Average Error: 0.2 → 0.3
Time: 5.5s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\left(1 - \frac{\frac{\frac{1}{x}}{\sqrt[3]{9} \cdot \sqrt[3]{9}}}{\sqrt[3]{9}}\right) - \frac{y}{3 \cdot \sqrt{x}}
double f(double x, double y) {
        double r511434 = 1.0;
        double r511435 = x;
        double r511436 = 9.0;
        double r511437 = r511435 * r511436;
        double r511438 = r511434 / r511437;
        double r511439 = r511434 - r511438;
        double r511440 = y;
        double r511441 = 3.0;
        double r511442 = sqrt(r511435);
        double r511443 = r511441 * r511442;
        double r511444 = r511440 / r511443;
        double r511445 = r511439 - r511444;
        return r511445;
}
↓
double f(double x, double y) {
        double r511446 = 1.0;
        double r511447 = x;
        double r511448 = r511446 / r511447;
        double r511449 = 9.0;
        double r511450 = cbrt(r511449);
        double r511451 = r511450 * r511450;
        double r511452 = r511448 / r511451;
        double r511453 = r511452 / r511450;
        double r511454 = r511446 - r511453;
        double r511455 = y;
        double r511456 = 3.0;
        double r511457 = sqrt(r511447);
        double r511458 = r511456 * r511457;
        double r511459 = r511455 / r511458;
        double r511460 = r511454 - r511459;
        return r511460;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

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-cube-cbrt0.2

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

  6. Applied associate-/r*0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020089 
(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)))))