Average Error: 0.4 → 0.2
Time: 35.3s
Precision: 64
\[\frac{x \cdot 100}{x + y}\]
↓
\[\frac{x}{\frac{y + x}{\sqrt{100}}} \cdot \sqrt{100}\]
\frac{x \cdot 100}{x + y}
↓
\frac{x}{\frac{y + x}{\sqrt{100}}} \cdot \sqrt{100}
double f(double x, double y) {
        double r35187748 = x;
        double r35187749 = 100.0;
        double r35187750 = r35187748 * r35187749;
        double r35187751 = y;
        double r35187752 = r35187748 + r35187751;
        double r35187753 = r35187750 / r35187752;
        return r35187753;
}
↓
double f(double x, double y) {
        double r35187754 = x;
        double r35187755 = y;
        double r35187756 = r35187755 + r35187754;
        double r35187757 = 100.0;
        double r35187758 = sqrt(r35187757);
        double r35187759 = r35187756 / r35187758;
        double r35187760 = r35187754 / r35187759;
        double r35187761 = r35187760 * r35187758;
        return r35187761;
}

Error

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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. Using strategy rm

  3. Applied associate-/l*0.2

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

  5. Applied add-sqr-sqrt0.2

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

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

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

  7. Applied times-frac0.4

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

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

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

  9. Applied times-frac0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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)))