Average Error: 0.1 → 0.1
Time: 14.8s
Precision: binary64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
↓
\[\log t - \mathsf{fma}\left(x, -\log y, y + z\right) \]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\log t - \mathsf{fma}\left(x, -\log y, y + z\right)

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))
↓
(FPCore (x y z t) :precision binary64 (- (log t) (fma x (- (log y)) (+ y z))))
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 log(t) - fma(x, -log(y), (y + z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Applied add-cube-cbrt_binary641.3

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

  4. Taylor expanded in y around inf 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022077 
(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)))