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

Percentage Accurate: 96.4% → 96.4%

Time: 16.0s

Alternatives: 20

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: 96.4% 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: 96.4% 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: 50.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(-0.5, b \cdot \left(a \cdot a\right), a\right) \cdot \left(-x\right), x\right)\\ \mathbf{elif}\;t\_2 \leq 5.5 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* 0.5 (* a (* a (* b b))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0)
     t_1
     (if (<= t_2 1e+58)
       (fma b (* (fma -0.5 (* b (* a a)) a) (- x)) x)
       (if (<= t_2 5.5e+250)
         t_1
         (* x (fma y (fma 0.5 (* y (* t t)) (- t)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (0.5 * (a * (a * (b * b))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+58) {
		tmp = fma(b, (fma(-0.5, (b * (a * a)), a) * -x), x);
	} else if (t_2 <= 5.5e+250) {
		tmp = t_1;
	} else {
		tmp = x * fma(y, fma(0.5, (y * (t * t)), -t), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+58)
		tmp = fma(b, Float64(fma(-0.5, Float64(b * Float64(a * a)), a) * Float64(-x)), x);
	elseif (t_2 <= 5.5e+250)
		tmp = t_1;
	else
		tmp = Float64(x * fma(y, fma(0.5, Float64(y * Float64(t * t)), Float64(-t)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 1e+58], N[(b * N[(N[(-0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] * (-x)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 5.5e+250], t$95$1, N[(x * N[(y * N[(0.5 * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $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))) < -2e3 or 9.99999999999999944e57 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.4999999999999999e250

   1. Initial program 98.1%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 21.3

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 21.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 21.4

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 21.4%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 46.7

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 46.7%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999944e57

   1. Initial program 94.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x + b \cdot \left(-1 \cdot \left(a \cdot x\right) + \frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, -1 \cdot \left(a \cdot x\right) + \frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right), x\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + -1 \cdot \left(a \cdot x\right)}, x\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)}, x\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) - a \cdot x}, x\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right) - a \cdot x}, x\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot x\right)} - a \cdot x, x\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot x\right)} - a \cdot x, x\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right)} \cdot \left(b \cdot x\right) - a \cdot x, x\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(b \cdot x\right) - a \cdot x, x\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot \left(b \cdot x\right) - a \cdot x, x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(x \cdot b\right)} - a \cdot x, x\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(x \cdot b\right)} - a \cdot x, x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \left(\frac{1}{2} \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot b\right) - \color{blue}{x \cdot a}, x\right) \]

      15. *-lowering-*.f64 70.2

          \[\leadsto \mathsf{fma}\left(b, \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot b\right) - \color{blue}{x \cdot a}, x\right) \]

   8. Simplified 70.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \left(0.5 \cdot \left(a \cdot a\right)\right) \cdot \left(x \cdot b\right) - x \cdot a, x\right)} \]

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) - -1 \cdot a\right)\right)}, x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) - -1 \cdot a\right)\right)}, x\right) \]

       2. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) - -1 \cdot a\right)\right)}, x\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(\color{blue}{x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) - -1 \cdot a\right)}\right), x\right) \]

       4. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot a\right)}\right), x\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{1} \cdot a\right)\right), x\right) \]

       6. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \left(\frac{-1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{a}\right)\right), x\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {a}^{2} \cdot b, a\right)}\right), x\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{b \cdot {a}^{2}}, a\right)\right), x\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{b \cdot {a}^{2}}, a\right)\right), x\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(b, \mathsf{neg}\left(x \cdot \mathsf{fma}\left(\frac{-1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right), x\right) \]

       11. *-lowering-*.f64 83.7

           \[\leadsto \mathsf{fma}\left(b, -x \cdot \mathsf{fma}\left(-0.5, b \cdot \color{blue}{\left(a \cdot a\right)}, a\right), x\right) \]

   11. Simplified 83.7%

       \[\leadsto \mathsf{fma}\left(b, \color{blue}{-x \cdot \mathsf{fma}\left(-0.5, b \cdot \left(a \cdot a\right), a\right)}, x\right) \]

   if 5.4999999999999999e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 91.8%

