Average Error: 0.1 → 0.1
Time: 31.0s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)
double f(double x, double y, double z) {
        double r10928493 = x;
        double r10928494 = 0.5;
        double r10928495 = r10928493 * r10928494;
        double r10928496 = y;
        double r10928497 = 1.0;
        double r10928498 = z;
        double r10928499 = r10928497 - r10928498;
        double r10928500 = log(r10928498);
        double r10928501 = r10928499 + r10928500;
        double r10928502 = r10928496 * r10928501;
        double r10928503 = r10928495 + r10928502;
        return r10928503;
}
↓
double f(double x, double y, double z) {
        double r10928504 = x;
        double r10928505 = 0.5;
        double r10928506 = 1.0;
        double r10928507 = -2.0;
        double r10928508 = z;
        double r10928509 = cbrt(r10928508);
        double r10928510 = log(r10928509);
        double r10928511 = fma(r10928507, r10928510, r10928508);
        double r10928512 = r10928511 - r10928510;
        double r10928513 = r10928506 - r10928512;
        double r10928514 = y;
        double r10928515 = r10928513 * r10928514;
        double r10928516 = fma(r10928504, r10928505, r10928515);
        return r10928516;
}

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

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

  4. Applied add-cube-cbrt0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(z - \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\right) \cdot y\right)\]

  5. Applied log-prod0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(z - \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\right) \cdot y\right)\]

  6. Applied associate--r+0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \color{blue}{\left(\left(z - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) - \log \left(\sqrt[3]{z}\right)\right)}\right) \cdot y\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(\color{blue}{\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right)} - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"

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

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