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

Percentage Accurate: 96.3% → 96.3%

Time: 15.1s

Alternatives: 18

Speedup: 1.0×

• 42.1% 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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 97.3%

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

Alternative 2: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0.0002:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+255}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (* y (* y (* 0.5 (* t t))))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000000.0)
     t_1
     (if (<= t_2 0.0002)
       (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
       (if (<= t_2 4e+255)
         t_1
         (* x (fma t (- (* 0.5 (* t (* y y))) y) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (y * (y * (0.5 * (t * t))));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000000.0) {
		tmp = t_1;
	} else if (t_2 <= 0.0002) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (t_2 <= 4e+255) {
		tmp = t_1;
	} else {
		tmp = x * fma(t, ((0.5 * (t * (y * y))) - y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(y * Float64(y * Float64(0.5 * Float64(t * t)))))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000000.0)
		tmp = t_1;
	elseif (t_2 <= 0.0002)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (t_2 <= 4e+255)
		tmp = t_1;
	else
		tmp = Float64(x * fma(t, Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(y * N[(y * N[(0.5 * N[(t * t), $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, -2000000.0], t$95$1, If[LessEqual[t$95$2, 0.0002], 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, 4e+255], t$95$1, N[(x * N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $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))) < -2e6 or 2.0000000000000001e-4 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 3.99999999999999995e255

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

      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 49.3

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

   5. Simplified 49.3%

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

   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 23.8

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

   8. Simplified 23.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 42.7

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 42.7%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -2e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e-4

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 98.3

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

   5. Simplified 98.3%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 97.1

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

   8. Simplified 97.1%

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

   if 3.99999999999999995e255 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 94.6%

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

   3. Taylor expanded in t around inf

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

      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 53.0

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

   5. Simplified 53.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 69.1

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

   8. Simplified 69.1%

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

Alternative 3: 45.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := t \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))
        (t_2 (* t (* x (* 0.5 (* y (* y t)))))))
   (if (<= t_1 -2e+209)
     (* y (/ x y))
     (if (<= t_1 -2000000.0)
       t_2
       (if (<= t_1 2e+36) (* x (- 1.0 (* a b))) t_2)))))

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 t_2 = t * (x * (0.5 * (y * (y * t))));
	double tmp;
	if (t_1 <= -2e+209) {
		tmp = y * (x / y);
	} else if (t_1 <= -2000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+36) {
		tmp = x * (1.0 - (a * b));
	} 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 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    t_2 = t * (x * (0.5d0 * (y * (y * t))))
    if (t_1 <= (-2d+209)) then
        tmp = y * (x / y)
    else if (t_1 <= (-2000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2d+36) then
        tmp = x * (1.0d0 - (a * b))
    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 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double t_2 = t * (x * (0.5 * (y * (y * t))));
	double tmp;
	if (t_1 <= -2e+209) {
		tmp = y * (x / y);
	} else if (t_1 <= -2000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+36) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	t_2 = t * (x * (0.5 * (y * (y * t))))
	tmp = 0
	if t_1 <= -2e+209:
		tmp = y * (x / y)
	elif t_1 <= -2000000.0:
		tmp = t_2
	elif t_1 <= 2e+36:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_2
	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)))
	t_2 = Float64(t * Float64(x * Float64(0.5 * Float64(y * Float64(y * t)))))
	tmp = 0.0
	if (t_1 <= -2e+209)
		tmp = Float64(y * Float64(x / y));
	elseif (t_1 <= -2000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e+36)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	t_2 = t * (x * (0.5 * (y * (y * t))));
	tmp = 0.0;
	if (t_1 <= -2e+209)
		tmp = y * (x / y);
	elseif (t_1 <= -2000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e+36)
		tmp = x * (1.0 - (a * b));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x * N[(0.5 * N[(y * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+209], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], t$95$2, If[LessEqual[t$95$1, 2e+36], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.0000000000000001e209

   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 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 63.5

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

   5. Simplified 63.5%

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

   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 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in y around inf

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

       1. remove-double-neg N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. distribute-lft-out-- N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. mul-1-neg N/A

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

       10. +-commutative N/A

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

       11. mul-1-neg N/A

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

       12. *-commutative N/A

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

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

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

       14. mul-1-neg N/A

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

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

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

       16. mul-1-neg N/A

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

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

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

       18. /-lowering-/.f64 6.1

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

   11. Simplified 6.1%

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

   12. Taylor expanded in t around 0

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 26.0

          \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]