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

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

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

      5. neg-lowering-neg.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 68.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 68.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(-0.5, b \cdot \left(a \cdot a\right), a\right) \cdot \left(-x\right), x\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5.5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+58}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_2 \leq 5.5 \cdot 10^{+250}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* 0.5 (* a (* a (* b b))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0)
     t_1
     (if (<= t_2 1e+58)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (if (<= t_2 5.5e+250)
         t_1
         (* x (fma y (fma 0.5 (* y (* t t)) (- t)) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (0.5 * (a * (a * (b * b))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+58) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (t_2 <= 5.5e+250) {
		tmp = t_1;
	} else {
		tmp = x * fma(y, fma(0.5, (y * (t * t)), -t), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+58)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_2 <= 5.5e+250)
		tmp = t_1;
	else
		tmp = Float64(x * fma(y, fma(0.5, Float64(y * Float64(t * t)), Float64(-t)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 1e+58], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5.5e+250], t$95$1, N[(x * N[(y * N[(0.5 * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $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))) < -2e3 or 9.99999999999999944e57 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.4999999999999999e250

   1. Initial program 98.1%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 44.8

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

   5. Simplified 44.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 21.3

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 21.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 21.4

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 21.4%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 46.7

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 46.7%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 9.99999999999999944e57

   1. Initial program 94.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) + 1\right) \]

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) - a, 1\right)} \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \left(\mathsf{neg}\left(a\right)\right)}, 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{-1 \cdot a}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      13. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      14. neg-lowering-neg.f64 82.4

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 82.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 5.4999999999999999e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 91.8%

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

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

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

      5. neg-lowering-neg.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 68.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 68.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 42.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+194}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))))
   (if (<= t_1 0.0)
     (* x (* 0.5 (* a (* a (* b b)))))
     (if (<= t_1 1e+194)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (* x (fma a (* a (* 0.5 (* b b))) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x * (0.5 * (a * (a * (b * b))));
	} else if (t_1 <= 1e+194) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else {
		tmp = x * fma(a, (a * (0.5 * (b * b))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))));
	elseif (t_1 <= 1e+194)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = Float64(x * fma(a, Float64(a * Float64(0.5 * Float64(b * b))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+194], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(a * N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 32.5

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 32.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 32.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 32.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 40.0

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 40.0%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if 0.0 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 9.99999999999999945e193

   1. Initial program 81.6%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) + 1\right) \]

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) - a, 1\right)} \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \left(\mathsf{neg}\left(a\right)\right)}, 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{-1 \cdot a}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      13. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      14. neg-lowering-neg.f64 93.9

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 93.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]

   if 9.99999999999999945e193 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 92.3%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 55.7

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 55.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 55.7

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 55.7%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 42.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ t_2 := a \cdot \left(b \cdot b\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot t\_2\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(b \cdot 0.5\right), b, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, t\_2, -b\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))
        (t_2 (* a (* b b))))
   (if (<= t_1 0.0)
     (* x (* 0.5 (* a t_2)))
     (if (<= t_1 5e-67)
       (* x (fma (* (* a a) (* b 0.5)) b 1.0))
       (* x (fma a (fma 0.5 t_2 (- b)) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double t_2 = a * (b * b);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x * (0.5 * (a * t_2));
	} else if (t_1 <= 5e-67) {
		tmp = x * fma(((a * a) * (b * 0.5)), b, 1.0);
	} else {
		tmp = x * fma(a, fma(0.5, t_2, -b), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
	t_2 = Float64(a * Float64(b * b))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x * Float64(0.5 * Float64(a * t_2)));
	elseif (t_1 <= 5e-67)
		tmp = Float64(x * fma(Float64(Float64(a * a) * Float64(b * 0.5)), b, 1.0));
	else
		tmp = Float64(x * fma(a, fma(0.5, t_2, Float64(-b)), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x * N[(0.5 * N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-67], N[(x * N[(N[(N[(a * a), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(0.5 * t$95$2 + (-b)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 32.5

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 32.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 32.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 32.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 40.0

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 40.0%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if 0.0 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 4.9999999999999999e-67

   1. Initial program 67.3%

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

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

      3. *-lowering-*.f64 67.3

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

   5. Simplified 67.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 45.8

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 45.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 45.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 45.8%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(b \cdot b\right)\right)} + 1\right) \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot b\right)} + 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot b} + 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right), b, 1\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right)}, b, 1\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{2} \cdot b\right), b, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}, b, 1\right) \]

