Average Error: 0.4 → 0.2
Time: 9.4s
Precision: 64
\[\frac{x \cdot 100}{x + y}\]
↓
\[\frac{\frac{x}{x + y}}{\frac{1}{100}}\]
\frac{x \cdot 100}{x + y}
↓
\frac{\frac{x}{x + y}}{\frac{1}{100}}
double f(double x, double y) {
        double r856177 = x;
        double r856178 = 100.0;
        double r856179 = r856177 * r856178;
        double r856180 = y;
        double r856181 = r856177 + r856180;
        double r856182 = r856179 / r856181;
        return r856182;
}
↓
double f(double x, double y) {
        double r856183 = x;
        double r856184 = y;
        double r856185 = r856183 + r856184;
        double r856186 = r856183 / r856185;
        double r856187 = 1.0;
        double r856188 = 100.0;
        double r856189 = r856187 / r856188;
        double r856190 = r856186 / r856189;
        return r856190;
}

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 associate-/l*0.2

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

  5. Applied div-inv0.2

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

  6. Applied associate-/r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))