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

Percentage Accurate: 96.8% → 96.8%

Time: 14.0s

Alternatives: 14

Speedup: 1.0×

• 41.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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: 96.8% 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

real(8) function code(x, y, z, t, a, b)
    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: 96.8% 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

real(8) function code(x, y, z, t, a, b)
    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 96.1%

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

Alternative 2: 33.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \log z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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 -2e+36)
     (- (* y (* x t)))
     (if (<= t_1 2e+187)
       (fma (* x y) (- t) x)
       (if (<= t_1 2e+304) (fma (* x y) (log z) x) (* x (* y (- t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+36) {
		tmp = -(y * (x * t));
	} else if (t_1 <= 2e+187) {
		tmp = fma((x * y), -t, x);
	} else if (t_1 <= 2e+304) {
		tmp = fma((x * y), log(z), x);
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+36)
		tmp = Float64(-Float64(y * Float64(x * t)));
	elseif (t_1 <= 2e+187)
		tmp = fma(Float64(x * y), Float64(-t), x);
	elseif (t_1 <= 2e+304)
		tmp = fma(Float64(x * y), log(z), x);
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return 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, -2e+36], (-N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 2e+187], N[(N[(x * y), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(N[(x * y), $MachinePrecision] * N[Log[z], $MachinePrecision] + x), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.00000000000000008e36

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.9%

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

   if -2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.99999999999999981e187

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 52.9%

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

   if 1.99999999999999981e187 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999999e304

   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 y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 25.6%

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

   if 1.9999999999999999e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 39.9%

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

   13. Applied rewrites 59.5%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \log z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x \cdot \left(y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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 -2e+36)
     (- (* y (* x t)))
     (if (<= t_1 2e+254)
       (fma (* x y) (- t) x)
       (if (<= t_1 2e+304) (* x (* y (log z))) (* x (* y (- t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+36) {
		tmp = -(y * (x * t));
	} else if (t_1 <= 2e+254) {
		tmp = fma((x * y), -t, x);
	} else if (t_1 <= 2e+304) {
		tmp = x * (y * log(z));
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+36)
		tmp = Float64(-Float64(y * Float64(x * t)));
	elseif (t_1 <= 2e+254)
		tmp = fma(Float64(x * y), Float64(-t), x);
	elseif (t_1 <= 2e+304)
		tmp = Float64(x * Float64(y * log(z)));
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return 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, -2e+36], (-N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 2e+254], N[(N[(x * y), $MachinePrecision] * (-t) + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(x * N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.00000000000000008e36

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.9%

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

   if -2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999999e254

   1. Initial program 94.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 45.8%

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

   if 1.9999999999999999e254 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999999e304

   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 y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 45.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 45.7%

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

   12. Taylor expanded in t around 0

       \[\leadsto x \cdot \left(y \cdot \log z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 44.3%

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

   if 1.9999999999999999e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 39.9%

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

   13. Applied rewrites 59.5%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x \cdot \left(y \cdot \log z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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 -2e+36)
     (- (* y (* x t)))
     (if (<= t_1 2e+304) (fma (* x y) (- t) x) (* x (* y (- t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+36) {
		tmp = -(y * (x * t));
	} else if (t_1 <= 2e+304) {
		tmp = fma((x * y), -t, x);
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+36)
		tmp = Float64(-Float64(y * Float64(x * t)));
	elseif (t_1 <= 2e+304)
		tmp = fma(Float64(x * y), Float64(-t), x);
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return 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, -2e+36], (-N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 2e+304], N[(N[(x * y), $MachinePrecision] * (-t) + x), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]]

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))) < -2.00000000000000008e36

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.9%

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

   if -2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.9999999999999999e304

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 42.0%

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

   if 1.9999999999999999e304 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 39.9%

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

   13. Applied rewrites 59.5%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(-x \cdot t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

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 -2e+36)
     (- (* y (* x t)))
     (if (<= t_1 1e+282) (fma (- (* x t)) y x) (* x (* y (- t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+36) {
		tmp = -(y * (x * t));
	} else if (t_1 <= 1e+282) {
		tmp = fma(-(x * t), y, x);
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+36)
		tmp = Float64(-Float64(y * Float64(x * t)));
	elseif (t_1 <= 1e+282)
		tmp = fma(Float64(-Float64(x * t)), y, x);
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return 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, -2e+36], (-N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 1e+282], N[((-N[(x * t), $MachinePrecision]) * y + x), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]]

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))) < -2.00000000000000008e36

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.9%

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

   if -2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1.00000000000000003e282

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.1%

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

   10. Applied rewrites 44.0%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 39.9%

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

   if 1.00000000000000003e282 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 33.3%