       8. *-lowering-*.f64 97.3

          \[\leadsto x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot 0.5\right)}, b, 1\right) \]

   13. Applied egg-rr 97.3%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(b \cdot 0.5\right), b, 1\right)} \]

   if 4.9999999999999999e-67 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 93.2%

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

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

      3. *-lowering-*.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 60.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 60.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 42.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(b \cdot 0.5\right), b, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))))
   (if (<= t_1 0.0)
     (* x (* 0.5 (* a (* a (* b b)))))
     (if (<= t_1 5e-67)
       (* x (fma (* (* a a) (* b 0.5)) b 1.0))
       (* x (fma a (* a (* 0.5 (* b b))) 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x * (0.5 * (a * (a * (b * b))));
	} else if (t_1 <= 5e-67) {
		tmp = x * fma(((a * a) * (b * 0.5)), b, 1.0);
	} else {
		tmp = x * fma(a, (a * (0.5 * (b * b))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))));
	elseif (t_1 <= 5e-67)
		tmp = Float64(x * fma(Float64(Float64(a * a) * Float64(b * 0.5)), b, 1.0));
	else
		tmp = Float64(x * fma(a, Float64(a * Float64(0.5 * Float64(b * b))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-67], N[(x * N[(N[(N[(a * a), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(a * N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 32.5

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 32.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 32.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 32.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 40.0

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 40.0%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if 0.0 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 4.9999999999999999e-67

   1. Initial program 67.3%

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

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

      3. *-lowering-*.f64 67.3

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

   5. Simplified 67.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 45.8

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 45.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 45.8

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 45.8%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot \left(b \cdot b\right)\right)} + 1\right) \]

       2. associate-*r* N/A

          \[\leadsto x \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot b\right)} + 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot b} + 1\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right), b, 1\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(\frac{1}{2} \cdot b\right)}, b, 1\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{2} \cdot b\right), b, 1\right) \]

       7. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot \frac{1}{2}\right)}, b, 1\right) \]

       8. *-lowering-*.f64 97.3

          \[\leadsto x \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \color{blue}{\left(b \cdot 0.5\right)}, b, 1\right) \]

   13. Applied egg-rr 97.3%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(b \cdot 0.5\right), b, 1\right)} \]

   if 4.9999999999999999e-67 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 93.2%

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

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

      3. *-lowering-*.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 60.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 60.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 60.9

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 60.9%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ t_2 := x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
        (t_2 (* x (* 0.5 (* a (* a (* b b)))))))
   (if (<= t_1 0.0) t_2 (if (<= t_1 10.0) (fma (* a (+ z b)) (- x) x) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	double t_2 = x * (0.5 * (a * (a * (b * b))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 10.0) {
		tmp = fma((a * (z + b)), -x, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))))
	t_2 = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 10.0)
		tmp = fma(Float64(a * Float64(z + b)), Float64(-x), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 10.0], N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 0.0 or 10 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))

   1. Initial program 96.7%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 47.6

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

   5. Simplified 47.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 29.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 29.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 30.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 30.0%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 48.3

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 48.3%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if 0.0 < (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))) < 10

   1. Initial program 93.6%

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

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

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

      2. --lowering--.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. accelerator-lowering-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. neg-lowering-neg.f64 96.2

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

   5. Simplified 96.2%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. +-lowering-+.f64 96.2

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

   8. Simplified 96.2%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. neg-mul-1 N/A

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

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -1 \cdot x, x\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(b + z\right)}, -1 \cdot x, x\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(b + z\right)}, -1 \cdot x, x\right) \]

       10. neg-mul-1 N/A

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

       11. neg-lowering-neg.f64 89.6

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

   11. Simplified 89.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5.5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(z + b\right) \cdot \left(0.5 \cdot \left(z + b\right)\right), -b\right) - z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \end{array} \end{array} \]

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 -2000.0)
     (* x (* 0.5 (* a (* a (* b b)))))
     (if (<= t_1 5.5e+250)
       (* x (fma a (- (fma a (* (+ z b) (* 0.5 (+ z b))) (- b)) z) 1.0))
       (* x (fma y (fma 0.5 (* y (* t t)) (- t)) 1.0))))))

