Average Error: 0.4 → 0.4
Time: 20.4s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\mathsf{fma}\left(3, y, \frac{0.333333333333333315}{x} - 3\right) \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\mathsf{fma}\left(3, y, \frac{0.333333333333333315}{x} - 3\right) \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 - 1\right)
double f(double x, double y) {
        double r428939 = 3.0;
        double r428940 = x;
        double r428941 = sqrt(r428940);
        double r428942 = r428939 * r428941;
        double r428943 = y;
        double r428944 = 1.0;
        double r428945 = 9.0;
        double r428946 = r428940 * r428945;
        double r428947 = r428944 / r428946;
        double r428948 = r428943 + r428947;
        double r428949 = r428948 - r428944;
        double r428950 = r428942 * r428949;
        return r428950;
}
↓
double f(double x, double y) {
        double r428951 = 3.0;
        double r428952 = y;
        double r428953 = 0.3333333333333333;
        double r428954 = x;
        double r428955 = r428953 / r428954;
        double r428956 = r428955 - r428951;
        double r428957 = fma(r428951, r428952, r428956);
        double r428958 = sqrt(r428954);
        double r428959 = r428957 * r428958;
        double r428960 = r428951 * r428958;
        double r428961 = 1.0;
        double r428962 = r428961 - r428961;
        double r428963 = r428960 * r428962;
        double r428964 = r428959 + r428963;
        return r428964;
}

Error

Bits error versus x

Bits error versus y

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-sqrt16.0

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

  5. Applied prod-diff16.0

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

  6. Applied distribute-lft-in16.0

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left(\left(3 \cdot y + 0.333333333333333315 \cdot \frac{1}{x}\right) - 3\right)} \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 - 1\right)\]

  10. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, y, \frac{0.333333333333333315}{x} - 3\right)} \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 - 1\right)\]

  11. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(3, y, \frac{0.333333333333333315}{x} - 3\right) \cdot \sqrt{x} + \left(3 \cdot \sqrt{x}\right) \cdot \left(1 - 1\right)\]

Reproduce

herbie shell --seed 2020045 +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)))