Average Error: 0.2 → 0.2
Time: 6.0s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r363113 = 1.0;
        double r363114 = x;
        double r363115 = 9.0;
        double r363116 = r363114 * r363115;
        double r363117 = r363113 / r363116;
        double r363118 = r363113 - r363117;
        double r363119 = y;
        double r363120 = 3.0;
        double r363121 = sqrt(r363114);
        double r363122 = r363120 * r363121;
        double r363123 = r363119 / r363122;
        double r363124 = r363118 - r363123;
        return r363124;
}
↓
double f(double x, double y) {
        double r363125 = 1.0;
        double r363126 = x;
        double r363127 = r363125 / r363126;
        double r363128 = 9.0;
        double r363129 = sqrt(r363128);
        double r363130 = r363127 / r363129;
        double r363131 = r363130 / r363129;
        double r363132 = r363125 - r363131;
        double r363133 = y;
        double r363134 = 3.0;
        double r363135 = sqrt(r363126);
        double r363136 = r363134 * r363135;
        double r363137 = r363133 / r363136;
        double r363138 = r363132 - r363137;
        return r363138;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

  1. Initial program 0.2

    \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.2

    \[\leadsto \left(1 - \frac{\frac{1}{x}}{\color{blue}{\sqrt{9} \cdot \sqrt{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  6. Applied associate-/r*0.2

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

  7. Final simplification0.2

    \[\leadsto \left(1 - \frac{\frac{\frac{1}{x}}{\sqrt{9}}}{\sqrt{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))