Average Error: 0.1 → 0.1
Time: 4.2s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(y, 1 - z, \mathsf{fma}\left(\log z, y, 0.5 \cdot x\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(y, 1 - z, \mathsf{fma}\left(\log z, y, 0.5 \cdot x\right)\right)
double f(double x, double y, double z) {
        double r246573 = x;
        double r246574 = 0.5;
        double r246575 = r246573 * r246574;
        double r246576 = y;
        double r246577 = 1.0;
        double r246578 = z;
        double r246579 = r246577 - r246578;
        double r246580 = log(r246578);
        double r246581 = r246579 + r246580;
        double r246582 = r246576 * r246581;
        double r246583 = r246575 + r246582;
        return r246583;
}
↓
double f(double x, double y, double z) {
        double r246584 = y;
        double r246585 = 1.0;
        double r246586 = z;
        double r246587 = r246585 - r246586;
        double r246588 = log(r246586);
        double r246589 = 0.5;
        double r246590 = x;
        double r246591 = r246589 * r246590;
        double r246592 = fma(r246588, r246584, r246591);
        double r246593 = fma(r246584, r246587, r246592);
        return r246593;
}

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. Using strategy rm

  3. Applied +-commutative0.1

    \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(\log z + \left(1 - z\right)\right)}\]
  4. Using strategy rm

  5. Applied distribute-rgt-in0.1

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

  6. Applied associate-+r+0.1

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

  7. Simplified0.1

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

  8. Taylor expanded around 0 0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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 z) (log z)))))