Average Error: 0.0 → 0.0
Time: 13.8s
Precision: 64
\[\frac{x + y}{x - y}\]
↓
\[\sqrt[3]{\frac{y + x}{x - y} \cdot \left(\frac{y + x}{x - y} \cdot \frac{y + x}{x - y}\right)}\]
\frac{x + y}{x - y}
↓
\sqrt[3]{\frac{y + x}{x - y} \cdot \left(\frac{y + x}{x - y} \cdot \frac{y + x}{x - y}\right)}
double f(double x, double y) {
        double r21878430 = x;
        double r21878431 = y;
        double r21878432 = r21878430 + r21878431;
        double r21878433 = r21878430 - r21878431;
        double r21878434 = r21878432 / r21878433;
        return r21878434;
}
↓
double f(double x, double y) {
        double r21878435 = y;
        double r21878436 = x;
        double r21878437 = r21878435 + r21878436;
        double r21878438 = r21878436 - r21878435;
        double r21878439 = r21878437 / r21878438;
        double r21878440 = r21878439 * r21878439;
        double r21878441 = r21878439 * r21878440;
        double r21878442 = cbrt(r21878441);
        return r21878442;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.0
Target0.0
Herbie0.0
\[\frac{1}{\frac{x}{x + y} - \frac{y}{x + y}}\]

Derivation

  1. Initial program 0.0

    \[\frac{x + y}{x - y}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube40.5

    \[\leadsto \frac{x + y}{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}}\]

  4. Applied add-cbrt-cube40.7

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}\]

  5. Applied cbrt-undiv40.7

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}}}\]

  6. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{\left(\frac{x + y}{x - y} \cdot \frac{x + y}{x - y}\right) \cdot \frac{x + y}{x - y}}}\]

  7. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{y + x}{x - y} \cdot \left(\frac{y + x}{x - y} \cdot \frac{y + x}{x - y}\right)}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"

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

  (/ (+ x y) (- x y)))