   14. Simplified 26.0%

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]

   if -2.0000000000000001e209 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e6 or 2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. neg-lowering-neg.f64 37.0

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

   8. Simplified 37.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

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

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

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

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

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

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

       18. *-commutative N/A

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

       19. *-lowering-*.f64 46.0

           \[\leadsto t \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \color{blue}{\left(y \cdot t\right)}\right)\right)\right) \]

   11. Simplified 46.0%

       \[\leadsto \color{blue}{t \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot t\right)\right)\right)\right)} \]

   if -2e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.00000000000000008e36

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 90.8

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

   5. Simplified 90.8%

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

   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 83.1

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

   8. Simplified 83.1%

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

Alternative 4: 49.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ t_2 := x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* y (* y (* 0.5 (* t t)))))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 2.0) (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0)) 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 * (y * (y * (0.5 * (t * t))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} 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(y * Float64(y * Float64(0.5 * Float64(t * t)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	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[(y * N[(y * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2.0], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $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 2 < (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 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 50.2

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

   5. Simplified 50.2%

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

   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 33.9

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

   8. Simplified 33.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 47.7

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 47.7%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   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)))) < 2

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 98.3

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

   5. Simplified 98.3%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 97.1

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

   8. Simplified 97.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 49.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\\ t_2 := x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* y (* y (* 0.5 (* t t)))))))
   (if (<= t_1 0.0) t_2 (if (<= t_1 2.0) (* x (- 1.0 (* a b))) 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 * (y * (y * (0.5 * (t * t))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = x * (1.0 - (a * b));
	} 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 = exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
    t_2 = x * (y * (y * (0.5d0 * (t * t))))
    if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = x * (1.0d0 - (a * b))
    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 = Math.exp(((y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b))));
	double t_2 = x * (y * (y * (0.5 * (t * t))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))
	t_2 = x * (y * (y * (0.5 * (t * t))))
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = x * (1.0 - (a * b))
	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(y * Float64(y * Float64(0.5 * Float64(t * t)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
	t_2 = x * (y * (y * (0.5 * (t * t))));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = x * (1.0 - (a * b));
	else
		tmp = t_2;
	end
	tmp_2 = 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[(y * N[(y * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2.0], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $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 2 < (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 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 50.2

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

   5. Simplified 50.2%

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

   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 33.9

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

   8. Simplified 33.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 47.7

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 47.7%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   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)))) < 2

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 98.3

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

   5. Simplified 98.3%

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

   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 96.0

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

   8. Simplified 96.0%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2000000:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, y \cdot \left(t \cdot t\right), -t\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t, 0.5 \cdot \left(t \cdot \left(y \cdot y\right)\right) - y, 1\right)\\ \end{array} \end{array} \]

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 -2000000.0)
     (* x (* y (* y (* 0.5 (* t t)))))
     (if (<= t_1 2e+152)
       (* x (fma y (fma 0.5 (* y (* t t)) (- t)) 1.0))
       (* x (fma t (- (* 0.5 (* t (* y y))) y) 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 <= -2000000.0) {
		tmp = x * (y * (y * (0.5 * (t * t))));
	} else if (t_1 <= 2e+152) {
		tmp = x * fma(y, fma(0.5, (y * (t * t)), -t), 1.0);
	} else {
		tmp = x * fma(t, ((0.5 * (t * (y * y))) - y), 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 <= -2000000.0)
		tmp = Float64(x * Float64(y * Float64(y * Float64(0.5 * Float64(t * t)))));
	elseif (t_1 <= 2e+152)
		tmp = Float64(x * fma(y, fma(0.5, Float64(y * Float64(t * t)), Float64(-t)), 1.0));
	else
		tmp = Float64(x * fma(t, Float64(Float64(0.5 * Float64(t * Float64(y * y))) - y), 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, -2000000.0], N[(x * N[(y * N[(y * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+152], N[(x * N[(y * N[(0.5 * N[(y * N[(t * t), $MachinePrecision]), $MachinePrecision] + (-t)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(t * N[(N[(0.5 * N[(t * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $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))) < -2e6

