Average Error: 0.1 → 0.1
Time: 5.6s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y \cdot \left(1 - z\right)\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y \cdot \left(1 - z\right)\right)\right)
double code(double x, double y, double z) {
	return ((x * 0.5) + (y * ((1.0 - z) + log(z))));
}
↓
double code(double x, double y, double z) {
	return fma(x, 0.5, fma(y, log(z), (y * (1.0 - z))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, y \cdot \left(\left(1 - z\right) + \log z\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(\left(1 - z\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\right)\]

  5. Applied log-prod0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(\left(1 - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\right)\]

  6. Applied associate-+r+0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \color{blue}{\left(\left(\left(1 - z\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right)} + \log \left(\sqrt[3]{z}\right)\right)\right)\]
  8. Using strategy rm

  9. Applied distribute-lft-in0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{y \cdot \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + y \cdot \log \left(\sqrt[3]{z}\right)}\right)\]

  10. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + \color{blue}{\log \left({z}^{\frac{1}{3}}\right) \cdot y}\right)\]

  11. Taylor expanded around 0 0.2

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\left(3 \cdot \left(\log \left({z}^{\frac{1}{3}}\right) \cdot y\right) + 1 \cdot y\right) - z \cdot y}\right)\]

  12. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\mathsf{fma}\left(y, \log z, y \cdot \left(1 - z\right)\right)}\right)\]

  13. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z, y \cdot \left(1 - z\right)\right)\right)\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))