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

Percentage Accurate: 96.6% → 96.6%

Time: 9.4s

Alternatives: 9

Speedup: 1.0×

• 41.3% 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: 96.6% 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: 96.6% 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]

Derivation

1. Initial program 97.7%

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

Alternative 2: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -400 \lor \neg \left(t \leq 10^{-97}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -400.0) (not (<= t 1e-97)))
   (* x (exp (fma (- b) a (* (- t) y))))
   (* x (exp (fma (- b) a (* (log z) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -400.0) || !(t <= 1e-97)) {
		tmp = x * exp(fma(-b, a, (-t * y)));
	} else {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -400.0) || !(t <= 1e-97))
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(Float64(-t) * y))));
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -400.0], N[Not[LessEqual[t, 1e-97]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -400 or 1.00000000000000004e-97 < t

   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. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 96.8%

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

   if -400 < t < 1.00000000000000004e-97

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, y \cdot \log z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -400 \lor \neg \left(t \leq 10^{-97}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -106000000000 \lor \neg \left(y \leq 3.65 \cdot 10^{-16}\right):\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -106000000000.0) (not (<= y 3.65e-16)))
   (* (exp (* (- (log z) t) y)) x)
   (* x (exp (fma (- b) a (* (- t) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -106000000000.0) || !(y <= 3.65e-16)) {
		tmp = exp(((log(z) - t) * y)) * x;
	} else {
		tmp = x * exp(fma(-b, a, (-t * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -106000000000.0) || !(y <= 3.65e-16))
		tmp = Float64(exp(Float64(Float64(log(z) - t) * y)) * x);
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(Float64(-t) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -106000000000.0], N[Not[LessEqual[y, 3.65e-16]], $MachinePrecision]], N[(N[Exp[N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.06e11 or 3.6500000000000001e-16 < y

   1. Initial program 97.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\frac{{\log z}^{2}}{t + \log z} - \frac{{t}^{2}}{t + \log z}\right)}} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{{\log z}^{2} - {t}^{2}}{t + \log z}}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot \left({\log z}^{2} - {t}^{2}\right)}{t + \log z}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot \left({\log z}^{2} - {t}^{2}\right)}{t + \log z}}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} - {t}^{2}\right) \cdot y}}{t + \log z}} \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} - {t}^{2}\right) \cdot y}}{t + \log z}} \]

      6. unpow2 N/A

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} - \color{blue}{t \cdot t}\right) \cdot y}{t + \log z}} \]

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

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} + \left(\mathsf{neg}\left(t\right)\right) \cdot t\right)} \cdot y}{t + \log z}} \]

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

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \color{blue}{\left(\mathsf{neg}\left(t \cdot t\right)\right)}\right) \cdot y}{t + \log z}} \]

      9. unpow2 N/A

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \left(\mathsf{neg}\left(\color{blue}{{t}^{2}}\right)\right)\right) \cdot y}{t + \log z}} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \color{blue}{-1 \cdot {t}^{2}}\right) \cdot y}{t + \log z}} \]

      11. +-commutative N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(-1 \cdot {t}^{2} + {\log z}^{2}\right)} \cdot y}{t + \log z}} \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)} + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      13. unpow2 N/A

          \[\leadsto x \cdot e^{\frac{\left(\left(\mathsf{neg}\left(\color{blue}{t \cdot t}\right)\right) + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

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

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot t} + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      15. lower-fma.f64 N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(t\right), t, {\log z}^{2}\right)} \cdot y}{t + \log z}} \]

      16. lower-neg.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(\color{blue}{-t}, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      17. lower-pow.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(-t, t, \color{blue}{{\log z}^{2}}\right) \cdot y}{t + \log z}} \]

      18. lower-log.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(-t, t, {\color{blue}{\log z}}^{2}\right) \cdot y}{t + \log z}} \]

   7. Applied rewrites 93.9%

      \[\leadsto x \cdot e^{\color{blue}{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \cdot x} \]

      3. lower-*.f64 93.9

         \[\leadsto \color{blue}{e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \cdot x} \]

   9. Applied rewrites 93.9%

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

   if -1.06e11 < y < 3.6500000000000001e-16

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 95.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -106000000000 \lor \neg \left(y \leq 3.65 \cdot 10^{-16}\right):\\ \;\;\;\;e^{\left(\log z - t\right) \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -43 \lor \neg \left(t \leq -3.55 \cdot 10^{-202}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -43.0) (not (<= t -3.55e-202)))
   (* x (exp (fma (- b) a (* (- t) y))))
   (* (exp (* (log z) y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -43.0) || !(t <= -3.55e-202)) {
		tmp = x * exp(fma(-b, a, (-t * y)));
	} else {
		tmp = exp((log(z) * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -43.0) || !(t <= -3.55e-202))
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(Float64(-t) * y))));
	else
		tmp = Float64(exp(Float64(log(z) * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -43.0], N[Not[LessEqual[t, -3.55e-202]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -43 or -3.55e-202 < t

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 92.3%

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

   if -43 < t < -3.55e-202

   1. Initial program 95.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto x \cdot e^{\color{blue}{y \cdot \left(\frac{{\log z}^{2}}{t + \log z} - \frac{{t}^{2}}{t + \log z}\right)}} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{{\log z}^{2} - {t}^{2}}{t + \log z}}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot \left({\log z}^{2} - {t}^{2}\right)}{t + \log z}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot \left({\log z}^{2} - {t}^{2}\right)}{t + \log z}}} \]

      4. *-commutative N/A

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} - {t}^{2}\right) \cdot y}}{t + \log z}} \]