   1. Initial program 98.9%

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

   3. Taylor expanded in 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 52.5

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

   5. Simplified 52.5%

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

   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 3.6

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

   8. Simplified 3.6%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

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

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

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

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 34.7

           \[\leadsto x \cdot \left(y \cdot \left(y \cdot \left(0.5 \cdot \color{blue}{\left(t \cdot t\right)}\right)\right)\right) \]

   11. Simplified 34.7%

       \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -2e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.0000000000000001e152

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

      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 73.0

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

   5. Simplified 73.0%

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

   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 69.4

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

   8. Simplified 69.4%

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

   if 2.0000000000000001e152 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 95.4%

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

   3. Taylor expanded in 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 48.9

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

   5. Simplified 48.9%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.4

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

   8. Simplified 67.4%

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

Alternative 7: 34.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_1 \leq -2000000:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_1 -2e+209)
     (* y (/ x y))
     (if (<= t_1 -2000000.0) (- (* x (* y t))) (* x (- 1.0 (* y t)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+209) {
		tmp = y * (x / y);
	} else if (t_1 <= -2000000.0) {
		tmp = -(x * (y * t));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_1 <= (-2d+209)) then
        tmp = y * (x / y)
    else if (t_1 <= (-2000000.0d0)) then
        tmp = -(x * (y * t))
    else
        tmp = x * (1.0d0 - (y * t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_1 <= -2e+209) {
		tmp = y * (x / y);
	} else if (t_1 <= -2000000.0) {
		tmp = -(x * (y * t));
	} else {
		tmp = x * (1.0 - (y * t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_1 <= -2e+209:
		tmp = y * (x / y)
	elif t_1 <= -2000000.0:
		tmp = -(x * (y * t))
	else:
		tmp = x * (1.0 - (y * t))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_1 <= -2e+209)
		tmp = Float64(y * Float64(x / y));
	elseif (t_1 <= -2000000.0)
		tmp = Float64(-Float64(x * Float64(y * t)));
	else
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_1 <= -2e+209)
		tmp = y * (x / y);
	elseif (t_1 <= -2000000.0)
		tmp = -(x * (y * t));
	else
		tmp = x * (1.0 - (y * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+209], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2000000.0], (-N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]), N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2.0000000000000001e209

   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 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 63.5

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

   5. Simplified 63.5%

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

   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 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in y around inf

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

       1. remove-double-neg N/A

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

       2. mul-1-neg N/A

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

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

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

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

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

       5. distribute-lft-out-- N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. mul-1-neg N/A

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

       10. +-commutative N/A

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

       11. mul-1-neg N/A

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

       12. *-commutative N/A

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

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

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

       14. mul-1-neg N/A

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

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

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

       16. mul-1-neg N/A

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

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

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

       18. /-lowering-/.f64 6.1

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

   11. Simplified 6.1%

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

   12. Taylor expanded in t around 0

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 26.0

          \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]

   14. Simplified 26.0%

       \[\leadsto y \cdot \color{blue}{\frac{x}{y}} \]

   if -2.0000000000000001e209 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -2e6

   1. Initial program 97.5%

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

   3. Taylor expanded in t around inf

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

      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 38.2

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

   5. Simplified 38.2%

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

   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 3.4

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

   8. Simplified 3.4%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 25.6

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

   11. Simplified 25.6%

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

   if -2e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   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 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 60.5

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

   5. Simplified 60.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 42.4

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

   8. Simplified 42.4%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2 \cdot 10^{+209}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000000:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -x \cdot \left(y \cdot t\right)\\ t_2 := y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)\\ \mathbf{if}\;t\_2 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* x (* y t))))
        (t_2 (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))))
   (if (<= t_2 -2000000.0) t_1 (if (<= t_2 2e+36) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (y * t));
	double t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+36) {
		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 * (y * t))
    t_2 = (y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))
    if (t_2 <= (-2000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 2d+36) 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 * (y * t));
	double t_2 = (y * (Math.log(z) - t)) + (a * (Math.log((1.0 - z)) - b));
	double tmp;
	if (t_2 <= -2000000.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+36) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x * (y * t))
	t_2 = (y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))
	tmp = 0
	if t_2 <= -2000000.0:
		tmp = t_1
	elif t_2 <= 2e+36:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x * Float64(y * t)))
	t_2 = Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))
	tmp = 0.0
	if (t_2 <= -2000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+36)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -(x * (y * t));
	t_2 = (y * (log(z) - t)) + (a * (log((1.0 - z)) - b));
	tmp = 0.0;
	if (t_2 <= -2000000.0)
		tmp = t_1;
	elseif (t_2 <= 2e+36)
		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[(y * t), $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, -2000000.0], t$95$1, If[LessEqual[t$95$2, 2e+36], 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))) < -2e6 or 2.00000000000000008e36 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.6%

