Average Error: 0.1 → 0.1
Time: 6.3s
Precision: binary64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
↓
\[\mathsf{fma}\left(y, \log z - z, \mathsf{fma}\left(0.5, x, y\right)\right) \]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(y, \log z - z, \mathsf{fma}\left(0.5, x, y\right)\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right) \]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

  2. Simplified0.1

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

  3. Applied fma-udef_binary640.1

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

  4. Taylor expanded in x around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))