Average Error: 0.4 → 0.4
Time: 25.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\sqrt{x} \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right) \cdot 3 + \sqrt{x} \cdot \left(3 \cdot \left(1 - 1\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\sqrt{x} \cdot \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right)\right) \cdot 3 + \sqrt{x} \cdot \left(3 \cdot \left(1 - 1\right)\right)
double f(double x, double y) {
        double r391096 = 3.0;
        double r391097 = x;
        double r391098 = sqrt(r391097);
        double r391099 = r391096 * r391098;
        double r391100 = y;
        double r391101 = 1.0;
        double r391102 = 9.0;
        double r391103 = r391097 * r391102;
        double r391104 = r391101 / r391103;
        double r391105 = r391100 + r391104;
        double r391106 = r391105 - r391101;
        double r391107 = r391099 * r391106;
        return r391107;
}
↓
double f(double x, double y) {
        double r391108 = x;
        double r391109 = sqrt(r391108);
        double r391110 = y;
        double r391111 = 1.0;
        double r391112 = r391111 / r391108;
        double r391113 = 9.0;
        double r391114 = r391112 / r391113;
        double r391115 = r391110 + r391114;
        double r391116 = r391115 - r391111;
        double r391117 = r391109 * r391116;
        double r391118 = 3.0;
        double r391119 = r391117 * r391118;
        double r391120 = r391111 - r391111;
        double r391121 = r391118 * r391120;
        double r391122 = r391109 * r391121;
        double r391123 = r391119 + r391122;
        return r391123;
}

Error

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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 add-cube-cbrt0.4

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

  4. Applied add-sqr-sqrt15.5

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

  5. Applied prod-diff15.5

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

  6. Applied distribute-lft-in15.5

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  10. Applied associate-/r*0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +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)))