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

   3. Taylor expanded in t around inf

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

      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 50.4

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

   5. Simplified 50.4%

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

   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 12.4

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

   8. Simplified 12.4%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 21.5

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

   11. Simplified 21.5%

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

   if -2e6 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < 2.00000000000000008e36

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 81.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq -2000000:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right) \leq 2 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-16}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* a (- (- z) b))))))
   (if (<= a -5.5e+100)
     t_1
     (if (<= a 1.45e-16) (* x (exp (* y (- (log z) t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((a * (-z - b)));
	double tmp;
	if (a <= -5.5e+100) {
		tmp = t_1;
	} else if (a <= 1.45e-16) {
		tmp = x * exp((y * (log(z) - t)));
	} 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((a * (-z - b)))
    if (a <= (-5.5d+100)) then
        tmp = t_1
    else if (a <= 1.45d-16) then
        tmp = x * exp((y * (log(z) - t)))
    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((a * (-z - b)));
	double tmp;
	if (a <= -5.5e+100) {
		tmp = t_1;
	} else if (a <= 1.45e-16) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp((a * (-z - b)))
	tmp = 0
	if a <= -5.5e+100:
		tmp = t_1
	elif a <= 1.45e-16:
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))))
	tmp = 0.0
	if (a <= -5.5e+100)
		tmp = t_1;
	elseif (a <= 1.45e-16)
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp((a * (-z - b)));
	tmp = 0.0;
	if (a <= -5.5e+100)
		tmp = t_1;
	elseif (a <= 1.45e-16)
		tmp = x * exp((y * (log(z) - t)));
	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[(a * N[((-z) - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.5e+100], t$95$1, If[LessEqual[a, 1.45e-16], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.5000000000000002e100 or 1.4499999999999999e-16 < a

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   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 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 85.5

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

   8. Simplified 85.5%

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

   if -5.5000000000000002e100 < a < 1.4499999999999999e-16

   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 88.6

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

   5. Simplified 88.6%

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

Alternative 10: 54.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.95 \cdot 10^{-167}:\\ \;\;\;\;y \cdot \left(y \cdot \left(x \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b \cdot \left(a \cdot a\right), -a\right), 1\right)\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \left(\left(t \cdot t\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -3.1e-34)
     t_1
     (if (<= y -2.95e-167)
       (* y (* y (* x (* 0.5 (* t t)))))
       (if (<= y 2.4e+19)
         (* x (fma b (fma 0.5 (* b (* a a)) (- a)) 1.0))
         (if (<= y 1.06e+143)
           (* x (* (* t t) (* -0.16666666666666666 (* t (* y (* y y))))))
           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 <= -3.1e-34) {
		tmp = t_1;
	} else if (y <= -2.95e-167) {
		tmp = y * (y * (x * (0.5 * (t * t))));
	} else if (y <= 2.4e+19) {
		tmp = x * fma(b, fma(0.5, (b * (a * a)), -a), 1.0);
	} else if (y <= 1.06e+143) {
		tmp = x * ((t * t) * (-0.16666666666666666 * (t * (y * (y * y)))));
	} 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 <= -3.1e-34)
		tmp = t_1;
	elseif (y <= -2.95e-167)
		tmp = Float64(y * Float64(y * Float64(x * Float64(0.5 * Float64(t * t)))));
	elseif (y <= 2.4e+19)
		tmp = Float64(x * fma(b, fma(0.5, Float64(b * Float64(a * a)), Float64(-a)), 1.0));
	elseif (y <= 1.06e+143)
		tmp = Float64(x * Float64(Float64(t * t) * Float64(-0.16666666666666666 * Float64(t * Float64(y * Float64(y * y))))));
	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, -3.1e-34], t$95$1, If[LessEqual[y, -2.95e-167], N[(y * N[(y * N[(x * N[(0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+19], N[(x * N[(b * N[(0.5 * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.06e+143], N[(x * N[(N[(t * t), $MachinePrecision] * N[(-0.16666666666666666 * N[(t * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.0999999999999998e-34 or 1.06e143 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 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 83.0

