Average Error: 12.3 → 1.8
Time: 16.3s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{y}} + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{y}} + x
double f(double x, double y, double z) {
        double r250466 = x;
        double r250467 = y;
        double r250468 = z;
        double r250469 = r250467 + r250468;
        double r250470 = r250466 * r250469;
        double r250471 = r250470 / r250468;
        return r250471;
}
↓
double f(double x, double y, double z) {
        double r250472 = x;
        double r250473 = cbrt(r250472);
        double r250474 = r250473 * r250473;
        double r250475 = z;
        double r250476 = cbrt(r250475);
        double r250477 = r250476 * r250476;
        double r250478 = r250474 / r250477;
        double r250479 = y;
        double r250480 = r250476 / r250479;
        double r250481 = r250473 / r250480;
        double r250482 = r250478 * r250481;
        double r250483 = r250482 + r250472;
        return r250483;
}

Error

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Bits error versus y

Bits error versus z

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Results

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Target

Original12.3
Target3.2
Herbie1.8
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.3

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

  2. Simplified3.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]

  3. Taylor expanded around 0 4.6

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

  4. Simplified5.2

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

  6. Applied fma-udef5.2

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

  7. Simplified4.6

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

  9. Applied associate-/l*3.1

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

  11. Applied *-un-lft-identity3.1

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

  12. Applied add-cube-cbrt3.5

    \[\leadsto \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot y}} + x\]

  13. Applied times-frac3.6

    \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}}} + x\]

  14. Applied add-cube-cbrt3.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}} + x\]

  15. Applied times-frac1.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{y}}} + x\]

  16. Simplified1.8

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

  17. Final simplification1.8

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{y}} + x\]

Reproduce

herbie shell --seed 2019323 +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))