Average Error: 0.4 → 0.2
Time: 2.7s
Precision: 64
\[\frac{x \cdot 100}{x + y}\]
↓
\[\sqrt{100} \cdot \frac{x}{\frac{x + y}{\sqrt{100}}}\]
\frac{x \cdot 100}{x + y}
↓
\sqrt{100} \cdot \frac{x}{\frac{x + y}{\sqrt{100}}}
double f(double x, double y) {
        double r817590 = x;
        double r817591 = 100.0;
        double r817592 = r817590 * r817591;
        double r817593 = y;
        double r817594 = r817590 + r817593;
        double r817595 = r817592 / r817594;
        return r817595;
}
↓
double f(double x, double y) {
        double r817596 = 100.0;
        double r817597 = sqrt(r817596);
        double r817598 = x;
        double r817599 = y;
        double r817600 = r817598 + r817599;
        double r817601 = r817600 / r817597;
        double r817602 = r817598 / r817601;
        double r817603 = r817597 * r817602;
        return r817603;
}

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

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y)
  :name "Development.Shake.Progress:message from shake-0.15.5"
  :precision binary64

  :herbie-target
  (* (/ x 1) (/ 100 (+ x y)))

  (/ (* x 100) (+ x y)))