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 <= -2000.0) {
		tmp = x * (0.5 * (a * (a * (b * b))));
	} else if (t_1 <= 5.5e+250) {
		tmp = x * fma(a, (fma(a, ((z + b) * (0.5 * (z + b))), -b) - z), 1.0);
	} else {
		tmp = x * fma(y, fma(0.5, (y * (t * t)), -t), 1.0);
	}
	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 <= -2000.0)
		tmp = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))));
	elseif (t_1 <= 5.5e+250)
		tmp = Float64(x * fma(a, Float64(fma(a, Float64(Float64(z + b) * Float64(0.5 * Float64(z + b))), Float64(-b)) - z), 1.0));
	else
		tmp = Float64(x * fma(y, fma(0.5, Float64(y * Float64(t * t)), Float64(-t)), 1.0));
	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, -2000.0], N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5.5e+250], N[(x * N[(a * N[(N[(a * N[(N[(z + b), $MachinePrecision] * N[(0.5 * N[(z + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $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))) < -2e3

   1. Initial program 98.0%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 2.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 2.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 2.9

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 2.9%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 42.1

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 42.1%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 5.4999999999999999e250

   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

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.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. accelerator-lowering-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. neg-lowering-neg.f64 73.6

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

   5. Simplified 73.6%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. +-lowering-+.f64 73.6

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

   8. Simplified 73.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot \left(b + z\right) + \frac{1}{2} \cdot \left(a \cdot {\left(b + z\right)}^{2}\right), 1\right)} \]

   11. Simplified 65.6%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(0.5 \cdot \left(b + z\right)\right) \cdot \left(b + z\right), -b\right) - z, 1\right)} \]

   if 5.4999999999999999e250 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 91.8%

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

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

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

      5. neg-lowering-neg.f64 64.4

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

   5. Simplified 64.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(-1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, -1 \cdot t + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} + \frac{1}{2} \cdot \left({t}^{2} \cdot y\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {t}^{2} \cdot y, \mathsf{neg}\left(t\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{y \cdot {t}^{2}}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      8. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(\frac{1}{2}, y \cdot \color{blue}{\left(t \cdot t\right)}, \mathsf{neg}\left(t\right)\right), 1\right) \]

      10. neg-lowering-neg.f64 68.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), \color{blue}{-t}\right), 1\right) \]

   8. Simplified 68.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 5.5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \left(z + b\right) \cdot \left(0.5 \cdot \left(z + b\right)\right), -b\right) - z, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(0.5 \cdot \left(x \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+94}:\\ \;\;\;\;x - t \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (* 0.5 (* x (* a (* a b))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0) t_1 (if (<= t_2 2e+94) (- x (* t (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (0.5 * (x * (a * (a * b))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x - (t * (x * y));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = b * (0.5d0 * (x * (a * (a * b))))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-2000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+94) then
        tmp = x - (t * (x * y))
    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 = b * (0.5 * (x * (a * (a * b))));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x - (t * (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = b * (0.5 * (x * (a * (a * b))))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -2000.0:
		tmp = t_1
	elif t_2 <= 2e+94:
		tmp = x - (t * (x * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(0.5 * Float64(x * Float64(a * Float64(a * b)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = Float64(x - Float64(t * Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = b * (0.5 * (x * (a * (a * b))));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = x - (t * (x * y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(0.5 * N[(x * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 2e+94], N[(x - N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3 or 2e94 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 49.0

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

   5. Simplified 49.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 30.0

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 30.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 30.0

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 30.0%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot \left({b}^{2} \cdot x\right)\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left({b}^{2} \cdot x\right)} \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot x\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot x\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \color{blue}{\left(\left(b \cdot x\right) \cdot b\right)} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot {a}^{2}\right) \cdot \left(b \cdot x\right)\right) \cdot b} \]

       6. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \cdot b \]

       7. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot \left(b \cdot x\right)\right)\right)} \]

       10. associate-*r* N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\left({a}^{2} \cdot b\right) \cdot x\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot \left({a}^{2} \cdot b\right)\right)}\right) \]

       13. unpow2 N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto b \cdot \left(\frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)}\right)\right) \]

       16. *-lowering-*.f64 44.7

           \[\leadsto b \cdot \left(0.5 \cdot \left(x \cdot \left(a \cdot \color{blue}{\left(a \cdot b\right)}\right)\right)\right) \]

