Average Error: 0.4 → 0.2
Time: 3.1s
Precision: 64
\[\frac{x \cdot 100}{x + y}\]
↓
\[100 \cdot \frac{x}{x + y}\]
\frac{x \cdot 100}{x + y}
↓
100 \cdot \frac{x}{x + y}
double f(double x, double y) {
        double r4004 = x;
        double r4005 = 100.0;
        double r4006 = r4004 * r4005;
        double r4007 = y;
        double r4008 = r4004 + r4007;
        double r4009 = r4006 / r4008;
        return r4009;
}
↓
double f(double x, double y) {
        double r4010 = 100.0;
        double r4011 = x;
        double r4012 = y;
        double r4013 = r4011 + r4012;
        double r4014 = r4011 / r4013;
        double r4015 = r4010 * r4014;
        return r4015;
}

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. Using strategy rm

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

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  7. Applied div-inv0.3

    \[\leadsto x \cdot \color{blue}{\left(100 \cdot \frac{1}{x + y}\right)}\]
  8. Using strategy rm

  9. Applied associate-*r*0.5

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

  10. Simplified0.5

    \[\leadsto \color{blue}{\left(100 \cdot x\right)} \cdot \frac{1}{x + y}\]
  11. Using strategy rm

  12. Applied associate-*l*0.3

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +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)))