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

Percentage Accurate: 96.8% → 99.7%

Time: 14.6s

Alternatives: 9

Speedup: 1.5×

• 41.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 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. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 2: 33.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0.998:\\ \;\;\;\;\frac{\left(-t\right) \cdot \left(x \cdot y\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t \cdot \left(x \cdot y\right), x\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b)))) 0.998)
   (/ (* (- t) (* x y)) 1.0)
   (/ (fma -1.0 (* t (* x y)) x) 1.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (exp(((y * (log(z) - t)) + (a * (log((1.0 - z)) - b)))) <= 0.998) {
		tmp = (-t * (x * y)) / 1.0;
	} else {
		tmp = fma(-1.0, (t * (x * y)), x) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (exp(Float64(Float64(y * Float64(log(z) - t)) + Float64(a * Float64(log(Float64(1.0 - z)) - b)))) <= 0.998)
		tmp = Float64(Float64(Float64(-t) * Float64(x * y)) / 1.0);
	else
		tmp = Float64(fma(-1.0, Float64(t * Float64(x * y)), x) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[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], 0.998], N[(N[((-t) * N[(x * y), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(-1.0 * N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision]]

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)))) < 0.998

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

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

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. exp-prod N/A

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

   7. Applied rewrites 38.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 3.0%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 19.3%

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

   if 0.998 < (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 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

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

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. exp-prod N/A

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

   7. Applied rewrites 64.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(-1, t \cdot \left(x \cdot y\right), x\right)}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \leq 0.998:\\ \;\;\;\;\frac{\left(-t\right) \cdot \left(x \cdot y\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t \cdot \left(x \cdot y\right), x\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+135} \lor \neg \left(t \leq 6.6 \cdot 10^{+139}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.5e+135) (not (<= t 6.6e+139)))
   (* x (exp (* (- y) t)))
   (* x (exp (fma (- b) a (* (log z) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -1.5e+135) || !(t <= 6.6e+139)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -1.5e+135) || !(t <= 6.6e+139))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -1.5e+135], N[Not[LessEqual[t, 6.6e+139]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5e135 or 6.6000000000000003e139 < t

   1. Initial program 97.1%

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

   3. Taylor expanded in 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 88.8

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

   5. Applied rewrites 88.8%

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

   if -1.5e135 < t < 6.6000000000000003e139

   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 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 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+135} \lor \neg \left(t \leq 6.6 \cdot 10^{+139}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-47} \lor \neg \left(y \leq 1.3 \cdot 10^{-61}\right):\\ \;\;\;\;e^{y \cdot \left(\log z - t\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -8.6e-47) (not (<= y 1.3e-61)))
   (* (exp (* y (- (log z) t))) x)
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -8.6e-47) || !(y <= 1.3e-61)) {
		tmp = exp((y * (log(z) - t))) * x;
	} 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 <= (-8.6d-47)) .or. (.not. (y <= 1.3d-61))) then
        tmp = exp((y * (log(z) - t))) * x
    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 <= -8.6e-47) || !(y <= 1.3e-61)) {
		tmp = Math.exp((y * (Math.log(z) - t))) * x;
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -8.6e-47) or not (y <= 1.3e-61):
		tmp = math.exp((y * (math.log(z) - t))) * x
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -8.6e-47) || !(y <= 1.3e-61))
		tmp = Float64(exp(Float64(y * Float64(log(z) - t))) * x);
	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 <= -8.6e-47) || ~((y <= 1.3e-61)))
		tmp = exp((y * (log(z) - t))) * x;
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -8.6e-47], N[Not[LessEqual[y, 1.3e-61]], $MachinePrecision]], N[(N[Exp[N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5999999999999995e-47 or 1.30000000000000005e-61 < y

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

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

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 83.6

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

   7. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 83.6%

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

   10. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-log.f64 83.6

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

   12. Applied rewrites 83.6%

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

   if -8.5999999999999995e-47 < y < 1.30000000000000005e-61

   1. Initial program 97.1%

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

   3. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-47} \lor \neg \left(y \leq 1.3 \cdot 10^{-61}\right):\\ \;\;\;\;e^{y \cdot \left(\log z - t\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% 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 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 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 97.3

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

5. Applied rewrites 97.3%

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

Alternative 6: 70.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+138} \lor \neg \left(t \leq 9.2 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -1.25e+138) (not (<= t 9.2e-71)))
   (* 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 ((t <= -1.25e+138) || !(t <= 9.2e-71)) {
		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 ((t <= (-1.25d+138)) .or. (.not. (t <= 9.2d-71))) 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 ((t <= -1.25e+138) || !(t <= 9.2e-71)) {
		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 (t <= -1.25e+138) or not (t <= 9.2e-71):
		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 ((t <= -1.25e+138) || !(t <= 9.2e-71))
		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 ((t <= -1.25e+138) || ~((t <= 9.2e-71)))
		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[t, -1.25e+138], N[Not[LessEqual[t, 9.2e-71]], $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 t < -1.25000000000000004e138 or 9.1999999999999994e-71 < 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-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 82.4

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

   5. Applied rewrites 82.4%

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

   if -1.25000000000000004e138 < t < 9.1999999999999994e-71

   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 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. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+138} \lor \neg \left(t \leq 9.2 \cdot 10^{-71}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, t \cdot \left(x \cdot y\right), x\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-z\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.8e+64)
   (/ (fma -1.0 (* t (* x y)) x) 1.0)
   (* x (exp (* (- z) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.8e+64) {
		tmp = fma(-1.0, (t * (x * y)), x) / 1.0;
	} else {
		tmp = x * exp((-z * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.8e+64)
		tmp = Float64(fma(-1.0, Float64(t * Float64(x * y)), x) / 1.0);
	else
		tmp = Float64(x * exp(Float64(Float64(-z) * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.8e+64], N[(N[(-1.0 * N[(t * N[(x * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / 1.0), $MachinePrecision], N[(x * N[Exp[N[((-z) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999999e64

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

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

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. exp-prod N/A

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 34.1%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 27.9%

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

   if -4.79999999999999999e64 < y

   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

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right) + \left(-1 \cdot \left(a \cdot z\right) + y \cdot \left(\log z - t\right)\right)}} \]
   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. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-out N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 41.8%

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

Alternative 8: 59.3% 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 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 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. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 60.8

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

5. Applied rewrites 60.8%

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

Alternative 9: 17.6% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-t\right) \cdot \left(x \cdot y\right)}{1} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ (* (- t) (* x y)) 1.0))

C

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

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 = (-t * (x * y)) / 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (-t * (x * y)) / 1.0;
}

Python

def code(x, y, z, t, a, b):
	return (-t * (x * y)) / 1.0

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[((-t) * N[(x * y), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{x \cdot 1}{e^{-\mathsf{fma}\left(\mathsf{log1p}\left(-z\right) - b, a, \left(\log z - t\right) \cdot y\right)}}} \]

5. Taylor expanded in y around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r* N/A

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

   13. exp-prod N/A

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

7. Applied rewrites 54.3%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 32.1%

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

11. Taylor expanded in t around inf

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

13. Applied rewrites 16.5%

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

14. Final simplification 16.5%

    \[\leadsto \frac{\left(-t\right) \cdot \left(x \cdot y\right)}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))))))