      5. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} - {t}^{2}\right) \cdot y}}{t + \log z}} \]

      6. unpow2 N/A

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} - \color{blue}{t \cdot t}\right) \cdot y}{t + \log z}} \]

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

         \[\leadsto x \cdot e^{\frac{\color{blue}{\left({\log z}^{2} + \left(\mathsf{neg}\left(t\right)\right) \cdot t\right)} \cdot y}{t + \log z}} \]

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

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \color{blue}{\left(\mathsf{neg}\left(t \cdot t\right)\right)}\right) \cdot y}{t + \log z}} \]

      9. unpow2 N/A

         \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \left(\mathsf{neg}\left(\color{blue}{{t}^{2}}\right)\right)\right) \cdot y}{t + \log z}} \]

      10. mul-1-neg N/A

          \[\leadsto x \cdot e^{\frac{\left({\log z}^{2} + \color{blue}{-1 \cdot {t}^{2}}\right) \cdot y}{t + \log z}} \]

      11. +-commutative N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(-1 \cdot {t}^{2} + {\log z}^{2}\right)} \cdot y}{t + \log z}} \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\left(\mathsf{neg}\left({t}^{2}\right)\right)} + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      13. unpow2 N/A

          \[\leadsto x \cdot e^{\frac{\left(\left(\mathsf{neg}\left(\color{blue}{t \cdot t}\right)\right) + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

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

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot t} + {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      15. lower-fma.f64 N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(t\right), t, {\log z}^{2}\right)} \cdot y}{t + \log z}} \]

      16. lower-neg.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(\color{blue}{-t}, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \]

      17. lower-pow.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(-t, t, \color{blue}{{\log z}^{2}}\right) \cdot y}{t + \log z}} \]

      18. lower-log.f64 N/A

          \[\leadsto x \cdot e^{\frac{\mathsf{fma}\left(-t, t, {\color{blue}{\log z}}^{2}\right) \cdot y}{t + \log z}} \]

   7. Applied rewrites 71.0%

      \[\leadsto x \cdot e^{\color{blue}{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \cdot x} \]

      3. lower-*.f64 71.0

         \[\leadsto \color{blue}{e^{\frac{\mathsf{fma}\left(-t, t, {\log z}^{2}\right) \cdot y}{t + \log z}} \cdot x} \]

   9. Applied rewrites 71.0%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 71.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -43 \lor \neg \left(t \leq -3.55 \cdot 10^{-202}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 97.0

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

5. Applied rewrites 97.0%

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

Alternative 6: 72.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+48} \lor \neg \left(y \leq 1.4 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.75e+48) (not (<= y 1.4e-12)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- (- z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.75e+48) || !(y <= 1.4e-12)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp(((-z - 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 ((y <= (-1.75d+48)) .or. (.not. (y <= 1.4d-12))) then
        tmp = x * exp((-y * t))
    else
        tmp = x * exp(((-z - 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 ((y <= -1.75e+48) || !(y <= 1.4e-12)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp(((-z - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.75e+48) or not (y <= 1.4e-12):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp(((-z - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.75e+48) || !(y <= 1.4e-12))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(-z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.75e+48) || ~((y <= 1.4e-12)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp(((-z - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.75e+48], N[Not[LessEqual[y, 1.4e-12]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7499999999999999e48 or 1.4000000000000001e-12 < 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 t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -1.7499999999999999e48 < y < 1.4000000000000001e-12

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub N/A

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

      7. mul-1-neg N/A

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

      8. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      9. lower-neg.f64 82.4

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 82.4%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+48} \lor \neg \left(y \leq 1.4 \cdot 10^{-12}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-z\right) - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-79} \lor \neg \left(b \leq 1.2 \cdot 10^{+145}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.2e-79) (not (<= b 1.2e+145)))
   (* x (exp (* (- b) a)))
   (* x (exp (* (- y) t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e-79) || !(b <= 1.2e+145)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x * exp((-y * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-5.2d-79)) .or. (.not. (b <= 1.2d+145))) then
        tmp = x * exp((-b * a))
    else
        tmp = x * exp((-y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.2e-79) || !(b <= 1.2e+145)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x * Math.exp((-y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -5.2e-79) or not (b <= 1.2e+145):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x * math.exp((-y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.2e-79) || !(b <= 1.2e+145))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.2e-79) || ~((b <= 1.2e+145)))
		tmp = x * exp((-b * a));
	else
		tmp = x * exp((-y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -5.2e-79], N[Not[LessEqual[b, 1.2e+145]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.19999999999999987e-79 or 1.19999999999999996e145 < b

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub N/A

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

      7. mul-1-neg N/A

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

      8. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      9. lower-neg.f64 78.9

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 78.9%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.2%

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

   if -5.19999999999999987e-79 < b < 1.19999999999999996e145

   1. Initial program 96.5%

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

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-79} \lor \neg \left(b \leq 1.2 \cdot 10^{+145}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 97.0

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

5. Applied rewrites 97.0%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 83.8%

   \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)} \]
9. Add Preprocessing

Alternative 9: 58.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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((-b * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-lft-identity N/A

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

   5. metadata-eval N/A

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

   6. cancel-sign-sub N/A

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

   7. mul-1-neg N/A

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

   8. lower-log1p.f64 N/A

      \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

   9. lower-neg.f64 59.9

      \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

5. Applied rewrites 59.9%

   \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

6. Taylor expanded in z around 0

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

8. Applied rewrites 54.7%

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

Reproduce

herbie shell --seed 2025017 
(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))))))