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

   5. Simplified 83.0%

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

   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 64.0

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

   8. Simplified 64.0%

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

   if -3.0999999999999998e-34 < y < -2.95000000000000011e-167

   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 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 51.7

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

   5. Simplified 51.7%

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

   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 55.9

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

   8. Simplified 55.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*r* N/A

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

       18. associate-*r* N/A

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

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

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

       20. associate-*r* N/A

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

   11. Simplified 59.3%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(x \cdot \left(0.5 \cdot \left(t \cdot t\right)\right)\right)\right)} \]

   if -2.95000000000000011e-167 < y < 2.4e19

   1. Initial program 99.0%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 83.0

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

   5. Simplified 83.0%

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

   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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 61.5

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

   8. Simplified 61.5%

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

   if 2.4e19 < y < 1.06e143

   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 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 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + t \cdot \left(-1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(t \cdot \left(-1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right)\right) + 1\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot y + t \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right) + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

   8. Simplified 55.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(t, t \cdot \mathsf{fma}\left(0.5, y \cdot y, -0.16666666666666666 \cdot \left(t \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) - y, 1\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({t}^{3} \cdot {y}^{3}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {t}^{3}\right) \cdot {y}^{3}\right)} \]

       2. *-commutative N/A

          \[\leadsto x \cdot \left(\color{blue}{\left({t}^{3} \cdot \frac{-1}{6}\right)} \cdot {y}^{3}\right) \]

       3. associate-*r* N/A

          \[\leadsto x \cdot \color{blue}{\left({t}^{3} \cdot \left(\frac{-1}{6} \cdot {y}^{3}\right)\right)} \]

       4. unpow3 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(\left(t \cdot t\right) \cdot t\right)} \cdot \left(\frac{-1}{6} \cdot {y}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{{t}^{2}} \cdot t\right) \cdot \left(\frac{-1}{6} \cdot {y}^{3}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left({t}^{2} \cdot \left(t \cdot \left(\frac{-1}{6} \cdot {y}^{3}\right)\right)\right)} \]

       7. associate-*r* N/A

          \[\leadsto x \cdot \left({t}^{2} \cdot \color{blue}{\left(\left(t \cdot \frac{-1}{6}\right) \cdot {y}^{3}\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto x \cdot \left({t}^{2} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot t\right)} \cdot {y}^{3}\right)\right) \]

       9. associate-*r* N/A

          \[\leadsto x \cdot \left({t}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right)\right)}\right) \]

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

           \[\leadsto x \cdot \color{blue}{\left({t}^{2} \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right)\right)\right)} \]

       11. unpow2 N/A

           \[\leadsto x \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(t \cdot {y}^{3}\right)\right)}\right) \]

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

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(t \cdot {y}^{3}\right)}\right)\right) \]

       15. cube-mult N/A

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot \left(t \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot \left(t \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

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

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot \left(t \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right) \]

       18. unpow2 N/A

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(\frac{-1}{6} \cdot \left(t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

       19. *-lowering-*.f64 67.3

           \[\leadsto x \cdot \left(\left(t \cdot t\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

   11. Simplified 67.3%

       \[\leadsto x \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot \left(-0.16666666666666666 \cdot \left(t \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{-y \cdot t}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- (* y t))))))
   (if (<= t -1.5e+65)
     t_1
     (if (<= t 2.25e+136) (* x (exp (* a (- (- z) b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(-(y * t));
	double tmp;
	if (t <= -1.5e+65) {
		tmp = t_1;
	} else if (t <= 2.25e+136) {
		tmp = x * exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp(-(y * t))
    if (t <= (-1.5d+65)) then
        tmp = t_1
    else if (t <= 2.25d+136) then
        tmp = x * exp((a * (-z - b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp(-(y * t));
	double tmp;
	if (t <= -1.5e+65) {
		tmp = t_1;
	} else if (t <= 2.25e+136) {
		tmp = x * Math.exp((a * (-z - b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp(-(y * t))
	tmp = 0
	if t <= -1.5e+65:
		tmp = t_1
	elif t <= 2.25e+136:
		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(-Float64(y * t))))
	tmp = 0.0
	if (t <= -1.5e+65)
		tmp = t_1;
	elseif (t <= 2.25e+136)
		tmp = Float64(x * exp(Float64(a * Float64(Float64(-z) - b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp(-(y * t));
	tmp = 0.0;
	if (t <= -1.5e+65)
		tmp = t_1;
	elseif (t <= 2.25e+136)
		tmp = x * exp((a * (-z - b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+65], t$95$1, If[LessEqual[t, 2.25e+136], 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 < -1.5000000000000001e65 or 2.25e136 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in 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 82.6

