Average Error: 0.1 → 0.1
Time: 13.5s
Precision: binary64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
↓
\[\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({y}^{0.3333333333333333}\right)\right) - y\right) - z\right) + \log t \]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), x \cdot \log \left({y}^{0.3333333333333333}\right)\right) - y\right) - z\right) + \log t

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  (-
   (- (fma x (* 2.0 (log (cbrt y))) (* x (log (pow y 0.3333333333333333)))) y)
   z)
  (log t)))
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, (2.0 * log(cbrt(y))), (x * log(pow(y, 0.3333333333333333)))) - y) - z) + log(t);
}

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. Applied add-cube-cbrt_binary640.1

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

  3. Applied log-prod_binary640.1

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

  4. Applied distribute-rgt-in_binary640.1

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

  5. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot x\right) - y\right) - z\right) + \log t \]

  6. Simplified0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{x \cdot \log \left(\sqrt[3]{y}\right)}\right) - y\right) - z\right) + \log t \]

  7. Applied fma-def_binary640.1

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

  8. Taylor expanded in y around 0 0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022068 
(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)))