Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B

Percentage Accurate: 96.5% → 99.2%

Time: 9.8s

Alternatives: 10

Speedup: 1.1×

• 40.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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)) + (a * (log((1.0d0 - z)) - b))))
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)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) + (a * (log((1.0d0 - z)) - b))))
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)) + (a * (Math.log((1.0 - z)) - b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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)) + (a * (((-1.0d0) * z) - b))))
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)) + (a * ((-1.0 * z) - b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in z around 0

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

   1. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 25500000000:\\ \;\;\;\;\frac{x}{e^{a \cdot \left(b + z\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -2.4e+83)
     t_1
     (if (<= y 25500000000.0) (/ x (exp (* a (+ b z)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -2.4e+83) {
		tmp = t_1;
	} else if (y <= 25500000000.0) {
		tmp = x / exp((a * (b + z)));
	} 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((y * (log(z) - t)))
    if (y <= (-2.4d+83)) then
        tmp = t_1
    else if (y <= 25500000000.0d0) then
        tmp = x / exp((a * (b + z)))
    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((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -2.4e+83) {
		tmp = t_1;
	} else if (y <= 25500000000.0) {
		tmp = x / Math.exp((a * (b + z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -2.4e+83:
		tmp = t_1
	elif y <= 25500000000.0:
		tmp = x / math.exp((a * (b + z)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -2.4e+83)
		tmp = t_1;
	elseif (y <= 25500000000.0)
		tmp = Float64(x / exp(Float64(a * Float64(b + z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -2.4e+83)
		tmp = t_1;
	elseif (y <= 25500000000.0)
		tmp = x / exp((a * (b + z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+83], t$95$1, If[LessEqual[y, 25500000000.0], N[(x / N[Exp[N[(a * N[(b + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991e83 or 2.55e10 < y

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   if -2.39999999999999991e83 < y < 2.55e10

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 63.4

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

   9. Applied rewrites 63.4%

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

Alternative 3: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{e^{t \cdot y}}\\ \mathbf{elif}\;y \leq 25500000000:\\ \;\;\;\;\frac{x}{e^{a \cdot \left(b + z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.2e+89)
   (/ x (exp (* t y)))
   (if (<= y 25500000000.0) (/ x (exp (* a (+ b z)))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.2e+89) {
		tmp = x / exp((t * y));
	} else if (y <= 25500000000.0) {
		tmp = x / exp((a * (b + z)));
	} else {
		tmp = x * pow(z, 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 <= (-7.2d+89)) then
        tmp = x / exp((t * y))
    else if (y <= 25500000000.0d0) then
        tmp = x / exp((a * (b + z)))
    else
        tmp = x * (z ** 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 <= -7.2e+89) {
		tmp = x / Math.exp((t * y));
	} else if (y <= 25500000000.0) {
		tmp = x / Math.exp((a * (b + z)));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.2e+89:
		tmp = x / math.exp((t * y))
	elif y <= 25500000000.0:
		tmp = x / math.exp((a * (b + z)))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.2e+89)
		tmp = Float64(x / exp(Float64(t * y)));
	elseif (y <= 25500000000.0)
		tmp = Float64(x / exp(Float64(a * Float64(b + z))));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.2e+89)
		tmp = x / exp((t * y));
	elseif (y <= 25500000000.0)
		tmp = x / exp((a * (b + z)));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.2e+89], N[(x / N[Exp[N[(t * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 25500000000.0], N[(x / N[Exp[N[(a * N[(b + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e89

