Average Error: 0.4 → 0.4
Time: 3.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}} - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\sqrt{\frac{1}{x \cdot 9}} \cdot \sqrt{\frac{1}{x \cdot 9}} - 1\right)
double f(double x, double y) {
        double r403371 = 3.0;
        double r403372 = x;
        double r403373 = sqrt(r403372);
        double r403374 = r403371 * r403373;
        double r403375 = y;
        double r403376 = 1.0;
        double r403377 = 9.0;
        double r403378 = r403372 * r403377;
        double r403379 = r403376 / r403378;
        double r403380 = r403375 + r403379;
        double r403381 = r403380 - r403376;
        double r403382 = r403374 * r403381;
        return r403382;
}
↓
double f(double x, double y) {
        double r403383 = 3.0;
        double r403384 = x;
        double r403385 = sqrt(r403384);
        double r403386 = r403383 * r403385;
        double r403387 = y;
        double r403388 = r403386 * r403387;
        double r403389 = 1.0;
        double r403390 = 9.0;
        double r403391 = r403384 * r403390;
        double r403392 = r403389 / r403391;
        double r403393 = sqrt(r403392);
        double r403394 = r403393 * r403393;
        double r403395 = r403394 - r403389;
        double r403396 = r403386 * r403395;
        double r403397 = r403388 + r403396;
        return r403397;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

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

  3. Applied associate--l+0.4

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

  4. Applied distribute-lft-in0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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