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

Percentage Accurate: 96.3% → 99.6%

Time: 10.1s

Alternatives: 11

Speedup: 1.5×

• 40.4% 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: 96.3% 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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 99.6%

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

Alternative 2: 50.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (or (<= t_1 -2e+17) (not (<= t_1 5e+23)))
     (exp (* (- a) b))
     (* x (exp (* (- a) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23)) {
		tmp = exp((-a * b));
	} else {
		tmp = x * exp((-a * z));
	}
	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 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if ((t_1 <= (-2d+17)) .or. (.not. (t_1 <= 5d+23))) then
        tmp = exp((-a * b))
    else
        tmp = x * exp((-a * z))
    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 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23)) {
		tmp = Math.exp((-a * b));
	} else {
		tmp = x * Math.exp((-a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if (t_1 <= -2e+17) or not (t_1 <= 5e+23):
		tmp = math.exp((-a * b))
	else:
		tmp = x * math.exp((-a * z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23))
		tmp = exp(Float64(Float64(-a) * b));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if ((t_1 <= -2e+17) || ~((t_1 <= 5e+23)))
		tmp = exp((-a * b));
	else
		tmp = x * exp((-a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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[Or[LessEqual[t$95$1, -2e+17], N[Not[LessEqual[t$95$1, 5e+23]], $MachinePrecision]], N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision], N[(x * N[Exp[N[((-a) * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

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))) < -2e17 or 4.9999999999999999e23 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.2%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 22.3

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

   7. Applied rewrites 9.2%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 45.8%

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

   if -2e17 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.9999999999999999e23

   1. Initial program 89.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 90.6%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+17} \lor \neg \left(y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+17} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;e^{b \cdot a} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (or (<= t_1 -2e+17) (not (<= t_1 5e+23)))
     (exp (* (- a) b))
     (* (exp (* b a)) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23)) {
		tmp = exp((-a * b));
	} else {
		tmp = exp((b * a)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if ((t_1 <= (-2d+17)) .or. (.not. (t_1 <= 5d+23))) then
        tmp = exp((-a * b))
    else
        tmp = exp((b * a)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23)) {
		tmp = Math.exp((-a * b));
	} else {
		tmp = Math.exp((b * a)) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if (t_1 <= -2e+17) or not (t_1 <= 5e+23):
		tmp = math.exp((-a * b))
	else:
		tmp = math.exp((b * a)) * x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if ((t_1 <= -2e+17) || !(t_1 <= 5e+23))
		tmp = exp(Float64(Float64(-a) * b));
	else
		tmp = Float64(exp(Float64(b * a)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if ((t_1 <= -2e+17) || ~((t_1 <= 5e+23)))
		tmp = exp((-a * b));
	else
		tmp = exp((b * a)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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[Or[LessEqual[t$95$1, -2e+17], N[Not[LessEqual[t$95$1, 5e+23]], $MachinePrecision]], N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(b * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]

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))) < -2e17 or 4.9999999999999999e23 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 45.2%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 22.3

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

   7. Applied rewrites 9.2%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 45.8%

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

   if -2e17 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 4.9999999999999999e23

   1. Initial program 89.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.2

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

   7. Applied rewrites 82.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+17} \lor \neg \left(y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;e^{b \cdot a} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+138}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+48}:\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{b \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1e+138)
   (* x (exp (* (- b) a)))
   (if (<= a 1.3e+48) (* x (exp (* (- (log z) t) y))) (/ x (exp (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1e+138) {
		tmp = x * exp((-b * a));
	} else if (a <= 1.3e+48) {
		tmp = x * exp(((log(z) - t) * y));
	} else {
		tmp = x / exp((b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1d+138)) then
        tmp = x * exp((-b * a))
    else if (a <= 1.3d+48) then
        tmp = x * exp(((log(z) - t) * y))
    else
        tmp = x / exp((b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1e+138) {
		tmp = x * Math.exp((-b * a));
	} else if (a <= 1.3e+48) {
		tmp = x * Math.exp(((Math.log(z) - t) * y));
	} else {
		tmp = x / Math.exp((b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1e+138:
		tmp = x * math.exp((-b * a))
	elif a <= 1.3e+48:
		tmp = x * math.exp(((math.log(z) - t) * y))
	else:
		tmp = x / math.exp((b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1e+138)
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	elseif (a <= 1.3e+48)
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	else
		tmp = Float64(x / exp(Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1e+138)
		tmp = x * exp((-b * a));
	elseif (a <= 1.3e+48)
		tmp = x * exp(((log(z) - t) * y));
	else
		tmp = x / exp((b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1e+138], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3e+48], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(b * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e138