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x}{e^{\mathsf{neg}\left(y \cdot \left(\log z - t\right)\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

   if -7.2e89 < y < 2.55e10

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in z around 0

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

      1. lower-+.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if 2.55e10 < y

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 52.9

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

   7. Applied rewrites 52.9%

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

Alternative 4: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+89}:\\ \;\;\;\;\frac{x}{e^{t \cdot y}}\\ \mathbf{elif}\;y \leq 25500000000:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -6e+89)
   (/ x (exp (* t y)))
   (if (<= y 25500000000.0) (/ x (exp (* a b))) (* x (pow z y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -6e+89) {
		tmp = x / exp((t * y));
	} else if (y <= 25500000000.0) {
		tmp = x / exp((a * b));
	} else {
		tmp = x * pow(z, 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 <= (-6d+89)) then
        tmp = x / exp((t * y))
    else if (y <= 25500000000.0d0) then
        tmp = x / exp((a * b))
    else
        tmp = x * (z ** 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 <= -6e+89) {
		tmp = x / Math.exp((t * y));
	} else if (y <= 25500000000.0) {
		tmp = x / Math.exp((a * b));
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -6e+89:
		tmp = x / math.exp((t * y))
	elif y <= 25500000000.0:
		tmp = x / math.exp((a * b))
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -6e+89)
		tmp = Float64(x / exp(Float64(t * y)));
	elseif (y <= 25500000000.0)
		tmp = Float64(x / exp(Float64(a * b)));
	else
		tmp = Float64(x * (z ^ y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -6e+89)
		tmp = x / exp((t * y));
	elseif (y <= 25500000000.0)
		tmp = x / exp((a * b));
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -6e+89], N[(x / N[Exp[N[(t * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 25500000000.0], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.00000000000000025e89

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x}{e^{\mathsf{neg}\left(y \cdot \left(\log z - t\right)\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

   if -6.00000000000000025e89 < y < 2.55e10

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 59.0%

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

   if 2.55e10 < y

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in t around 0

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

      1. lower-pow.f64 52.9

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

   7. Applied rewrites 52.9%

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

Alternative 5: 69.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{e^{t \cdot y}}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;\frac{x}{e^{a \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (exp (* t y)))))
   (if (<= y -6e+89) t_1 (if (<= y 40000000000.0) (/ x (exp (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / exp((t * y));
	double tmp;
	if (y <= -6e+89) {
		tmp = t_1;
	} else if (y <= 40000000000.0) {
		tmp = x / exp((a * b));
	} 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((t * y))
    if (y <= (-6d+89)) then
        tmp = t_1
    else if (y <= 40000000000.0d0) then
        tmp = x / exp((a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / Math.exp((t * y));
	double tmp;
	if (y <= -6e+89) {
		tmp = t_1;
	} else if (y <= 40000000000.0) {
		tmp = x / Math.exp((a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / math.exp((t * y))
	tmp = 0
	if y <= -6e+89:
		tmp = t_1
	elif y <= 40000000000.0:
		tmp = x / math.exp((a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / exp(Float64(t * y)))
	tmp = 0.0
	if (y <= -6e+89)
		tmp = t_1;
	elseif (y <= 40000000000.0)
		tmp = Float64(x / exp(Float64(a * b)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / exp((t * y));
	tmp = 0.0;
	if (y <= -6e+89)
		tmp = t_1;
	elseif (y <= 40000000000.0)
		tmp = x / exp((a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[Exp[N[(t * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+89], t$95$1, If[LessEqual[y, 40000000000.0], N[(x / N[Exp[N[(a * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000025e89 or 4e10 < y

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x}{e^{\mathsf{neg}\left(y \cdot \left(\log z - t\right)\right)}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in t around inf

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

      1. lower-*.f64 57.9

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

   9. Applied rewrites 57.9%

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

   if -6.00000000000000025e89 < y < 4e10

   1. Initial program 96.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. sub-negate-rev N/A

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

      5. exp-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 59.7

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

   6. Applied rewrites 59.7%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 59.0%

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

Alternative 6: 57.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{e^{t \cdot y}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. add-flip N/A

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

   4. sub-negate-rev N/A

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

   5. exp-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower--.f64 N/A

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

3. Applied rewrites 96.5%

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

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{x}{e^{\mathsf{neg}\left(y \cdot \left(\log z - t\right)\right)}}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-log.f64 72.4

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

6. Applied rewrites 72.4%

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

7. Taylor expanded in t around inf

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

   1. lower-*.f64 57.9

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

9. Applied rewrites 57.9%

   \[\leadsto \frac{x}{e^{t \cdot y}} \]
10. Add Preprocessing

Alternative 7: 33.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lower-log.f64 31.4

