Average Error: 0.1 → 0.1
Time: 5.8s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\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(x, log(y), log(t)) - (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. Using strategy rm

  3. Applied flip3-+45.4

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

  4. Simplified45.4

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

  6. Applied clear-num45.4

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

  8. Applied sum-cubes45.4

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

  9. Applied associate-/r*37.2

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

  10. Simplified0.2

    \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t}}\]
  11. Using strategy rm

  12. Applied div-inv0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020071 +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)))