Average Error: 12.5 → 1.1
Time: 8.9s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r1616372 = x;
        double r1616373 = y;
        double r1616374 = z;
        double r1616375 = r1616373 + r1616374;
        double r1616376 = r1616372 * r1616375;
        double r1616377 = r1616376 / r1616374;
        return r1616377;
}
↓
double f(double x, double y, double z) {
        double r1616378 = x;
        double r1616379 = y;
        double r1616380 = z;
        double r1616381 = r1616379 + r1616380;
        double r1616382 = cbrt(r1616381);
        double r1616383 = r1616382 * r1616382;
        double r1616384 = cbrt(r1616380);
        double r1616385 = r1616384 * r1616384;
        double r1616386 = r1616383 / r1616385;
        double r1616387 = r1616378 * r1616386;
        double r1616388 = r1616382 / r1616384;
        double r1616389 = r1616387 * r1616388;
        return r1616389;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.5
Target3.0
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.5

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

  3. Applied *-un-lft-identity12.5

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

  4. Applied times-frac3.2

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

  5. Simplified3.2

    \[\leadsto \color{blue}{x} \cdot \frac{y + z}{z}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt4.5

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

  8. Applied add-cube-cbrt3.8

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

  9. Applied times-frac3.8

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

  10. Applied associate-*r*1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019235 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))