Average Error: 1.6 → 0.7
Time: 14.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{4 + x}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{4 + x}{y} - \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 r1091380 = x;
        double r1091381 = 4.0;
        double r1091382 = r1091380 + r1091381;
        double r1091383 = y;
        double r1091384 = r1091382 / r1091383;
        double r1091385 = r1091380 / r1091383;
        double r1091386 = z;
        double r1091387 = r1091385 * r1091386;
        double r1091388 = r1091384 - r1091387;
        double r1091389 = fabs(r1091388);
        return r1091389;
}
↓
double f(double x, double y, double z) {
        double r1091390 = 4.0;
        double r1091391 = x;
        double r1091392 = r1091390 + r1091391;
        double r1091393 = y;
        double r1091394 = r1091392 / r1091393;
        double r1091395 = cbrt(r1091391);
        double r1091396 = r1091395 * r1091395;
        double r1091397 = cbrt(r1091393);
        double r1091398 = r1091397 * r1091397;
        double r1091399 = r1091396 / r1091398;
        double r1091400 = r1091395 / r1091397;
        double r1091401 = z;
        double r1091402 = r1091400 * r1091401;
        double r1091403 = r1091399 * r1091402;
        double r1091404 = r1091394 - r1091403;
        double r1091405 = fabs(r1091404);
        return r1091405;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Initial program 1.6

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

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

  4. Applied add-cube-cbrt2.0

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

  5. Applied times-frac2.0

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

  6. Applied associate-*l*0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))