Average Error: 0.4 → 0.2
Time: 6.0s
Precision: 64
\[\frac{x \cdot 100}{x + y}\]
↓
\[\left(\frac{x}{y + x} \cdot \sqrt{100}\right) \cdot \sqrt{100}\]
\frac{x \cdot 100}{x + y}
↓
\left(\frac{x}{y + x} \cdot \sqrt{100}\right) \cdot \sqrt{100}
double f(double x, double y) {
        double r623522 = x;
        double r623523 = 100.0;
        double r623524 = r623522 * r623523;
        double r623525 = y;
        double r623526 = r623522 + r623525;
        double r623527 = r623524 / r623526;
        return r623527;
}
↓
double f(double x, double y) {
        double r623528 = x;
        double r623529 = y;
        double r623530 = r623529 + r623528;
        double r623531 = r623528 / r623530;
        double r623532 = 100.0;
        double r623533 = sqrt(r623532);
        double r623534 = r623531 * r623533;
        double r623535 = r623534 * r623533;
        return r623535;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.2
Herbie0.2
\[\frac{x}{1} \cdot \frac{100}{x + y}\]

Derivation

  1. Initial program 0.4

    \[\frac{x \cdot 100}{x + y}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{x \cdot 100}{y + x}}\]
  3. Using strategy rm

  4. Applied associate-/l*0.2

    \[\leadsto \color{blue}{\frac{x}{\frac{y + x}{100}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{x}{\frac{y + x}{\color{blue}{\sqrt{100} \cdot \sqrt{100}}}}\]

  7. Applied *-un-lft-identity0.2

    \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot \left(y + x\right)}}{\sqrt{100} \cdot \sqrt{100}}}\]

  8. Applied times-frac0.4

    \[\leadsto \frac{x}{\color{blue}{\frac{1}{\sqrt{100}} \cdot \frac{y + x}{\sqrt{100}}}}\]

  9. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{1}{\sqrt{100}} \cdot \frac{y + x}{\sqrt{100}}}\]

  10. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{\sqrt{100}}} \cdot \frac{x}{\frac{y + x}{\sqrt{100}}}}\]

  11. Simplified0.2

    \[\leadsto \color{blue}{\sqrt{100}} \cdot \frac{x}{\frac{y + x}{\sqrt{100}}}\]

  12. Simplified0.2

    \[\leadsto \sqrt{100} \cdot \color{blue}{\left(\sqrt{100} \cdot \frac{x}{y + x}\right)}\]

  13. Final simplification0.2

    \[\leadsto \left(\frac{x}{y + x} \cdot \sqrt{100}\right) \cdot \sqrt{100}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Development.Shake.Progress:message from shake-0.15.5"

  :herbie-target
  (* (/ x 1.0) (/ 100.0 (+ x y)))

  (/ (* x 100.0) (+ x y)))