Average Error: 1.6 → 0.7
Time: 4.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|
double f(double x, double y, double z) {
        double r36743 = x;
        double r36744 = 4.0;
        double r36745 = r36743 + r36744;
        double r36746 = y;
        double r36747 = r36745 / r36746;
        double r36748 = r36743 / r36746;
        double r36749 = z;
        double r36750 = r36748 * r36749;
        double r36751 = r36747 - r36750;
        double r36752 = fabs(r36751);
        return r36752;
}
↓
double f(double x, double y, double z) {
        double r36753 = 4.0;
        double r36754 = 1.0;
        double r36755 = y;
        double r36756 = r36754 / r36755;
        double r36757 = x;
        double r36758 = r36757 / r36755;
        double r36759 = fma(r36753, r36756, r36758);
        double r36760 = cbrt(r36757);
        double r36761 = r36760 * r36760;
        double r36762 = cbrt(r36755);
        double r36763 = r36762 * r36762;
        double r36764 = r36761 / r36763;
        double r36765 = r36760 / r36762;
        double r36766 = z;
        double r36767 = r36765 * r36766;
        double r36768 = r36764 * r36767;
        double r36769 = r36759 - r36768;
        double r36770 = fabs(r36769);
        return r36770;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.6

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

  2. Taylor expanded around 0 1.6

    \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]

  3. Simplified1.6

    \[\leadsto \left|\color{blue}{\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.9

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot z\right|\]

  6. Applied add-cube-cbrt2.0

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \cdot z\right|\]

  7. Applied times-frac2.0

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot z\right|\]

  8. Applied associate-*l*0.7

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)}\right|\]

  9. Final simplification0.7

    \[\leadsto \left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))