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

Percentage Accurate: 96.3% → 99.2%

Time: 9.1s

Alternatives: 9

Speedup: 1.4×

• 41.7% 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: 99.2% accurate, 1.4× speedup

Math

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

FPCore

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

Python

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

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (-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[((-z) - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 37.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;t\_1 \leq -1000000 \lor \neg \left(t\_1 \leq 10^{-6}\right):\\ \;\;\;\;\left(-t\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -1e+189)
     (/ x (* y t))
     (if (or (<= t_1 -1000000.0) (not (<= t_1 1e-6))) (* (- t) (* x y)) 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 <= -1e+189) {
		tmp = x / (y * t);
	} else if ((t_1 <= -1000000.0) || !(t_1 <= 1e-6)) {
		tmp = -t * (x * y);
	} else {
		tmp = 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 <= (-1d+189)) then
        tmp = x / (y * t)
    else if ((t_1 <= (-1000000.0d0)) .or. (.not. (t_1 <= 1d-6))) then
        tmp = -t * (x * y)
    else
        tmp = 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 <= -1e+189) {
		tmp = x / (y * t);
	} else if ((t_1 <= -1000000.0) || !(t_1 <= 1e-6)) {
		tmp = -t * (x * y);
	} else {
		tmp = 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 <= -1e+189:
		tmp = x / (y * t)
	elif (t_1 <= -1000000.0) or not (t_1 <= 1e-6):
		tmp = -t * (x * y)
	else:
		tmp = 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 <= -1e+189)
		tmp = Float64(x / Float64(y * t));
	elseif ((t_1 <= -1000000.0) || !(t_1 <= 1e-6))
		tmp = Float64(Float64(-t) * Float64(x * y));
	else
		tmp = 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 <= -1e+189)
		tmp = x / (y * t);
	elseif ((t_1 <= -1000000.0) || ~((t_1 <= 1e-6)))
		tmp = -t * (x * y);
	else
		tmp = 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[LessEqual[t$95$1, -1e+189], N[(x / N[(y * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -1000000.0], N[Not[LessEqual[t$95$1, 1e-6]], $MachinePrecision]], N[((-t) * N[(x * y), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e189

   1. Initial program 96.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \left(\mathsf{neg}\left(y \cdot \left(\log z - t\right)\right)\right)} \]

      2. distribute-lft-neg-out N/A

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

      3. +-commutative N/A

         \[\leadsto \frac{x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\log z - t\right) + 1} \]

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 38.7

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

   10. Applied rewrites 38.7%

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

   11. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 30.1

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

   13. Applied rewrites 30.1%

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

   if -1e189 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e6 or 9.99999999999999955e-7 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 66.6

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

   7. Applied rewrites 66.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 21.8

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

   10. Applied rewrites 21.8%

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

   11. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \left(x \cdot y\right) \]

       3. lift-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 21.9

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

   13. Applied rewrites 21.9%

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

   if -1e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999955e-7

   1. Initial program 86.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 84.0

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 80.1%

       \[\leadsto x \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1 \cdot 10^{+189}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -1000000 \lor \neg \left(y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{-6}\right):\\ \;\;\;\;\left(-t\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 24.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{if}\;t\_1 \leq 0 \lor \neg \left(t\_1 \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\left(-t\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))))
   (if (or (<= t_1 0.0) (not (<= t_1 5e+300))) (* (- t) (* x y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 5e+300)) {
		tmp = -t * (x * y);
	} else {
		tmp = 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 = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
    if ((t_1 <= 0.0d0) .or. (.not. (t_1 <= 5d+300))) then
        tmp = -t * (x * y)
    else
        tmp = 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 = x * Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 5e+300)) {
		tmp = -t * (x * y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))
	tmp = 0
	if (t_1 <= 0.0) or not (t_1 <= 5e+300):
		tmp = -t * (x * y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
	tmp = 0.0
	if ((t_1 <= 0.0) || !(t_1 <= 5e+300))
		tmp = Float64(Float64(-t) * Float64(x * y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	tmp = 0.0;
	if ((t_1 <= 0.0) || ~((t_1 <= 5e+300)))
		tmp = -t * (x * y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 5e+300]], $MachinePrecision]], N[((-t) * N[(x * y), $MachinePrecision]), $MachinePrecision], x]]

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 or 5.00000000000000026e300 < (*.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 95.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 21.1

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

   10. Applied rewrites 21.1%

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

   11. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \left(x \cdot y\right) \]

       3. lift-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 18.5

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

   13. Applied rewrites 18.5%

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

   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))))) < 5.00000000000000026e300

   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. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \]
   9. Step-by-step derivation

   10. Applied rewrites 93.1%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0 \lor \neg \left(x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\left(-t\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+41} \lor \neg \left(y \leq 0.072\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\left(-z\right) - b, a, \left(-t\right) \cdot y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2e+41) (not (<= y 0.072)))
   (* x (exp (* (- (log z) t) y)))
   (* (exp (fma (- (- z) b) a (* (- t) y))) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2e+41) || !(y <= 0.072)) {
		tmp = x * exp(((log(z) - t) * y));
	} else {
		tmp = exp(fma((-z - b), a, (-t * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2e+41) || !(y <= 0.072))
		tmp = Float64(x * exp(Float64(Float64(log(z) - t) * y)));
	else
		tmp = Float64(exp(fma(Float64(Float64(-z) - b), a, Float64(Float64(-t) * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2e+41], N[Not[LessEqual[y, 0.072]], $MachinePrecision]], N[(x * N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000001e41 or 0.0719999999999999946 < y

   1. Initial program 98.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.00000000000000001e41 < y < 0.0719999999999999946

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   8. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.9

