) * log(a)) - b))) / y;
	} else {
		tmp = (x * exp(((log(z) * y) - 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 <= (-1.18d+91)) then
        tmp = (((z ** y) / a) * x) / y
    else if (y <= 0.00092d0) then
        tmp = (x * exp((((t - 1.0d0) * log(a)) - b))) / y
    else
        tmp = (x * exp(((log(z) * y) - 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 <= -1.18e+91) {
		tmp = ((Math.pow(z, y) / a) * x) / y;
	} else if (y <= 0.00092) {
		tmp = (x * Math.exp((((t - 1.0) * Math.log(a)) - b))) / y;
	} else {
		tmp = (x * Math.exp(((Math.log(z) * y) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.18e+91:
		tmp = ((math.pow(z, y) / a) * x) / y
	elif y <= 0.00092:
		tmp = (x * math.exp((((t - 1.0) * math.log(a)) - b))) / y
	else:
		tmp = (x * math.exp(((math.log(z) * y) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.18e+91)
		tmp = Float64(Float64(Float64((z ^ y) / a) * x) / y);
	elseif (y <= 0.00092)
		tmp = Float64(Float64(x * exp(Float64(Float64(Float64(t - 1.0) * log(a)) - b))) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(log(z) * y) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.18e+91)
		tmp = (((z ^ y) / a) * x) / y;
	elseif (y <= 0.00092)
		tmp = (x * exp((((t - 1.0) * log(a)) - b))) / y;
	else
		tmp = (x * exp(((log(z) * y) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.18e+91], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 0.00092], N[(N[(x * N[Exp[N[(N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.18000000000000008e91

   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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.0%

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

   10. Applied rewrites 97.6%

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

   if -1.18000000000000008e91 < y < 9.2000000000000003e-4

   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

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   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 96.0

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

   5. Applied rewrites 96.0%

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

   if 9.2000000000000003e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.0%

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

Alternative 4: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot e^{\log z \cdot y - b}}{y}\\ \mathbf{elif}\;b \leq 1000:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t - 1\right)} \cdot {z}^{y}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\log a \cdot t - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.6e-11)
   (/ (* x (exp (- (* (log z) y) b))) y)
   (if (<= b 1000.0)
     (/ (* x (* (pow a (- t 1.0)) (pow z y))) y)
     (/ (* x (exp (- (* (log a) t) b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.6e-11) {
		tmp = (x * exp(((log(z) * y) - b))) / y;
	} else if (b <= 1000.0) {
		tmp = (x * (pow(a, (t - 1.0)) * pow(z, y))) / y;
	} else {
		tmp = (x * exp(((log(a) * t) - 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 (b <= (-1.6d-11)) then
        tmp = (x * exp(((log(z) * y) - b))) / y
    else if (b <= 1000.0d0) then
        tmp = (x * ((a ** (t - 1.0d0)) * (z ** y))) / y
    else
        tmp = (x * exp(((log(a) * t) - 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 (b <= -1.6e-11) {
		tmp = (x * Math.exp(((Math.log(z) * y) - b))) / y;
	} else if (b <= 1000.0) {
		tmp = (x * (Math.pow(a, (t - 1.0)) * Math.pow(z, y))) / y;
	} else {
		tmp = (x * Math.exp(((Math.log(a) * t) - b))) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.6e-11:
		tmp = (x * math.exp(((math.log(z) * y) - b))) / y
	elif b <= 1000.0:
		tmp = (x * (math.pow(a, (t - 1.0)) * math.pow(z, y))) / y
	else:
		tmp = (x * math.exp(((math.log(a) * t) - b))) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.6e-11)
		tmp = Float64(Float64(x * exp(Float64(Float64(log(z) * y) - b))) / y);
	elseif (b <= 1000.0)
		tmp = Float64(Float64(x * Float64((a ^ Float64(t - 1.0)) * (z ^ y))) / y);
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(log(a) * t) - b))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.6e-11)
		tmp = (x * exp(((log(z) * y) - b))) / y;
	elseif (b <= 1000.0)
		tmp = (x * ((a ^ (t - 1.0)) * (z ^ y))) / y;
	else
		tmp = (x * exp(((log(a) * t) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.6e-11], N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1000.0], N[(N[(x * N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * N[Power[z, y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.59999999999999997e-11