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

   5. Simplified 82.6%

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

   if -1.5000000000000001e65 < t < 2.25e136

   1. Initial program 97.4%

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

   3. Taylor expanded in 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 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in z around 0

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

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

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

      2. mul-1-neg N/A

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

      3. neg-lowering-neg.f64 78.7

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

   8. Simplified 78.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+65}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;x \cdot e^{a \cdot \left(\left(-z\right) - b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{-y \cdot t}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+84}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (- (* y t))))))
   (if (<= t -1.1e+27) t_1 (if (<= t 2.1e+84) (* x (exp (* a (- b)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp(-(y * t));
	double tmp;
	if (t <= -1.1e+27) {
		tmp = t_1;
	} else if (t <= 2.1e+84) {
		tmp = x * exp((a * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * exp(-(y * t))
    if (t <= (-1.1d+27)) then
        tmp = t_1
    else if (t <= 2.1d+84) then
        tmp = x * exp((a * -b))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * Math.exp(-(y * t));
	double tmp;
	if (t <= -1.1e+27) {
		tmp = t_1;
	} else if (t <= 2.1e+84) {
		tmp = x * Math.exp((a * -b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * math.exp(-(y * t))
	tmp = 0
	if t <= -1.1e+27:
		tmp = t_1
	elif t <= 2.1e+84:
		tmp = x * math.exp((a * -b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(-Float64(y * t))))
	tmp = 0.0
	if (t <= -1.1e+27)
		tmp = t_1;
	elseif (t <= 2.1e+84)
		tmp = Float64(x * exp(Float64(a * Float64(-b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * exp(-(y * t));
	tmp = 0.0;
	if (t <= -1.1e+27)
		tmp = t_1;
	elseif (t <= 2.1e+84)
		tmp = x * exp((a * -b));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[(-N[(y * t), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.1e+27], t$95$1, If[LessEqual[t, 2.1e+84], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0999999999999999e27 or 2.10000000000000019e84 < t

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-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 81.0

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

   5. Simplified 81.0%

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

   if -1.0999999999999999e27 < t < 2.10000000000000019e84

   1. Initial program 97.9%

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

   3. Taylor expanded in 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 76.2

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

   5. Simplified 76.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+84}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-y \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot {z}^{y}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+164}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (pow z y))))
   (if (<= y -3.2e+28) t_1 (if (<= y 8.9e+164) (* 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 <= -3.2e+28) {
		tmp = t_1;
	} else if (y <= 8.9e+164) {
		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 <= (-3.2d+28)) then
        tmp = t_1
    else if (y <= 8.9d+164) 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 <= -3.2e+28) {
		tmp = t_1;
	} else if (y <= 8.9e+164) {
		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 <= -3.2e+28:
		tmp = t_1
	elif y <= 8.9e+164:
		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 <= -3.2e+28)
		tmp = t_1;
	elseif (y <= 8.9e+164)
		tmp = Float64(x * exp(Float64(a * Float64(-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 <= -3.2e+28)
		tmp = t_1;
	elseif (y <= 8.9e+164)
		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, -3.2e+28], t$95$1, If[LessEqual[y, 8.9e+164], N[(x * N[Exp[N[(a * (-b)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e28 or 8.8999999999999999e164 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 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 88.2

