Average Error: 0.1 → 0.1
Time: 8.5s
Precision: binary64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
↓
\[\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z + 1, -y \cdot z\right)\right) \]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z + 1, -y \cdot z\right)\right)

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

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. Taylor expanded in z around 0 0.1

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

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

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

  5. Applied distribute-rgt-out_binary640.1

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

  6. Applied fma-neg_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022097 
(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)))))