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 89.9%

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

   if -1.59999999999999997e-11 < b < 1e3

   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 88.1

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

   5. Applied rewrites 88.1%

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

   if 1e3 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 90.2

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

   5. Applied rewrites 90.2%

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

Alternative 5: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -72 \lor \neg \left(y \leq 0.00092\right):\\ \;\;\;\;\frac{x \cdot e^{\log z \cdot y - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -72.0) (not (<= y 0.00092)))
   (/ (* x (exp (- (* (log z) y) b))) y)
   (/ (* (pow a (- t 1.0)) x) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -72.0) || !(y <= 0.00092)) {
		tmp = (x * exp(((log(z) * y) - b))) / y;
	} else {
		tmp = (pow(a, (t - 1.0)) * x) / 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 <= (-72.0d0)) .or. (.not. (y <= 0.00092d0))) then
        tmp = (x * exp(((log(z) * y) - b))) / y
    else
        tmp = ((a ** (t - 1.0d0)) * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -72.0) || !(y <= 0.00092)) {
		tmp = (x * Math.exp(((Math.log(z) * y) - b))) / y;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -72.0) or not (y <= 0.00092):
		tmp = (x * math.exp(((math.log(z) * y) - b))) / y
	else:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -72.0) || !(y <= 0.00092))
		tmp = Float64(Float64(x * exp(Float64(Float64(log(z) * y) - b))) / y);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -72.0) || ~((y <= 0.00092)))
		tmp = (x * exp(((log(z) * y) - b))) / y;
	else
		tmp = ((a ^ (t - 1.0)) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -72.0], N[Not[LessEqual[y, 0.00092]], $MachinePrecision]], N[(N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -72 or 9.2000000000000003e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.0%

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

   if -72 < y < 9.2000000000000003e-4

   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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -72 \lor \neg \left(y \leq 0.00092\right):\\ \;\;\;\;\frac{x \cdot e^{\log z \cdot y - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.45 \lor \neg \left(y \leq 0.00092\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -3.45) (not (<= y 0.00092)))
   (/ (* (/ (pow z y) a) x) y)
   (/ (* (pow a (- t 1.0)) x) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.45) || !(y <= 0.00092)) {
		tmp = ((pow(z, y) / a) * x) / y;
	} else {
		tmp = (pow(a, (t - 1.0)) * x) / 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 <= (-3.45d0)) .or. (.not. (y <= 0.00092d0))) then
        tmp = (((z ** y) / a) * x) / y
    else
        tmp = ((a ** (t - 1.0d0)) * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -3.45) || !(y <= 0.00092)) {
		tmp = ((Math.pow(z, y) / a) * x) / y;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -3.45) or not (y <= 0.00092):
		tmp = ((math.pow(z, y) / a) * x) / y
	else:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -3.45) || !(y <= 0.00092))
		tmp = Float64(Float64(Float64((z ^ y) / a) * x) / y);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -3.45) || ~((y <= 0.00092)))
		tmp = (((z ^ y) / a) * x) / y;
	else
		tmp = ((a ^ (t - 1.0)) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -3.45], N[Not[LessEqual[y, 0.00092]], $MachinePrecision]], N[(N[(N[(N[Power[z, y], $MachinePrecision] / a), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4500000000000002 or 9.2000000000000003e-4 < 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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.9%

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

   10. Applied rewrites 83.0%

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

   if -3.4500000000000002 < y < 9.2000000000000003e-4

   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

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.45 \lor \neg \left(y \leq 0.00092\right):\\ \;\;\;\;\frac{\frac{{z}^{y}}{a} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-21} \lor \neg \left(b \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.5e-21) (not (<= b 4.5e+82)))
   (/ x (* (* (exp b) y) a))
   (/ (* (pow a (- t 1.0)) x) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.5e-21) || !(b <= 4.5e+82)) {
		tmp = x / ((exp(b) * y) * a);
	} else {
		tmp = (pow(a, (t - 1.0)) * x) / 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 <= (-5.5d-21)) .or. (.not. (b <= 4.5d+82))) then
        tmp = x / ((exp(b) * y) * a)
    else
        tmp = ((a ** (t - 1.0d0)) * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.5e-21) || !(b <= 4.5e+82)) {
		tmp = x / ((Math.exp(b) * y) * a);
	} else {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -5.5e-21) or not (b <= 4.5e+82):
		tmp = x / ((math.exp(b) * y) * a)
	else:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.5e-21) || !(b <= 4.5e+82))
		tmp = Float64(x / Float64(Float64(exp(b) * y) * a));
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.5e-21) || ~((b <= 4.5e+82)))
		tmp = x / ((exp(b) * y) * a);
	else
		tmp = ((a ^ (t - 1.0)) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -5.5e-21], N[Not[LessEqual[b, 4.5e+82]], $MachinePrecision]], N[(x / N[(N[(N[Exp[b], $MachinePrecision] * y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.49999999999999977e-21 or 4.4999999999999997e82 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 95.7