   1. Initial program 93.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 91.0%

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

   if -1e138 < a < 1.29999999999999998e48

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 85.6%

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

   if 1.29999999999999998e48 < a

   1. Initial program 93.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 77.0%

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

Alternative 5: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 1.45 \cdot 10^{-5}\right):\\ \;\;\;\;e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\mathsf{E}\left(\right)}^{\left(b \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.5) (not (<= y 1.45e-5)))
   (exp (* y (- (log z) t)))
   (/ x (pow (E) (* b a)))))

Derivation

1. Split input into 2 regimes

2. if y < -2.5 or 1.45e-5 < y

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 54.3%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 27.8

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

   7. Applied rewrites 11.6%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 68.1%

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

   if -2.5 < y < 1.45e-5

   1. Initial program 95.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 95.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{e^{\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 79.8%

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

   12. Applied rewrites 79.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \lor \neg \left(y \leq 1.45 \cdot 10^{-5}\right):\\ \;\;\;\;e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\mathsf{E}\left(\right)}^{\left(b \cdot a\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\mathsf{E}\left(\right)}^{\left(b \cdot a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+200) (not (<= t 3.6e-25)))
   (* x (exp (* (- t) y)))
   (/ x (pow (E) (* b a)))))

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001e200 or 3.5999999999999999e-25 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 84.3%

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

   if -1.8500000000000001e200 < t < 3.5999999999999999e-25

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x}{e^{\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

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

   12. Applied rewrites 72.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\mathsf{E}\left(\right)}^{\left(b \cdot a\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{b \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+200) (not (<= t 3.6e-25)))
   (* x (exp (* (- t) y)))
   (/ x (exp (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 3.6e-25)) {
		tmp = x * exp((-t * y));
	} else {
		tmp = x / exp((b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-1.85d+200)) .or. (.not. (t <= 3.6d-25))) then
        tmp = x * exp((-t * y))
    else
        tmp = x / exp((b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 3.6e-25)) {
		tmp = x * Math.exp((-t * y));
	} else {
		tmp = x / Math.exp((b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.85e+200) or not (t <= 3.6e-25):
		tmp = x * math.exp((-t * y))
	else:
		tmp = x / math.exp((b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.85e+200) || !(t <= 3.6e-25))
		tmp = Float64(x * exp(Float64(Float64(-t) * y)));
	else
		tmp = Float64(x / exp(Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.85e+200) || ~((t <= 3.6e-25)))
		tmp = x * exp((-t * y));
	else
		tmp = x / exp((b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.85e+200], N[Not[LessEqual[t, 3.6e-25]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[N[(b * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001e200 or 3.5999999999999999e-25 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 84.3%

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

   if -1.8500000000000001e200 < t < 3.5999999999999999e-25

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. sinh-+-cosh-rev N/A

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

      3. flip-+ N/A

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

      4. sinh-cosh N/A

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

      5. sinh---cosh-rev N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 72.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{b \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+200) (not (<= t 3.6e-25)))
   (* x (exp (* (- t) y)))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 3.6e-25)) {
		tmp = x * exp((-t * y));
	} else {
		tmp = x * exp((-b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-1.85d+200)) .or. (.not. (t <= 3.6d-25))) then
        tmp = x * exp((-t * y))
    else
        tmp = x * exp((-b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 3.6e-25)) {
		tmp = x * Math.exp((-t * y));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.85e+200) or not (t <= 3.6e-25):
		tmp = x * math.exp((-t * y))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.85e+200) || !(t <= 3.6e-25))
		tmp = Float64(x * exp(Float64(Float64(-t) * y)));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.85e+200) || ~((t <= 3.6e-25)))
		tmp = x * exp((-t * y));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.85e+200], N[Not[LessEqual[t, 3.6e-25]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001e200 or 3.5999999999999999e-25 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 84.3%

