Average Error: 0.1 → 0.1
Time: 7.9s
Precision: binary64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
↓
\[\mathsf{fma}\left(y, \left(1 + \log z\right) - z, 0.5 \cdot x\right) \]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(y, \left(1 + \log z\right) - z, 0.5 \cdot x\right)

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

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. Taylor expanded in x around 0 0.1

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

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\log z + 1\right) - z, 0.5 \cdot x\right)} \]

  4. Applied *-un-lft-identity_binary640.1

    \[\leadsto \mathsf{fma}\left(y, \left(\log \color{blue}{\left(1 \cdot z\right)} + 1\right) - z, 0.5 \cdot x\right) \]

  5. Applied log-prod_binary640.1

    \[\leadsto \mathsf{fma}\left(y, \left(\color{blue}{\left(\log 1 + \log z\right)} + 1\right) - z, 0.5 \cdot x\right) \]

  6. Applied associate-+l+_binary640.1

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\log 1 + \left(\log z + 1\right)\right)} - z, 0.5 \cdot x\right) \]

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \left(1 + \log z\right) - z, 0.5 \cdot x\right) \]

Reproduce

herbie shell --seed 2021329 
(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)))))