Average Error: 23.5 → 23.5
Time: 4.1m
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 r2628744 = x;
        double r2628745 = y;
        double r2628746 = r2628744 * r2628745;
        double r2628747 = z;
        double r2628748 = t;
        double r2628749 = a;
        double r2628750 = r2628748 - r2628749;
        double r2628751 = r2628747 * r2628750;
        double r2628752 = r2628746 + r2628751;
        double r2628753 = b;
        double r2628754 = r2628753 - r2628745;
        double r2628755 = r2628747 * r2628754;
        double r2628756 = r2628745 + r2628755;
        double r2628757 = r2628752 / r2628756;
        return r2628757;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r2628758 = x;
        double r2628759 = y;
        double r2628760 = z;
        double r2628761 = t;
        double r2628762 = a;
        double r2628763 = r2628761 - r2628762;
        double r2628764 = r2628760 * r2628763;
        double r2628765 = fma(r2628758, r2628759, r2628764);
        double r2628766 = cbrt(r2628762);
        double r2628767 = -r2628766;
        double r2628768 = r2628766 * r2628766;
        double r2628769 = r2628766 * r2628768;
        double r2628770 = fma(r2628767, r2628768, r2628769);
        double r2628771 = r2628770 * r2628760;
        double r2628772 = r2628765 + r2628771;
        double r2628773 = b;
        double r2628774 = r2628773 - r2628759;
        double r2628775 = r2628760 * r2628774;
        double r2628776 = r2628759 + r2628775;
        double r2628777 = r2628772 / r2628776;
        return r2628777;
}

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)))))