Average Error: 9.3 → 0.4
Time: 17.1s
Precision: binary64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
↓
\[\left(\log y \cdot x - \mathsf{fma}\left(y, z, \left(z \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right) - t \]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log y \cdot x - \mathsf{fma}\left(y, z, \left(z \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right) - t

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))
↓
(FPCore (x y z t)
 :precision binary64
 (- (- (* (log y) x) (fma y z (* (* z (* y y)) 0.5))) t))
double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log(1.0 - y))) - t;
}
↓
double code(double x, double y, double z, double t) {
	return ((log(y) * x) - fma(y, z, ((z * (y * y)) * 0.5))) - t;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \]

Derivation

  1. Initial program 9.3

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, z \cdot \mathsf{log1p}\left(-y\right)\right) - t} \]

  3. Applied add-cube-cbrt_binary640.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x, \log y, z \cdot \mathsf{log1p}\left(-y\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, \log y, z \cdot \mathsf{log1p}\left(-y\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, \log y, z \cdot \mathsf{log1p}\left(-y\right)\right)}} - t \]

  4. Taylor expanded in y around 0 0.4

    \[\leadsto \color{blue}{\left(\log y \cdot x - \left(0.5 \cdot \left({y}^{2} \cdot z\right) + y \cdot z\right)\right)} - t \]

  5. Simplified0.4

    \[\leadsto \color{blue}{\left(\log y \cdot x - \mathsf{fma}\left(y, z, \left(z \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right)} - t \]

  6. Final simplification0.4

    \[\leadsto \left(\log y \cdot x - \mathsf{fma}\left(y, z, \left(z \cdot \left(y \cdot y\right)\right) \cdot 0.5\right)\right) - t \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))