Average Error: 0.1 → 0.1
Time: 11.2s
Precision: binary64
\[\left(x \cdot \log y - z\right) - y \]
↓
\[\left(\mathsf{fma}\left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right), x, x \cdot \log \left({\left(\frac{1}{y}\right)}^{-0.3333333333333333}\right)\right) - z\right) - y \]
\left(x \cdot \log y - z\right) - y
↓
\left(\mathsf{fma}\left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right), x, x \cdot \log \left({\left(\frac{1}{y}\right)}^{-0.3333333333333333}\right)\right) - z\right) - y

(FPCore (x y z) :precision binary64 (- (- (* x (log y)) z) y))
↓
(FPCore (x y z)
 :precision binary64
 (-
  (-
   (fma
    (log (pow (cbrt y) 2.0))
    x
    (* x (log (pow (/ 1.0 y) -0.3333333333333333))))
   z)
  y))
double code(double x, double y, double z) {
	return ((x * log(y)) - z) - y;
}
↓
double code(double x, double y, double z) {
	return (fma(log(pow(cbrt(y), 2.0)), x, (x * log(pow((1.0 / y), -0.3333333333333333)))) - z) - y;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Applied add-cube-cbrt_binary640.1

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

  3. Applied log-prod_binary640.1

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

  4. Applied distribute-rgt-in_binary640.1

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

  5. Applied fma-def_binary640.1

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

  6. Taylor expanded in y around inf 0.1

    \[\leadsto \left(\mathsf{fma}\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right), x, \color{blue}{\log \left({\left(\frac{1}{y}\right)}^{-0.3333333333333333}\right) \cdot x}\right) - z\right) - y \]

  7. Applied pow2_binary640.1

    \[\leadsto \left(\mathsf{fma}\left(\log \color{blue}{\left({\left(\sqrt[3]{y}\right)}^{2}\right)}, x, \log \left({\left(\frac{1}{y}\right)}^{-0.3333333333333333}\right) \cdot x\right) - z\right) - y \]

  8. Final simplification0.1

    \[\leadsto \left(\mathsf{fma}\left(\log \left({\left(\sqrt[3]{y}\right)}^{2}\right), x, x \cdot \log \left({\left(\frac{1}{y}\right)}^{-0.3333333333333333}\right)\right) - z\right) - y \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))