   14. Simplified 44.7%

       \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \left(x \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)\right)} \]

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e94

   1. Initial program 93.5%

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

   3. 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. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 76.7

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

   5. Simplified 76.7%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 69.1

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

   8. Simplified 69.1%

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

Alternative 10: 33.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -x \cdot \left(a \cdot b\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* x (* a b))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0) t_1 (if (<= t_2 2e+94) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (a * b));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = -(x * (a * b))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-2000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+94) then
        tmp = x
    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 * (a * b));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x * (a * b))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -2000.0:
		tmp = t_1
	elif t_2 <= 2e+94:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x * Float64(a * b)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(x * (a * b));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 2e+94], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3 or 2e94 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 49.0

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

   5. Simplified 49.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 30.0

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 30.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot b}, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b\right)}, 1\right) \]

       2. neg-lowering-neg.f64 15.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   11. Simplified 15.3%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   12. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 N/A

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

       3. *-lowering-*.f64 26.3

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

   14. Simplified 26.3%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e94

   1. Initial program 93.5%

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

   3. 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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 67.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -a \cdot \left(x \cdot b\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* a (* x b))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000.0) t_1 (if (<= t_2 2e+94) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(a * (x * b));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = -(a * (x * b))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-2000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+94) then
        tmp = x
    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 = -(a * (x * b));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+94) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(a * (x * b))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -2000.0:
		tmp = t_1
	elif t_2 <= 2e+94:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(a * Float64(x * b)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(a * (x * b));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -2000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+94)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2000.0], t$95$1, If[LessEqual[t$95$2, 2e+94], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3 or 2e94 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 49.0

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

   5. Simplified 49.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 30.0

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 30.0%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot b}, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b\right)}, 1\right) \]

       2. neg-lowering-neg.f64 15.3

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   11. Simplified 15.3%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   12. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. *-lowering-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. distribute-rgt-neg-in N/A

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

       7. mul-1-neg N/A

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

       8. *-lowering-*.f64 N/A

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

       9. mul-1-neg N/A

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

       10. neg-lowering-neg.f64 25.8

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

   14. Simplified 25.8%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2e94

   1. Initial program 93.5%

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

   3. 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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 86.0

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

   5. Simplified 86.0%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 67.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-a \cdot \left(x \cdot b\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a \cdot \left(x \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))) 0.0)
   (* x (* 0.5 (* a (* a (* b b)))))
   (* x (fma a (* a (* 0.5 (* b b))) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))))) <= 0.0) {
		tmp = x * (0.5 * (a * (a * (b * b))));
	} else {
		tmp = x * fma(a, (a * (0.5 * (b * b))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b))))) <= 0.0)
		tmp = Float64(x * Float64(0.5 * Float64(a * Float64(a * Float64(b * b)))));
	else
		tmp = Float64(x * fma(a, Float64(a * Float64(0.5 * Float64(b * b))), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(x * N[(0.5 * N[(a * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(a * N[(a * N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 98.4%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 32.5

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 32.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 32.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 32.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]

   12. Taylor expanded in a around inf

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} \]

       2. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {b}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot \left(a \cdot {b}^{2}\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \color{blue}{\left(a \cdot {b}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 40.0

          \[\leadsto x \cdot \left(0.5 \cdot \left(a \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

   14. Simplified 40.0%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)} \]

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

   1. Initial program 89.8%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 58.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 58.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot {b}^{2}}, 1\right) \]

       2. *-commutative N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(a \cdot \frac{1}{2}\right)} \cdot {b}^{2}, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}, 1\right) \]

       6. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

       7. *-lowering-*.f64 58.9

          \[\leadsto x \cdot \mathsf{fma}\left(a, a \cdot \left(0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right), 1\right) \]

   11. Simplified 58.9%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{a \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 34.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -2000.0)
   (- (* x (* a b)))
   (fma (* a (+ z b)) (- x) x)))

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))) <= -2000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = fma((a * (z + b)), -x, x);
	}
	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))) <= -2000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = fma(Float64(a * Float64(z + b)), Float64(-x), x);
	end
	return 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], -2000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), N[(N[(a * N[(z + b), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3

   1. Initial program 98.0%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 2.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 2.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot b}, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b\right)}, 1\right) \]

       2. neg-lowering-neg.f64 3.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   11. Simplified 3.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   12. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 N/A

