Average Error: 2.0 → 0.5
Time: 9.4s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + \mathsf{fma}\left(a, \log 1, -\mathsf{fma}\left(a, b, 1 \cdot \left(a \cdot z\right)\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{y \cdot \left(\log z - t\right) + \mathsf{fma}\left(a, \log 1, -\mathsf{fma}\left(a, b, 1 \cdot \left(a \cdot z\right)\right)\right)}
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)))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp(((y * (log(z) - t)) + fma(a, log(1.0), -fma(a, b, (1.0 * (a * z)))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.0

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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 z)) b))))))