Average Error: 1.5 → 0.7
Time: 3.6s
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 r29268 = x;
        double r29269 = 4.0;
        double r29270 = r29268 + r29269;
        double r29271 = y;
        double r29272 = r29270 / r29271;
        double r29273 = r29268 / r29271;
        double r29274 = z;
        double r29275 = r29273 * r29274;
        double r29276 = r29272 - r29275;
        double r29277 = fabs(r29276);
        return r29277;
}
↓
double f(double x, double y, double z) {
        double r29278 = 4.0;
        double r29279 = 1.0;
        double r29280 = y;
        double r29281 = r29279 / r29280;
        double r29282 = x;
        double r29283 = r29282 / r29280;
        double r29284 = fma(r29278, r29281, r29283);
        double r29285 = cbrt(r29282);
        double r29286 = r29285 * r29285;
        double r29287 = cbrt(r29280);
        double r29288 = r29287 * r29287;
        double r29289 = r29286 / r29288;
        double r29290 = r29285 / r29287;
        double r29291 = z;
        double r29292 = r29290 * r29291;
        double r29293 = r29289 * r29292;
        double r29294 = r29284 - r29293;
        double r29295 = fabs(r29294);
        return r29295;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.5

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

  2. Taylor expanded around 0 1.5

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

  3. Simplified1.5

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

  5. Applied add-cube-cbrt1.8

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

  6. Applied add-cube-cbrt1.9

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

  7. Applied times-frac1.9

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

  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 2019353 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))