Average Error: 12.3 → 1.1
Time: 2.7s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y + z}}{\sqrt[3]{z}} \cdot x\right)\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y + z}}{\sqrt[3]{z}} \cdot x\right)
double f(double x, double y, double z) {
        double r457468 = x;
        double r457469 = y;
        double r457470 = z;
        double r457471 = r457469 + r457470;
        double r457472 = r457468 * r457471;
        double r457473 = r457472 / r457470;
        return r457473;
}
↓
double f(double x, double y, double z) {
        double r457474 = y;
        double r457475 = z;
        double r457476 = r457474 + r457475;
        double r457477 = cbrt(r457476);
        double r457478 = r457477 * r457477;
        double r457479 = cbrt(r457475);
        double r457480 = r457479 * r457479;
        double r457481 = r457478 / r457480;
        double r457482 = r457477 / r457479;
        double r457483 = x;
        double r457484 = r457482 * r457483;
        double r457485 = r457481 * r457484;
        return r457485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.3
Target3.1
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.3

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

  3. Applied associate-/l*3.1

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

  5. Applied clear-num3.3

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z}{y + z}}{x}}}\]
  6. Using strategy rm

  7. Applied div-inv3.3

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

  8. Applied add-sqr-sqrt3.3

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

  9. Applied times-frac3.6

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

  10. Simplified3.6

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

  11. Simplified3.4

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

  13. Applied add-cube-cbrt4.6

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

  14. Applied add-cube-cbrt3.9

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

  15. Applied times-frac3.9

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

  16. Applied associate-*l*1.1

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

  17. Final simplification1.1

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

Reproduce

herbie shell --seed 2020046 
(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))