Average Error: 12.1 → 1.0
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right) + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right) + x
double f(double x, double y, double z) {
        double r511259 = x;
        double r511260 = y;
        double r511261 = z;
        double r511262 = r511260 + r511261;
        double r511263 = r511259 * r511262;
        double r511264 = r511263 / r511261;
        return r511264;
}
↓
double f(double x, double y, double z) {
        double r511265 = y;
        double r511266 = cbrt(r511265);
        double r511267 = r511266 * r511266;
        double r511268 = z;
        double r511269 = cbrt(r511268);
        double r511270 = r511269 * r511269;
        double r511271 = r511267 / r511270;
        double r511272 = r511266 / r511269;
        double r511273 = x;
        double r511274 = r511272 * r511273;
        double r511275 = r511271 * r511274;
        double r511276 = r511275 + r511273;
        return r511276;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.1
Target2.9
Herbie1.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.1

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

  2. Simplified3.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef3.2

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

  6. Applied add-cube-cbrt3.6

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

  7. Applied add-cube-cbrt3.7

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

  8. Applied times-frac3.7

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

  9. Applied associate-*l*1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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))