Average Error: 1.7 → 0.2
Time: 10.9s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\mathsf{fma}\left(y, \log z - t, \mathsf{fma}\left(\sqrt{\log 1} + \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -b \cdot 1\right) \cdot a\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}\]
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, \mathsf{fma}\left(\sqrt{\log 1} + \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, \sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z}, -b \cdot 1\right) \cdot a\right) + a \cdot \mathsf{fma}\left(-b, 1, b \cdot 1\right)}
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) exp(((double) (((double) (y * ((double) (((double) log(z)) - t)))) + ((double) (a * ((double) (((double) log(((double) (1.0 - z)))) - b))))))))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) exp(((double) (((double) fma(y, ((double) (((double) log(z)) - t)), ((double) (((double) fma(((double) (((double) sqrt(((double) log(1.0)))) + ((double) sqrt(((double) (((double) (0.5 * ((double) (((double) pow(z, 2.0)) / ((double) pow(1.0, 2.0)))))) + ((double) (1.0 * z)))))))), ((double) (((double) sqrt(((double) log(1.0)))) - ((double) sqrt(((double) (((double) (0.5 * ((double) (((double) pow(z, 2.0)) / ((double) pow(1.0, 2.0)))))) + ((double) (1.0 * z)))))))), ((double) -(((double) (b * 1.0)))))) * a)))) + ((double) (a * ((double) fma(((double) -(b)), 1.0, ((double) (b * 1.0))))))))))));
}

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 1.7

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

  2. Taylor expanded around 0 0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied difference-of-squares0.4

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

  8. Applied prod-diff0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Applied associate-+r+0.4

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2020114 +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))))))