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

       3. *-lowering-*.f64 25.9

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

   14. Simplified 25.9%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.0%

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

   3. 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. *-lowering-*.f64 N/A

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

      2. --lowering--.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. accelerator-lowering-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. neg-lowering-neg.f64 68.3

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

   5. Simplified 68.3%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. +-lowering-+.f64 68.3

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

   8. Simplified 68.3%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. neg-mul-1 N/A

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

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(b + z\right), -1 \cdot x, x\right)} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \left(b + z\right)}, -1 \cdot x, x\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(b + z\right)}, -1 \cdot x, x\right) \]

       10. neg-mul-1 N/A

           \[\leadsto \mathsf{fma}\left(a \cdot \left(b + z\right), \color{blue}{\mathsf{neg}\left(x\right)}, x\right) \]

       11. neg-lowering-neg.f64 43.3

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

   11. Simplified 43.3%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(z + b\right), -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))) -2000.0)
   (- (* x (* a b)))
   (* x (- 1.0 (* a b)))))

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))) <= -2000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	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))) <= (-2000.0d0)) then
        tmp = -(x * (a * b))
    else
        tmp = x * (1.0d0 - (a * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))) <= -2000.0) {
		tmp = -(x * (a * b));
	} else {
		tmp = x * (1.0 - (a * b));
	}
	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))) <= -2000.0:
		tmp = -(x * (a * b))
	else:
		tmp = x * (1.0 - (a * b))
	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))) <= -2000.0)
		tmp = Float64(-Float64(x * Float64(a * b)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	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))) <= -2000.0)
		tmp = -(x * (a * b));
	else
		tmp = x * (1.0 - (a * b));
	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], -2000.0], (-N[(x * N[(a * b), $MachinePrecision]), $MachinePrecision]), N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e3

   1. Initial program 98.0%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 42.0

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

   5. Simplified 42.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + a \cdot \left(-1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, -1 \cdot b + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} + \frac{1}{2} \cdot \left(a \cdot {b}^{2}\right), 1\right) \]

      4. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot \left(a \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(b\right)\right)}, 1\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, a \cdot {b}^{2}, \mathsf{neg}\left(b\right)\right)}, 1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{a \cdot {b}^{2}}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{1}{2}, a \cdot \color{blue}{\left(b \cdot b\right)}, \mathsf{neg}\left(b\right)\right), 1\right) \]

      9. neg-lowering-neg.f64 2.9

         \[\leadsto x \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \color{blue}{-b}\right), 1\right) \]

   8. Simplified 2.9%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), -b\right), 1\right)} \]

   9. Taylor expanded in a around 0

      \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-1 \cdot b}, 1\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{\mathsf{neg}\left(b\right)}, 1\right) \]

       2. neg-lowering-neg.f64 3.2

          \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   11. Simplified 3.2%

       \[\leadsto x \cdot \mathsf{fma}\left(a, \color{blue}{-b}, 1\right) \]

   12. Taylor expanded in a around inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 N/A

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

       3. *-lowering-*.f64 25.9

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

   14. Simplified 25.9%

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

   if -2e3 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.0%

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

   3. 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 43.3

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

   8. Simplified 43.3%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000:\\ \;\;\;\;-x \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 86.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -1.62 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -1.62e-43)
     t_1
     (if (<= y 8.2e-5) (* 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 <= -1.62e-43) {
		tmp = t_1;
	} else if (y <= 8.2e-5) {
		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 <= (-1.62d-43)) then
        tmp = t_1
    else if (y <= 8.2d-5) 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 <= -1.62e-43) {
		tmp = t_1;
	} else if (y <= 8.2e-5) {
		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 <= -1.62e-43:
		tmp = t_1
	elif y <= 8.2e-5:
		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 <= -1.62e-43)
		tmp = t_1;
	elseif (y <= 8.2e-5)
		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 <= -1.62e-43)
		tmp = t_1;
	elseif (y <= 8.2e-5)
		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, -1.62e-43], t$95$1, If[LessEqual[y, 8.2e-5], 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 < -1.6199999999999999e-43 or 8.20000000000000009e-5 < y

   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 inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 87.8

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

   5. Simplified 87.8%

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

   if -1.6199999999999999e-43 < y < 8.20000000000000009e-5

   1. Initial program 94.7%

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

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

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

      2. --lowering--.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. accelerator-lowering-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. neg-lowering-neg.f64 92.2

