Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\mathsf{fma}\left(\log y, x, \log t - \left(y + z\right)\right) + \mathsf{fma}\left(-\sqrt[3]{y + z}, \sqrt[3]{y + z} \cdot \sqrt[3]{y + z}, \sqrt[3]{y + z} \cdot \left(\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}\right)\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(\log y, x, \log t - \left(y + z\right)\right) + \mathsf{fma}\left(-\sqrt[3]{y + z}, \sqrt[3]{y + z} \cdot \sqrt[3]{y + z}, \sqrt[3]{y + z} \cdot \left(\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}\right)\right)
double code(double x, double y, double z, double t) {
	return ((((x * log(y)) - y) - z) + log(t));
}
↓
double code(double x, double y, double z, double t) {
	return (fma(log(y), x, (log(t) - (y + z))) + fma(-cbrt((y + z)), (cbrt((y + z)) * cbrt((y + z))), (cbrt((y + z)) * (cbrt((y + z)) * cbrt((y + z))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{\log y \cdot x + \left(\log t - \left(y + z\right)\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

    \[\leadsto \log y \cdot x + \left(\log t - \color{blue}{\left(\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}\right) \cdot \sqrt[3]{y + z}}\right)\]

  7. Applied pow10.8

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

  8. Applied log-pow0.8

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

  9. Applied prod-diff0.8

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

  10. Applied associate-+r+0.8

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))