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

   5. Applied rewrites 95.7%

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

   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 79.4

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. exp-diff N/A

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

       5. lower-/.f64 N/A

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

       6. exp-to-pow N/A

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

       7. lower-pow.f64 N/A

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

       8. lower--.f64 N/A

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

       9. lower-exp.f64 62.2

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

   11. Applied rewrites 62.2%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 83.3%

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

   if -5.49999999999999977e-21 < b < 4.4999999999999997e82

   1. Initial program 97.8%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-21} \lor \neg \left(b \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{\left(e^{b} \cdot y\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+108} \lor \neg \left(b \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.9e+108) (not (<= b 4.5e+82)))
   (* (/ (exp (- b)) y) x)
   (/ (* (pow a (- t 1.0)) x) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.9e+108) || !(b <= 4.5e+82)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = (pow(a, (t - 1.0)) * x) / 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 <= (-1.9d+108)) .or. (.not. (b <= 4.5d+82))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = ((a ** (t - 1.0d0)) * x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.9e+108) || !(b <= 4.5e+82)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = (Math.pow(a, (t - 1.0)) * x) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -1.9e+108) or not (b <= 4.5e+82):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = (math.pow(a, (t - 1.0)) * x) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.9e+108) || !(b <= 4.5e+82))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64((a ^ Float64(t - 1.0)) * x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -1.9e+108) || ~((b <= 4.5e+82)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = ((a ^ (t - 1.0)) * x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.9e+108], N[Not[LessEqual[b, 4.5e+82]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.90000000000000004e108 or 4.4999999999999997e82 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 95.9

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

   5. Applied rewrites 95.9%

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

   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 89.1

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

   8. Applied rewrites 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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 89.1

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

   10. Applied rewrites 89.1%

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

   if -1.90000000000000004e108 < b < 4.4999999999999997e82

   1. Initial program 97.9%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+108} \lor \neg \left(b \leq 4.5 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+16} \lor \neg \left(b \leq 245\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.55e+16) (not (<= b 245.0)))
   (* (/ (exp (- b)) y) x)
   (* (/ 1.0 a) (/ x y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.55e+16) || !(b <= 245.0)) {
		tmp = (exp(-b) / y) * x;
	} else {
		tmp = (1.0 / a) * (x / 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 <= (-1.55d+16)) .or. (.not. (b <= 245.0d0))) then
        tmp = (exp(-b) / y) * x
    else
        tmp = (1.0d0 / a) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.55e+16) || !(b <= 245.0)) {
		tmp = (Math.exp(-b) / y) * x;
	} else {
		tmp = (1.0 / a) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -1.55e+16) or not (b <= 245.0):
		tmp = (math.exp(-b) / y) * x
	else:
		tmp = (1.0 / a) * (x / y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.55e+16) || !(b <= 245.0))
		tmp = Float64(Float64(exp(Float64(-b)) / y) * x);
	else
		tmp = Float64(Float64(1.0 / a) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -1.55e+16) || ~((b <= 245.0)))
		tmp = (exp(-b) / y) * x;
	else
		tmp = (1.0 / a) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.55e+16], N[Not[LessEqual[b, 245.0]], $MachinePrecision]], N[(N[(N[Exp[(-b)], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.55e16 or 245 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 96.0

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

   5. Applied rewrites 96.0%

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

   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 78.1

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

   8. Applied rewrites 78.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{-b}}}{y} \]
   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 78.1

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

   10. Applied rewrites 78.1%

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

   if -1.55e16 < b < 245

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. exp-sum N/A

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

      6. lower-*.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. exp-to-pow N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 36.4%

       \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+16} \lor \neg \left(b \leq 245\right):\\ \;\;\;\;\frac{e^{-b}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a} \cdot \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (/ 1.0 a) (/ x y)))

C

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

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (1.0 / a) * (x / y);
}

Python

def code(x, y, z, t, a, b):
	return (1.0 / a) * (x / y)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(1.0 / a), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. exp-sum N/A

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

   6. lower-*.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. exp-to-pow N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-/.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 53.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 31.5%

    \[\leadsto \frac{1}{a} \cdot \frac{x}{y} \]
12. Add Preprocessing

Developer Target 1: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

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 2025015 
(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))