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

   5. Simplified 92.2%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. +-lowering-+.f64 92.2

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

   8. Simplified 92.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{-43}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ t_2 := x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+157}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0015:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))) (t_2 (* x (exp (* t (- y))))))
   (if (<= y -1.25e+157)
     t_2
     (if (<= y -2.2e+55)
       t_1
       (if (<= y 0.0015)
         (* x (exp (- (* a b))))
         (if (<= y 4.6e+148) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double t_2 = x * exp((t * -y));
	double tmp;
	if (y <= -1.25e+157) {
		tmp = t_2;
	} else if (y <= -2.2e+55) {
		tmp = t_1;
	} else if (y <= 0.0015) {
		tmp = x * exp(-(a * b));
	} else if (y <= 4.6e+148) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x * (z ** y)
    t_2 = x * exp((t * -y))
    if (y <= (-1.25d+157)) then
        tmp = t_2
    else if (y <= (-2.2d+55)) then
        tmp = t_1
    else if (y <= 0.0015d0) then
        tmp = x * exp(-(a * b))
    else if (y <= 4.6d+148) then
        tmp = t_1
    else
        tmp = t_2
    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.pow(z, y);
	double t_2 = x * Math.exp((t * -y));
	double tmp;
	if (y <= -1.25e+157) {
		tmp = t_2;
	} else if (y <= -2.2e+55) {
		tmp = t_1;
	} else if (y <= 0.0015) {
		tmp = x * Math.exp(-(a * b));
	} else if (y <= 4.6e+148) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.pow(z, y)
	t_2 = x * math.exp((t * -y))
	tmp = 0
	if y <= -1.25e+157:
		tmp = t_2
	elif y <= -2.2e+55:
		tmp = t_1
	elif y <= 0.0015:
		tmp = x * math.exp(-(a * b))
	elif y <= 4.6e+148:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	t_2 = Float64(x * exp(Float64(t * Float64(-y))))
	tmp = 0.0
	if (y <= -1.25e+157)
		tmp = t_2;
	elseif (y <= -2.2e+55)
		tmp = t_1;
	elseif (y <= 0.0015)
		tmp = Float64(x * exp(Float64(-Float64(a * b))));
	elseif (y <= 4.6e+148)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (z ^ y);
	t_2 = x * exp((t * -y));
	tmp = 0.0;
	if (y <= -1.25e+157)
		tmp = t_2;
	elseif (y <= -2.2e+55)
		tmp = t_1;
	elseif (y <= 0.0015)
		tmp = x * exp(-(a * b));
	elseif (y <= 4.6e+148)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Exp[N[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+157], t$95$2, If[LessEqual[y, -2.2e+55], t$95$1, If[LessEqual[y, 0.0015], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+148], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.24999999999999994e157 or 4.6000000000000001e148 < y

   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 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. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 72.0

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

   5. Simplified 72.0%

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

   if -1.24999999999999994e157 < y < -2.2000000000000001e55 or 0.0015 < y < 4.6000000000000001e148

   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 inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 75.4

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

   8. Simplified 75.4%

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

   if -2.2000000000000001e55 < y < 0.0015

   1. Initial program 95.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+157}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+55}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 0.0015:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{if}\;t \leq -4200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+146}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* t (- y))))))
   (if (<= t -4200000.0)
     t_1
     (if (<= t 8e+146) (* 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((t * -y));
	double tmp;
	if (t <= -4200000.0) {
		tmp = t_1;
	} else if (t <= 8e+146) {
		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((t * -y))
    if (t <= (-4200000.0d0)) then
        tmp = t_1
    else if (t <= 8d+146) 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((t * -y));
	double tmp;
	if (t <= -4200000.0) {
		tmp = t_1;
	} else if (t <= 8e+146) {
		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((t * -y))
	tmp = 0
	if t <= -4200000.0:
		tmp = t_1
	elif t <= 8e+146:
		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(t * Float64(-y))))
	tmp = 0.0
	if (t <= -4200000.0)
		tmp = t_1;
	elseif (t <= 8e+146)
		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((t * -y));
	tmp = 0.0;
	if (t <= -4200000.0)
		tmp = t_1;
	elseif (t <= 8e+146)
		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[(t * (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4200000.0], t$95$1, If[LessEqual[t, 8e+146], 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 < -4.2e6 or 7.99999999999999947e146 < t

