Average Error: 2.0 → 0.3
Time: 13.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\mathsf{fma}\left(y, \log z - t, \left(\log 1 - \left(\mathsf{fma}\left(\frac{1}{2}, \frac{{z}^{2}}{{1}^{2}}, 1 \cdot z\right) + b\right)\right) \cdot a\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(fma(y, (log(z) - t), ((log(1.0) - (fma(0.5, (pow(z, 2.0) / pow(1.0, 2.0)), (1.0 * z)) + b)) * a))));
}

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

    \[\leadsto x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\color{blue}{\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right)} - b\right)}\]
  3. Using strategy rm

  4. Applied add-exp-log0.5

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020102 +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))))))