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

   10. Applied rewrites 98.9%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+41} \lor \neg \left(y \leq 0.072\right):\\ \;\;\;\;x \cdot e^{\left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\left(-z\right) - b, a, \left(-t\right) \cdot y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+229}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\left(-z\right) - b, a, \left(-t\right) \cdot y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2e+229)
   (* x (pow z y))
   (* (exp (fma (- (- z) b) a (* (- t) y))) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2e+229) {
		tmp = x * pow(z, y);
	} else {
		tmp = exp(fma((-z - b), a, (-t * y))) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2e+229)
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(exp(fma(Float64(Float64(-z) - b), a, Float64(Float64(-t) * y))) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2e+229], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e229

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-pow.f64 90.2

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

   8. Applied rewrites 90.2%

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

   if -2e229 < y

   1. Initial program 94.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.6

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

   8. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.6

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

   10. Applied rewrites 89.6%

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

Alternative 6: 71.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+90} \lor \neg \left(t \leq 1.75 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.3e+90) (not (<= t 1.75e-30)))
   (* x (exp (* (- t) y)))
   (* x (exp (* (- a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.3e+90) || !(t <= 1.75e-30)) {
		tmp = x * exp((-t * y));
	} else {
		tmp = x * 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 <= (-2.3d+90)) .or. (.not. (t <= 1.75d-30))) then
        tmp = x * exp((-t * y))
    else
        tmp = x * 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 <= -2.3e+90) || !(t <= 1.75e-30)) {
		tmp = x * Math.exp((-t * y));
	} else {
		tmp = x * Math.exp((-a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2.3e+90) or not (t <= 1.75e-30):
		tmp = x * math.exp((-t * y))
	else:
		tmp = x * math.exp((-a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.3e+90) || !(t <= 1.75e-30))
		tmp = Float64(x * exp(Float64(Float64(-t) * y)));
	else
		tmp = Float64(x * 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 <= -2.3e+90) || ~((t <= 1.75e-30)))
		tmp = x * exp((-t * y));
	else
		tmp = x * exp((-a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.3e+90], N[Not[LessEqual[t, 1.75e-30]], $MachinePrecision]], N[(x * N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3e90 or 1.7500000000000001e-30 < t

   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 t around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -2.3e90 < t < 1.7500000000000001e-30

   1. Initial program 93.8%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.5

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

   5. Applied rewrites 67.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+90} \lor \neg \left(t \leq 1.75 \cdot 10^{-30}\right):\\ \;\;\;\;x \cdot e^{\left(-t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+18} \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.85e+18) (not (<= y 0.17)))
   (* x (pow z y))
   (* x (exp (* (- a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.85e+18) || !(y <= 0.17)) {
		tmp = x * pow(z, y);
	} else {
		tmp = x * 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 ((y <= (-1.85d+18)) .or. (.not. (y <= 0.17d0))) then
        tmp = x * (z ** y)
    else
        tmp = x * 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 ((y <= -1.85e+18) || !(y <= 0.17)) {
		tmp = x * Math.pow(z, y);
	} else {
		tmp = x * Math.exp((-a * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.85e+18) or not (y <= 0.17):
		tmp = x * math.pow(z, y)
	else:
		tmp = x * math.exp((-a * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.85e+18) || !(y <= 0.17))
		tmp = Float64(x * (z ^ y));
	else
		tmp = Float64(x * exp(Float64(Float64(-a) * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.85e+18) || ~((y <= 0.17)))
		tmp = x * (z ^ y);
	else
		tmp = x * exp((-a * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.85e+18], N[Not[LessEqual[y, 0.17]], $MachinePrecision]], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-a) * b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e18 or 0.170000000000000012 < y

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 91.3

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-pow.f64 65.4

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

   8. Applied rewrites 65.4%

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

   if -1.85e18 < y < 0.170000000000000012

   1. Initial program 92.7%

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+18} \lor \neg \left(y \leq 0.17\right):\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+148}:\\ \;\;\;\;\left(-t\right) \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -9.5e+148) (* (- t) (* x y)) (* x (pow z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9.5e+148) {
		tmp = -t * (x * y);
	} 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 (t <= (-9.5d+148)) then
        tmp = -t * (x * y)
    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 (t <= -9.5e+148) {
		tmp = -t * (x * y);
	} else {
		tmp = x * Math.pow(z, y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -9.5e+148:
		tmp = -t * (x * y)
	else:
		tmp = x * math.pow(z, y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -9.5e+148)
		tmp = Float64(Float64(-t) * Float64(x * y));
	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 (t <= -9.5e+148)
		tmp = -t * (x * y);
	else
		tmp = x * (z ^ y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -9.5e+148], N[((-t) * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000002e148

   1. Initial program 97.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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. 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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

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

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

      12. exp-diff N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-exp.f64 87.1

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 28.5

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

   10. Applied rewrites 28.5%

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

   11. Taylor expanded in t around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(t\right)\right) \cdot \left(x \cdot y\right) \]

       3. lift-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 31.2

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

   13. Applied rewrites 31.2%

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

   if -9.5000000000000002e148 < t

   1. Initial program 94.7%

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

   3. Taylor expanded in a around 0

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in t around 0

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

      1. lower-pow.f64 52.6

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

   8. Applied rewrites 52.6%

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

Alternative 9: 18.8% accurate, 328.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 95.1%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
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. 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. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

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

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

   12. exp-diff N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 75.6%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. distribute-lft-neg-out N/A

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

   4. lift-log.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-exp.f64 69.4

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

7. Applied rewrites 69.4%

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

8. Taylor expanded in y around 0

   \[\leadsto x \]
9. Step-by-step derivation

10. Applied rewrites 14.2%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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))))))