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

   13. Applied rewrites 49.6%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+36}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(-x \cdot t, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 19.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+303}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) 1e+303)
   (* t (* y (- x)))
   (* x (* y (- t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= 1e+303) {
		tmp = t * (y * -x);
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= 1d+303) then
        tmp = t * (y * -x)
    else
        tmp = x * (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 (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= 1e+303) {
		tmp = t * (y * -x);
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= 1e+303:
		tmp = t * (y * -x)
	else:
		tmp = x * (y * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= 1e+303)
		tmp = Float64(t * Float64(y * Float64(-x)));
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= 1e+303)
		tmp = t * (y * -x);
	else
		tmp = x * (y * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 1e+303], N[(t * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 1e303

   1. Initial program 96.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 29.2%

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

   10. Applied rewrites 28.0%

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

   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

   13. Applied rewrites 15.1%

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

   if 1e303 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 42.1%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 38.8%

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

   13. Applied rewrites 57.9%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+303}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 19.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+147}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -2e+147)
   (- (* y (* x t)))
   (* x (* y (- t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -2e+147) {
		tmp = -(y * (x * t));
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))) <= (-2d+147)) then
        tmp = -(y * (x * t))
    else
        tmp = x * (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 (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -2e+147) {
		tmp = -(y * (x * t));
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))) <= -2e+147:
		tmp = -(y * (x * t))
	else:
		tmp = x * (y * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))) <= -2e+147)
		tmp = Float64(-Float64(y * Float64(x * t)));
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))) <= -2e+147)
		tmp = -(y * (x * t));
	else
		tmp = x * (y * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], -2e+147], (-N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]), N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e147

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.2%

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

   if -2e147 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 17.1%

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

   13. Applied rewrites 21.7%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+147}:\\ \;\;\;\;-y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -1.55e-19)
     t_1
     (if (<= y 8.4e-13) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -1.55e-19) {
		tmp = t_1;
	} else if (y <= 8.4e-13) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp((y * (log(z) - t)))
    if (y <= (-1.55d-19)) then
        tmp = t_1
    else if (y <= 8.4d-13) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * (Math.log(z) - t)));
	double tmp;
	if (y <= -1.55e-19) {
		tmp = t_1;
	} else if (y <= 8.4e-13) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * (math.log(z) - t)))
	tmp = 0
	if y <= -1.55e-19:
		tmp = t_1
	elif y <= 8.4e-13:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -1.55e-19)
		tmp = t_1;
	elseif (y <= 8.4e-13)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * (log(z) - t)));
	tmp = 0.0;
	if (y <= -1.55e-19)
		tmp = t_1;
	elseif (y <= 8.4e-13)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e-19], t$95$1, If[LessEqual[y, 8.4e-13], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5499999999999999e-19 or 8.39999999999999955e-13 < y

   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 y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if -1.5499999999999999e-19 < y < 8.39999999999999955e-13

   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. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-19}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -246:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-38}:\\ \;\;\;\;x \cdot e^{y \cdot \log z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -246.0) t_1 (if (<= t 1.65e-38) (* x (exp (* y (log z)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (t <= -246.0) {
		tmp = t_1;
	} else if (t <= 1.65e-38) {
		tmp = x * exp((y * log(z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 * -t))
    if (t <= (-246.0d0)) then
        tmp = t_1
    else if (t <= 1.65d-38) then
        tmp = x * exp((y * log(z)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -246.0) {
		tmp = t_1;
	} else if (t <= 1.65e-38) {
		tmp = x * Math.exp((y * Math.log(z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -246.0:
		tmp = t_1
	elif t <= 1.65e-38:
		tmp = x * math.exp((y * math.log(z)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -246.0)
		tmp = t_1;
	elseif (t <= 1.65e-38)
		tmp = Float64(x * exp(Float64(y * log(z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -246.0)
		tmp = t_1;
	elseif (t <= 1.65e-38)
		tmp = x * exp((y * log(z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -246.0], t$95$1, If[LessEqual[t, 1.65e-38], N[(x * N[Exp[N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -246 or 1.6500000000000001e-38 < t

   1. Initial program 95.3%

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -246 < t < 1.6500000000000001e-38

   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. 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.2%

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

Alternative 10: 74.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -2300:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -2300.0)
     t_1
     (if (<= t 2.2e+33) (* x (exp (- (* a (+ z b))))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (t <= -2300.0) {
		tmp = t_1;
	} else if (t <= 2.2e+33) {
		tmp = x * exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 * -t))
    if (t <= (-2300.0d0)) then
        tmp = t_1
    else if (t <= 2.2d+33) then
        tmp = x * exp(-(a * (z + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -2300.0) {
		tmp = t_1;
	} else if (t <= 2.2e+33) {
		tmp = x * Math.exp(-(a * (z + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -2300.0:
		tmp = t_1
	elif t <= 2.2e+33:
		tmp = x * math.exp(-(a * (z + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -2300.0)
		tmp = t_1;
	elseif (t <= 2.2e+33)
		tmp = Float64(x * exp(Float64(-Float64(a * Float64(z + b)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -2300.0)
		tmp = t_1;
	elseif (t <= 2.2e+33)
		tmp = x * exp(-(a * (z + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2300.0], t$95$1, If[LessEqual[t, 2.2e+33], N[(x * N[Exp[(-N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2300 or 2.19999999999999994e33 < t