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

   if -1.8500000000000001e200 < t < 3.5999999999999999e-25

   1. Initial program 96.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 72.6%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 3.6 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+201} \lor \neg \left(t \leq 4.7 \cdot 10^{+137}\right):\\ \;\;\;\;e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.12e+201) (not (<= t 4.7e+137)))
   (exp (* (- t) y))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.12e+201) || !(t <= 4.7e+137)) {
		tmp = exp((-t * y));
	} else {
		tmp = x * exp((-b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-1.12d+201)) .or. (.not. (t <= 4.7d+137))) then
        tmp = exp((-t * y))
    else
        tmp = x * exp((-b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.12e+201) || !(t <= 4.7e+137)) {
		tmp = Math.exp((-t * y));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.12e+201) or not (t <= 4.7e+137):
		tmp = math.exp((-t * y))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.12e+201) || !(t <= 4.7e+137))
		tmp = exp(Float64(Float64(-t) * y));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.12e+201) || ~((t <= 4.7e+137)))
		tmp = exp((-t * y));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.12e+201], N[Not[LessEqual[t, 4.7e+137]], $MachinePrecision]], N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.11999999999999994e201 or 4.6999999999999998e137 < t

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 92.3%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 46.9

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

   7. Applied rewrites 5.1%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 63.2%

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

   if -1.11999999999999994e201 < t < 4.6999999999999998e137

   1. Initial program 96.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 72.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+201} \lor \neg \left(t \leq 4.7 \cdot 10^{+137}\right):\\ \;\;\;\;e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 4.8 \cdot 10^{+125}\right):\\ \;\;\;\;e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.85e+200) (not (<= t 4.8e+125)))
   (exp (* (- t) y))
   (exp (* (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 4.8e+125)) {
		tmp = exp((-t * y));
	} else {
		tmp = exp((-a * b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-1.85d+200)) .or. (.not. (t <= 4.8d+125))) then
        tmp = exp((-t * y))
    else
        tmp = exp((-a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.85e+200) || !(t <= 4.8e+125)) {
		tmp = Math.exp((-t * y));
	} else {
		tmp = Math.exp((-a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -1.85e+200) or not (t <= 4.8e+125):
		tmp = math.exp((-t * y))
	else:
		tmp = math.exp((-a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.85e+200) || !(t <= 4.8e+125))
		tmp = exp(Float64(Float64(-t) * y));
	else
		tmp = exp(Float64(Float64(-a) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -1.85e+200) || ~((t <= 4.8e+125)))
		tmp = exp((-t * y));
	else
		tmp = exp((-a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.85e+200], N[Not[LessEqual[t, 4.8e+125]], $MachinePrecision]], N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision], N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8500000000000001e200 or 4.7999999999999999e125 < t

   1. Initial program 96.3%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 92.7%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 46.0

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

   7. Applied rewrites 6.5%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 59.8%

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

   if -1.8500000000000001e200 < t < 4.7999999999999999e125

   1. Initial program 96.6%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 41.6%

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

      1. add-exp-log N/A

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

      2. lower-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lift-exp.f64 N/A

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

      6. add-log-exp N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 22.3

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

   7. Applied rewrites 18.3%

      \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 45.0%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+200} \lor \neg \left(t \leq 4.8 \cdot 10^{+125}\right):\\ \;\;\;\;e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ e^{\left(-a\right) \cdot b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 52.4%

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

   1. add-exp-log N/A

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

   2. lower-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. log-prod N/A

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

   5. lift-exp.f64 N/A

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

   6. add-log-exp N/A

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

   7. lower-+.f64 N/A

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

   8. lower-log.f64 27.3

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

7. Applied rewrites 15.8%

   \[\leadsto \color{blue}{e^{\log x + t \cdot y}} \]

8. Taylor expanded in b around inf

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

10. Applied rewrites 38.3%

    \[\leadsto e^{\color{blue}{\left(-a\right) \cdot b}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025021 
(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))))))