Average Error: 0.2 → 0.2
Time: 22.3s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) + \frac{-1}{9 \cdot x} \cdot 1\right) + \frac{\frac{1}{x}}{9} \cdot \left(\left(-1\right) + 1\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(\left(1 - \frac{\frac{y}{\sqrt{x}}}{3}\right) + \frac{-1}{9 \cdot x} \cdot 1\right) + \frac{\frac{1}{x}}{9} \cdot \left(\left(-1\right) + 1\right)
double f(double x, double y) {
        double r280075 = 1.0;
        double r280076 = x;
        double r280077 = 9.0;
        double r280078 = r280076 * r280077;
        double r280079 = r280075 / r280078;
        double r280080 = r280075 - r280079;
        double r280081 = y;
        double r280082 = 3.0;
        double r280083 = sqrt(r280076);
        double r280084 = r280082 * r280083;
        double r280085 = r280081 / r280084;
        double r280086 = r280080 - r280085;
        return r280086;
}
↓
double f(double x, double y) {
        double r280087 = 1.0;
        double r280088 = y;
        double r280089 = x;
        double r280090 = sqrt(r280089);
        double r280091 = r280088 / r280090;
        double r280092 = 3.0;
        double r280093 = r280091 / r280092;
        double r280094 = r280087 - r280093;
        double r280095 = -1.0;
        double r280096 = 9.0;
        double r280097 = r280096 * r280089;
        double r280098 = r280095 / r280097;
        double r280099 = r280098 * r280087;
        double r280100 = r280094 + r280099;
        double r280101 = 1.0;
        double r280102 = r280101 / r280089;
        double r280103 = r280102 / r280096;
        double r280104 = -r280087;
        double r280105 = r280104 + r280087;
        double r280106 = r280103 * r280105;
        double r280107 = r280100 + r280106;
        return r280107;
}

Error

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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 *-un-lft-identity0.2

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

  4. Applied *-un-lft-identity0.2

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

  5. Applied distribute-lft-out--0.2

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

  6. Simplified0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied div-inv0.2

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

  10. Applied times-frac0.2

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

  11. Applied add-sqr-sqrt15.4

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

  12. Applied prod-diff15.4

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

  13. Simplified0.2

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

  14. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 +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)))))