   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 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-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -2300 < t < 2.19999999999999994e33

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2300:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -530:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+30}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- t))))))
   (if (<= t -530.0) t_1 (if (<= t 1.4e+30) (* x (exp (- (* a b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * -t));
	double tmp;
	if (t <= -530.0) {
		tmp = t_1;
	} else if (t <= 1.4e+30) {
		tmp = x * exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 * -t))
    if (t <= (-530.0d0)) then
        tmp = t_1
    else if (t <= 1.4d+30) then
        tmp = x * exp(-(a * b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp((y * -t));
	double tmp;
	if (t <= -530.0) {
		tmp = t_1;
	} else if (t <= 1.4e+30) {
		tmp = x * Math.exp(-(a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((y * -t))
	tmp = 0
	if t <= -530.0:
		tmp = t_1
	elif t <= 1.4e+30:
		tmp = x * math.exp(-(a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(-t))))
	tmp = 0.0
	if (t <= -530.0)
		tmp = t_1;
	elseif (t <= 1.4e+30)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((y * -t));
	tmp = 0.0;
	if (t <= -530.0)
		tmp = t_1;
	elseif (t <= 1.4e+30)
		tmp = x * exp(-(a * b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -530.0], t$95$1, If[LessEqual[t, 1.4e+30], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -530 or 1.39999999999999992e30 < t

   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 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-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -530 < t < 1.39999999999999992e30

   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 b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

Alternative 12: 36.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(y \cdot \log z\right)\\ \mathbf{elif}\;y \leq 0.34:\\ \;\;\;\;x \cdot e^{-z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.65e+121)
   (* x (* y (log z)))
   (if (<= y 0.34) (* x (exp (- (* z a)))) (* x (* y (- t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.65e+121) {
		tmp = x * (y * log(z));
	} else if (y <= 0.34) {
		tmp = x * exp(-(z * a));
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 <= (-1.65d+121)) then
        tmp = x * (y * log(z))
    else if (y <= 0.34d0) then
        tmp = x * exp(-(z * a))
    else
        tmp = x * (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 (y <= -1.65e+121) {
		tmp = x * (y * Math.log(z));
	} else if (y <= 0.34) {
		tmp = x * Math.exp(-(z * a));
	} else {
		tmp = x * (y * -t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -1.65e+121:
		tmp = x * (y * math.log(z))
	elif y <= 0.34:
		tmp = x * math.exp(-(z * a))
	else:
		tmp = x * (y * -t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.65e+121)
		tmp = Float64(x * Float64(y * log(z)));
	elseif (y <= 0.34)
		tmp = Float64(x * exp(Float64(-Float64(z * a))));
	else
		tmp = Float64(x * Float64(y * Float64(-t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -1.65e+121)
		tmp = x * (y * log(z));
	elseif (y <= 0.34)
		tmp = x * exp(-(z * a));
	else
		tmp = x * (y * -t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.65e+121], N[(x * N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.34], N[(x * N[Exp[(-N[(z * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(x * N[(y * (-t)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999e121

   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 y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 33.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 35.1%

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

   12. Taylor expanded in t around 0

       \[\leadsto x \cdot \left(y \cdot \log z\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 25.3%

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

   if -1.6499999999999999e121 < y < 0.340000000000000024

   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 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.1%

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

   if 0.340000000000000024 < 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 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 15.8%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 27.6%

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

   13. Applied rewrites 28.9%

       \[\leadsto \left(\left(-t\right) \cdot y\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+121}:\\ \;\;\;\;x \cdot \left(y \cdot \log z\right)\\ \mathbf{elif}\;y \leq 0.34:\\ \;\;\;\;x \cdot e^{-z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(-t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z 4.2e-49) (* x (exp (- (* a b)))) (* x (exp (- (* z a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= 4.2e-49) {
		tmp = x * exp(-(a * b));
	} else {
		tmp = x * exp(-(z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 (z <= 4.2d-49) then
        tmp = x * exp(-(a * b))
    else
        tmp = x * exp(-(z * 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 (z <= 4.2e-49) {
		tmp = x * Math.exp(-(a * b));
	} else {
		tmp = x * Math.exp(-(z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= 4.2e-49:
		tmp = x * math.exp(-(a * b))
	else:
		tmp = x * math.exp(-(z * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= 4.2e-49)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	else
		tmp = Float64(x * exp(Float64(-Float64(z * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= 4.2e-49)
		tmp = x * exp(-(a * b));
	else
		tmp = x * exp(-(z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, 4.2e-49], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-N[(z * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.1999999999999998e-49

   1. Initial program 97.3%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if 4.1999999999999998e-49 < z

   1. Initial program 88.0%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 56.8%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 18.3% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ -y \cdot \left(x \cdot t\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = -(y * (x * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 58.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 31.0%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 17.4%

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

12. Final simplification 17.4%

    \[\leadsto -y \cdot \left(x \cdot t\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))))