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

Percentage Accurate: 96.4% → 99.6%

Time: 10.6s

Alternatives: 8

Speedup: 1.5×

• 41.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * exp(((y * (log(z) - t)) + (a * (log((1.0d0 - z)) - b))))
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return x * math.exp(((y * (math.log(z) - t)) + (a * (math.log((1.0 - z)) - b))))

Julia

function code(x, y, z, t, a, b)
	return Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b))));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 2: 85.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+132}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* x (exp (* y (- (log z) t))))))
   (if (<= y -7.2e-37)
     t_1
     (if (<= y 5e-261)
       (* x (exp (* (- (* (- (* -0.5 z) 1.0) z) b) a)))
       (if (<= y 3e+132) (* x (exp (fma (- b) a (* (log z) y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x * exp((y * (log(z) - t)));
	double tmp;
	if (y <= -7.2e-37) {
		tmp = t_1;
	} else if (y <= 5e-261) {
		tmp = x * exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	} else if (y <= 3e+132) {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x * exp(Float64(y * Float64(log(z) - t))))
	tmp = 0.0
	if (y <= -7.2e-37)
		tmp = t_1;
	elseif (y <= 5e-261)
		tmp = Float64(x * exp(Float64(Float64(Float64(Float64(Float64(-0.5 * z) - 1.0) * z) - b) * a)));
	elseif (y <= 3e+132)
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-37], t$95$1, If[LessEqual[y, 5e-261], N[(x * N[Exp[N[(N[(N[(N[(N[(-0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+132], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000014e-37 or 2.9999999999999998e132 < y

   1. Initial program 96.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 16.9%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

   10. Applied rewrites 16.9%

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

   11. Taylor expanded in a around 0

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

       1. rec-exp N/A

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

       2. lower-exp.f64 N/A

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

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

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

       4. lower-neg.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-log.f64 95.2

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

   13. Applied rewrites 95.2%

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

   if -7.20000000000000014e-37 < y < 4.99999999999999981e-261

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.5%

      \[\leadsto x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a} \]

   if 4.99999999999999981e-261 < y < 2.9999999999999998e132

   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 z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-37}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-261}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+132}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-37} \lor \neg \left(y \leq 3 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.2e-37) (not (<= y 3e-15)))
   (* x (exp (* y (- (log z) t))))
   (* x (exp (* (- (* (- (* -0.5 z) 1.0) z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.2e-37) || !(y <= 3e-15)) {
		tmp = x * exp((y * (log(z) - t)));
	} else {
		tmp = x * exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7.2d-37)) .or. (.not. (y <= 3d-15))) then
        tmp = x * exp((y * (log(z) - t)))
    else
        tmp = x * exp(((((((-0.5d0) * z) - 1.0d0) * z) - b) * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.2e-37) || !(y <= 3e-15)) {
		tmp = x * Math.exp((y * (Math.log(z) - t)));
	} else {
		tmp = x * Math.exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7.2e-37) or not (y <= 3e-15):
		tmp = x * math.exp((y * (math.log(z) - t)))
	else:
		tmp = x * math.exp((((((-0.5 * z) - 1.0) * z) - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.2e-37) || !(y <= 3e-15))
		tmp = Float64(x * exp(Float64(y * Float64(log(z) - t))));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(Float64(Float64(-0.5 * z) - 1.0) * z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7.2e-37) || ~((y <= 3e-15)))
		tmp = x * exp((y * (log(z) - t)));
	else
		tmp = x * exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.2e-37], N[Not[LessEqual[y, 3e-15]], $MachinePrecision]], N[(x * N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(N[(N[(N[(-0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000014e-37 or 3e-15 < y

   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 z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 16.3%

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

      1. lift-exp.f64 N/A

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

      2. sinh-+-cosh-rev N/A

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

   10. Applied rewrites 16.3%

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

   11. Taylor expanded in a around 0

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

       1. rec-exp N/A

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

       2. lower-exp.f64 N/A

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

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

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

       4. lower-neg.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-log.f64 89.8

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

   13. Applied rewrites 89.8%

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

   if -7.20000000000000014e-37 < y < 3e-15

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.7%

      \[\leadsto x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-37} \lor \neg \left(y \leq 3 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot e^{y \cdot \left(\log z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 96.5

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

5. Applied rewrites 96.5%

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

Alternative 5: 71.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-37} \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.2e-37) (not (<= y 1950000000000.0)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- (* (- (* -0.5 z) 1.0) z) b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.2e-37) || !(y <= 1950000000000.0)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7.2d-37)) .or. (.not. (y <= 1950000000000.0d0))) then
        tmp = x * exp((-y * t))
    else
        tmp = x * exp(((((((-0.5d0) * z) - 1.0d0) * z) - b) * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7.2e-37) || !(y <= 1950000000000.0)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7.2e-37) or not (y <= 1950000000000.0):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp((((((-0.5 * z) - 1.0) * z) - b) * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7.2e-37) || !(y <= 1950000000000.0))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(Float64(Float64(Float64(-0.5 * z) - 1.0) * z) - b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7.2e-37) || ~((y <= 1950000000000.0)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp((((((-0.5 * z) - 1.0) * z) - b) * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7.2e-37], N[Not[LessEqual[y, 1950000000000.0]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[(N[(N[(N[(N[(-0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000014e-37 or 1.95e12 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -7.20000000000000014e-37 < y < 1.95e12

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.1%

      \[\leadsto x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-37} \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(\left(-0.5 \cdot z - 1\right) \cdot z - b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-42} \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6e-42) (not (<= y 1950000000000.0)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6e-42) || !(y <= 1950000000000.0)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp((-b * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6d-42)) .or. (.not. (y <= 1950000000000.0d0))) then
        tmp = x * exp((-y * t))
    else
        tmp = x * exp((-b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6e-42) || !(y <= 1950000000000.0)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6e-42) or not (y <= 1950000000000.0):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6e-42) || !(y <= 1950000000000.0))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6e-42) || ~((y <= 1950000000000.0)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6e-42], N[Not[LessEqual[y, 1950000000000.0]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000054e-42 or 1.95e12 < y

   1. Initial program 95.2%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 62.9

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

   5. Applied rewrites 62.9%

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

   if -6.00000000000000054e-42 < y < 1.95e12

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 85.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-42} \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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((-b * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 96.5

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 62.9%

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

Alternative 8: 34.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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((-z * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 33.3%

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

Reproduce

herbie shell --seed 2024337 
(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))))))