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

      2. lower-*.f64 17.7

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

   10. Applied rewrites 17.7%

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

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

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lower-log.f64 31.4

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

   7. Applied rewrites 31.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lift-log.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(y \cdot \left(\log z - t\right) + 1\right) \]

      6. lift-log.f64 N/A

         \[\leadsto x \cdot \left(y \cdot \left(\log z - t\right) + 1\right) \]

      7. lift--.f64 N/A

         \[\leadsto x \cdot \left(y \cdot \left(\log z - t\right) + 1\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\left(\log z - t\right) \cdot y + 1\right) \]

      9. lower-fma.f64 31.4

         \[\leadsto x \cdot \mathsf{fma}\left(\log z - t, y, 1\right) \]

   9. Applied rewrites 31.4%

      \[\leadsto x \cdot \mathsf{fma}\left(\log z - t, y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 26.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;x \cdot \left(-1 \cdot \left(t \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot \left(-1 \cdot t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))) 0.0)
   (* x (* -1.0 (* t y)))
   (* x (+ 1.0 (* y (* -1.0 t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))))) <= 0.0) {
		tmp = x * (-1.0 * (t * y));
	} else {
		tmp = x * (1.0 + (y * (-1.0 * t)));
	}
	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 ((x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))) <= 0.0d0) then
        tmp = x * ((-1.0d0) * (t * y))
    else
        tmp = x * (1.0d0 + (y * ((-1.0d0) * t)))
    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 ((x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))))) <= 0.0) {
		tmp = x * (-1.0 * (t * y));
	} else {
		tmp = x * (1.0 + (y * (-1.0 * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))) <= 0.0:
		tmp = x * (-1.0 * (t * y))
	else:
		tmp = x * (1.0 + (y * (-1.0 * t)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(x * N[(-1.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(y * N[(-1.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lower-log.f64 31.4

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

      2. lower-*.f64 17.7

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

   10. Applied rewrites 17.7%

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

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

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lower-log.f64 31.4

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in t around inf

      \[\leadsto x \cdot \left(1 + y \cdot \left(-1 \cdot t\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 29.2

         \[\leadsto x \cdot \left(1 + y \cdot \left(-1 \cdot t\right)\right) \]

   10. Applied rewrites 29.2%

       \[\leadsto x \cdot \left(1 + y \cdot \left(-1 \cdot t\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 25.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(-1 \cdot \left(t \cdot y\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* -1.0 (* t y))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -10000.0) t_1 (if (<= t_2 5e+140) (* x 1.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (-1.0 * (t * y));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+140) {
		tmp = x * 1.0;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x * ((-1.0d0) * (t * y))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-10000.0d0)) then
        tmp = t_1
    else if (t_2 <= 5d+140) then
        tmp = x * 1.0d0
    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 * (-1.0 * (t * y));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+140) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (-1.0 * (t * y))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -10000.0:
		tmp = t_1
	elif t_2 <= 5e+140:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(-1.0 * Float64(t * y)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+140)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (-1.0 * (t * y));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 5e+140)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(-1.0 * N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -10000.0], t$95$1, If[LessEqual[t$95$2, 5e+140], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e4 or 5.00000000000000008e140 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

      4. lower-log.f64 31.4

         \[\leadsto x \cdot \left(1 + y \cdot \left(\log z - t\right)\right) \]

   7. Applied rewrites 31.4%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

      2. lower-*.f64 17.7

         \[\leadsto x \cdot \left(-1 \cdot \left(t \cdot y\right)\right) \]

   10. Applied rewrites 17.7%

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

   if -1e4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.00000000000000008e140

   1. Initial program 96.5%

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

   2. Taylor expanded in a around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 72.4

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 20.7%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 20.7% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

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 * 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in a around 0

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

   1. lower-exp.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-log.f64 72.4

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

4. Applied rewrites 72.4%

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

5. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 20.7%

   \[\leadsto x \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))