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

   5. Simplified 88.2%

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

   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 67.9

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

   8. Simplified 67.9%

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

   if -3.2e28 < y < 8.8999999999999999e164

   1. Initial program 98.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 76.0

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

   5. Simplified 76.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+28}:\\ \;\;\;\;x \cdot {z}^{y}\\ \mathbf{elif}\;y \leq 8.9 \cdot 10^{+164}:\\ \;\;\;\;x \cdot e^{a \cdot \left(-b\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot {z}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+29}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, -y, \frac{x}{t}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.7e+29)
   (* t (fma x (- y) (/ x t)))
   (if (<= y 2.8e+20) (* x (- 1.0 (* a b))) (- (* x (* y t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.7e+29) {
		tmp = t * fma(x, -y, (x / t));
	} else if (y <= 2.8e+20) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = -(x * (y * t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.7e+29)
		tmp = Float64(t * fma(x, Float64(-y), Float64(x / t)));
	elseif (y <= 2.8e+20)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	else
		tmp = Float64(-Float64(x * Float64(y * t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.7e+29], N[(t * N[(x * (-y) + N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+20], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7000000000000002e29

   1. Initial program 93.4%

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

   3. Taylor expanded in 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 49.5

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

   5. Simplified 49.5%

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

   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 15.2

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

   8. Simplified 15.2%

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

   9. Taylor expanded in t around inf

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

       1. remove-double-neg N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

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

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

       5. mul-1-neg N/A

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

       6. associate-*r* N/A

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

       7. distribute-lft-out-- N/A

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

       8. mul-1-neg N/A

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

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

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

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

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

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

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

       12. distribute-lft-out-- N/A

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

       13. mul-1-neg N/A

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

       14. remove-double-neg N/A

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

       15. sub-neg N/A

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

       16. mul-1-neg N/A

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

       17. +-commutative N/A

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

       18. mul-1-neg N/A

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

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

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

       20. mul-1-neg N/A

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

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

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

       22. mul-1-neg N/A

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

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

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

       24. /-lowering-/.f64 23.3

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

   11. Simplified 23.3%

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

   if -4.7000000000000002e29 < y < 2.8e20

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 79.8

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

   5. Simplified 79.8%

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

   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 45.4

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

   8. Simplified 45.4%

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

   if 2.8e20 < y

   1. Initial program 96.6%

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

   3. Taylor expanded in t around inf

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

      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 69.5

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

   5. Simplified 69.5%

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

   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 21.5

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

   8. Simplified 21.5%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 32.9

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

   11. Simplified 32.9%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+29}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, -y, \frac{x}{t}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(x \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6.2e+112)
   (* x (- 1.0 (* a b)))
   (if (<= b 4.8e+85) (* x (- 1.0 (* y t))) (- x (* a (* x b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.2e+112) {
		tmp = x * (1.0 - (a * b));
	} else if (b <= 4.8e+85) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x - (a * (x * 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 (b <= (-6.2d+112)) then
        tmp = x * (1.0d0 - (a * b))
    else if (b <= 4.8d+85) then
        tmp = x * (1.0d0 - (y * t))
    else
        tmp = x - (a * (x * 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 (b <= -6.2e+112) {
		tmp = x * (1.0 - (a * b));
	} else if (b <= 4.8e+85) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = x - (a * (x * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6.2e+112:
		tmp = x * (1.0 - (a * b))
	elif b <= 4.8e+85:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = x - (a * (x * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.2e+112)
		tmp = Float64(x * Float64(1.0 - Float64(a * b)));
	elseif (b <= 4.8e+85)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = Float64(x - Float64(a * Float64(x * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6.2e+112)
		tmp = x * (1.0 - (a * b));
	elseif (b <= 4.8e+85)
		tmp = x * (1.0 - (y * t));
	else
		tmp = x - (a * (x * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.2e+112], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e+85], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(a * N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.19999999999999965e112

   1. Initial program 97.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 87.4

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

   5. Simplified 87.4%

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

   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 37.4

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

   8. Simplified 37.4%

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

   if -6.19999999999999965e112 < b < 4.79999999999999993e85

   1. Initial program 97.6%

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

   3. Taylor expanded in 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 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 36.9

