Average Error: 2.0 → 1.3
Time: 13.2s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{x \cdot \left(\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right)}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{x \cdot \left(\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right)}{y}
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((x * (sqrt((pow((1.0 / a), 1.0) / exp(fma(y, log((1.0 / z)), fma(log((1.0 / a)), t, b))))) * sqrt((pow((1.0 / a), 1.0) / exp(fma(y, log((1.0 / z)), fma(log((1.0 / a)), t, b))))))) / y);
}

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

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

  3. Applied add-cube-cbrt2.3

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

  4. Applied associate-*r*2.3

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

  6. Applied add-sqr-sqrt2.3

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

  7. Taylor expanded around inf 2.2

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

  8. Simplified2.2

    \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}} \cdot \sqrt{e^{\left(y \cdot \log z + \left(\left(t - 1\right) \cdot \left(\sqrt[3]{\log a} \cdot \sqrt[3]{\log a}\right)\right) \cdot \sqrt[3]{\log a}\right) - b}}\right)}{y}\]

  9. Taylor expanded around inf 1.9

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

  10. Simplified1.3

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

  11. Final simplification1.3

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

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))