   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 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. *-lowering-*.f64 N/A

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

      5. neg-lowering-neg.f64 85.5

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

   5. Simplified 85.5%

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

   if -4.2e6 < t < 7.99999999999999947e146

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.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. accelerator-lowering-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. neg-lowering-neg.f64 72.2

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

   5. Simplified 72.2%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. *-lowering-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. +-lowering-+.f64 72.2

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

   8. Simplified 72.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4200000:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+146}:\\ \;\;\;\;x \cdot e^{-a \cdot \left(z + b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{t \cdot \left(-y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0015:\\ \;\;\;\;x \cdot e^{-a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -2.2e+55) t_1 (if (<= y 0.0015) (* 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 * pow(z, y);
	double tmp;
	if (y <= -2.2e+55) {
		tmp = t_1;
	} else if (y <= 0.0015) {
		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 * (z ** y)
    if (y <= (-2.2d+55)) then
        tmp = t_1
    else if (y <= 0.0015d0) 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.pow(z, y);
	double tmp;
	if (y <= -2.2e+55) {
		tmp = t_1;
	} else if (y <= 0.0015) {
		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.pow(z, y)
	tmp = 0
	if y <= -2.2e+55:
		tmp = t_1
	elif y <= 0.0015:
		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 * (z ^ y))
	tmp = 0.0
	if (y <= -2.2e+55)
		tmp = t_1;
	elseif (y <= 0.0015)
		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 * (z ^ y);
	tmp = 0.0;
	if (y <= -2.2e+55)
		tmp = t_1;
	elseif (y <= 0.0015)
		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[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+55], t$95$1, If[LessEqual[y, 0.0015], N[(x * N[Exp[(-N[(a * b), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e55 or 0.0015 < y

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 89.8

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

   5. Simplified 89.8%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 63.8

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

   8. Simplified 63.8%

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

   if -2.2000000000000001e55 < y < 0.0015

   1. Initial program 95.5%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

Alternative 19: 58.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -5e+51)
     t_1
     (if (<= y 2.3e-9) (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * pow(z, y);
	double tmp;
	if (y <= -5e+51) {
		tmp = t_1;
	} else if (y <= 2.3e-9) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * (z ^ y))
	tmp = 0.0
	if (y <= -5e+51)
		tmp = t_1;
	elseif (y <= 2.3e-9)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Power[z, y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e+51], t$95$1, If[LessEqual[y, 2.3e-9], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e51 or 2.2999999999999999e-9 < y

   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. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 87.9

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

   5. Simplified 87.9%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

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

      2. pow-lowering-pow.f64 62.9

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

   8. Simplified 62.9%

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

   if -5e51 < y < 2.2999999999999999e-9

   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 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. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 80.9

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

   5. Simplified 80.9%

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

   6. Taylor expanded in b around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + b \cdot \left(-1 \cdot a + \frac{1}{2} \cdot \left({a}^{2} \cdot b\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

         \[\leadsto x \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right) + 1\right) \]

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) - a, 1\right)} \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \left(\mathsf{neg}\left(a\right)\right)}, 1\right) \]

      7. mul-1-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \frac{1}{2} \cdot \left({a}^{2} \cdot b\right) + \color{blue}{-1 \cdot a}, 1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot b, -1 \cdot a\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, \color{blue}{b \cdot {a}^{2}}, -1 \cdot a\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \color{blue}{\left(a \cdot a\right)}, -1 \cdot a\right), 1\right) \]

      13. mul-1-neg N/A

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(\frac{1}{2}, b \cdot \left(a \cdot a\right), \color{blue}{\mathsf{neg}\left(a\right)}\right), 1\right) \]

      14. neg-lowering-neg.f64 66.3

          \[\leadsto x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), \color{blue}{-a}\right), 1\right) \]

   8. Simplified 66.3%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 19.2% accurate, 328.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 inf

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

   1. *-lowering-*.f64 N/A

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

   2. --lowering--.f64 N/A

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

   3. log-lowering-log.f64 71.2

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

5. Simplified 71.2%

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

6. Taylor expanded in y around 0

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

8. Simplified 19.1%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))))