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

   8. Simplified 36.9%

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

   if 4.79999999999999993e85 < b

   1. Initial program 95.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 80.6

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

   5. Simplified 80.6%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 31.2

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

   8. Simplified 31.2%

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

Alternative 16: 31.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(1 - a \cdot b\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \left(1 - y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (- 1.0 (* a b)))))
   (if (<= b -1e+110) t_1 (if (<= b 5e+85) (* x (- 1.0 (* y t))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * (1.0 - (a * b));
	double tmp;
	if (b <= -1e+110) {
		tmp = t_1;
	} else if (b <= 5e+85) {
		tmp = x * (1.0 - (y * t));
	} 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 * (1.0d0 - (a * b))
    if (b <= (-1d+110)) then
        tmp = t_1
    else if (b <= 5d+85) then
        tmp = x * (1.0d0 - (y * t))
    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 * (1.0 - (a * b));
	double tmp;
	if (b <= -1e+110) {
		tmp = t_1;
	} else if (b <= 5e+85) {
		tmp = x * (1.0 - (y * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x * (1.0 - (a * b))
	tmp = 0
	if b <= -1e+110:
		tmp = t_1
	elif b <= 5e+85:
		tmp = x * (1.0 - (y * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * Float64(1.0 - Float64(a * b)))
	tmp = 0.0
	if (b <= -1e+110)
		tmp = t_1;
	elseif (b <= 5e+85)
		tmp = Float64(x * Float64(1.0 - Float64(y * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x * (1.0 - (a * b));
	tmp = 0.0;
	if (b <= -1e+110)
		tmp = t_1;
	elseif (b <= 5e+85)
		tmp = x * (1.0 - (y * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+110], t$95$1, If[LessEqual[b, 5e+85], N[(x * N[(1.0 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1e110 or 5.0000000000000001e85 < 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 84.0

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

   5. Simplified 84.0%

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

   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 34.3

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

   8. Simplified 34.3%

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

   if -1e110 < b < 5.0000000000000001e85

   1. Initial program 97.6%

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

   3. Taylor expanded in 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 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 36.9

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

   8. Simplified 36.9%

      \[\leadsto x \cdot \color{blue}{\left(1 - y \cdot t\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 32.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -x \cdot \left(y \cdot t\right)\\ \mathbf{if}\;y \leq -6.1 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (* x (* y t)))))
   (if (<= y -6.1e+29) t_1 (if (<= y 2.8e+20) (* x (- 1.0 (* a b))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(x * (y * t));
	double tmp;
	if (y <= -6.1e+29) {
		tmp = t_1;
	} else if (y <= 2.8e+20) {
		tmp = x * (1.0 - (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 * (y * t))
    if (y <= (-6.1d+29)) then
        tmp = t_1
    else if (y <= 2.8d+20) then
        tmp = x * (1.0d0 - (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 * (y * t));
	double tmp;
	if (y <= -6.1e+29) {
		tmp = t_1;
	} else if (y <= 2.8e+20) {
		tmp = x * (1.0 - (a * b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -(x * (y * t))
	tmp = 0
	if y <= -6.1e+29:
		tmp = t_1
	elif y <= 2.8e+20:
		tmp = x * (1.0 - (a * b))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(x * Float64(y * t)))
	tmp = 0.0
	if (y <= -6.1e+29)
		tmp = t_1;
	elseif (y <= 2.8e+20)
		tmp = Float64(x * Float64(1.0 - 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 * (y * t));
	tmp = 0.0;
	if (y <= -6.1e+29)
		tmp = t_1;
	elseif (y <= 2.8e+20)
		tmp = x * (1.0 - (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[(y * t), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y, -6.1e+29], t$95$1, If[LessEqual[y, 2.8e+20], N[(x * N[(1.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.0999999999999998e29 or 2.8e20 < y

   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 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 59.3

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

   5. Simplified 59.3%

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

   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 18.3

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

   8. Simplified 18.3%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

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

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

       6. mul-1-neg N/A

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

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

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

       8. mul-1-neg N/A

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

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

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

       10. mul-1-neg N/A

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

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

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

       12. mul-1-neg N/A

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

       13. neg-lowering-neg.f64 23.9

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

   11. Simplified 23.9%

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

   if -6.0999999999999998e29 < y < 2.8e20

   1. Initial program 99.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   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 79.8

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

   5. Simplified 79.8%

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

   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 45.4

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

   8. Simplified 45.4%

      \[\leadsto x \cdot \color{blue}{\left(1 - a \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+29}:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \left(1 - a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \left(y \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.4% 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 97.3%

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

3. Taylor expanded in 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 70.4

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

5. Simplified 70.4%

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

6. Taylor expanded in y around 0

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

8. Simplified 18.8%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))