Average Error: 23.5 → 23.5
Time: 3.6m
Precision: 64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z}{y + z \cdot \left(b - y\right)}\]
\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}
↓
\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z}{y + z \cdot \left(b - y\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r2508492 = x;
        double r2508493 = y;
        double r2508494 = r2508492 * r2508493;
        double r2508495 = z;
        double r2508496 = t;
        double r2508497 = a;
        double r2508498 = r2508496 - r2508497;
        double r2508499 = r2508495 * r2508498;
        double r2508500 = r2508494 + r2508499;
        double r2508501 = b;
        double r2508502 = r2508501 - r2508493;
        double r2508503 = r2508495 * r2508502;
        double r2508504 = r2508493 + r2508503;
        double r2508505 = r2508500 / r2508504;
        return r2508505;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2508506 = x;
        double r2508507 = y;
        double r2508508 = z;
        double r2508509 = t;
        double r2508510 = a;
        double r2508511 = r2508509 - r2508510;
        double r2508512 = r2508508 * r2508511;
        double r2508513 = fma(r2508506, r2508507, r2508512);
        double r2508514 = cbrt(r2508510);
        double r2508515 = -r2508514;
        double r2508516 = r2508514 * r2508514;
        double r2508517 = r2508514 * r2508516;
        double r2508518 = fma(r2508515, r2508516, r2508517);
        double r2508519 = r2508518 * r2508508;
        double r2508520 = r2508513 + r2508519;
        double r2508521 = b;
        double r2508522 = r2508521 - r2508507;
        double r2508523 = r2508508 * r2508522;
        double r2508524 = r2508507 + r2508523;
        double r2508525 = r2508520 / r2508524;
        return r2508525;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original23.5
Target18.4
Herbie23.5
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Initial program 23.5

    \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt23.6

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - \color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\right)}{y + z \cdot \left(b - y\right)}\]

  4. Applied add-sqr-sqrt43.9

    \[\leadsto \frac{x \cdot y + z \cdot \left(\color{blue}{\sqrt{t} \cdot \sqrt{t}} - \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right)}{y + z \cdot \left(b - y\right)}\]

  5. Applied prod-diff43.9

    \[\leadsto \frac{x \cdot y + z \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -\sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right)\right)}}{y + z \cdot \left(b - y\right)}\]

  6. Applied distribute-rgt-in43.9

    \[\leadsto \frac{x \cdot y + \color{blue}{\left(\mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -\sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z\right)}}{y + z \cdot \left(b - y\right)}\]

  7. Applied associate-+r+43.9

    \[\leadsto \frac{\color{blue}{\left(x \cdot y + \mathsf{fma}\left(\sqrt{t}, \sqrt{t}, -\sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z}}{y + z \cdot \left(b - y\right)}\]

  8. Simplified23.5

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z}{y + z \cdot \left(b - y\right)}\]

  9. Final simplification23.5

    \[\leadsto \frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{a}, \sqrt[3]{a} \cdot \sqrt[3]{a}, \sqrt[3]{a} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot z}{y + z \cdot \left(b - y\right)}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))