Average Error: 0.4 → 0.4
Time: 28.5s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{0.1111111111111111}{\sqrt{x}}, y - 1\right)\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{0.1111111111111111}{\sqrt{x}}, y - 1\right)\right)
double f(double x, double y) {
        double r528403 = 3.0;
        double r528404 = x;
        double r528405 = sqrt(r528404);
        double r528406 = r528403 * r528405;
        double r528407 = y;
        double r528408 = 1.0;
        double r528409 = 9.0;
        double r528410 = r528404 * r528409;
        double r528411 = r528408 / r528410;
        double r528412 = r528407 + r528411;
        double r528413 = r528412 - r528408;
        double r528414 = r528406 * r528413;
        return r528414;
}
↓
double f(double x, double y) {
        double r528415 = 3.0;
        double r528416 = x;
        double r528417 = sqrt(r528416);
        double r528418 = 1.0;
        double r528419 = r528418 / r528417;
        double r528420 = 0.1111111111111111;
        double r528421 = r528420 / r528417;
        double r528422 = y;
        double r528423 = 1.0;
        double r528424 = r528422 - r528423;
        double r528425 = fma(r528419, r528421, r528424);
        double r528426 = r528417 * r528425;
        double r528427 = r528415 * r528426;
        return r528427;
}

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. Taylor expanded around 0 0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right)} - 1\right)\]

  3. Simplified0.4

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\frac{0.1111111111111111}{x} + y\right)} - 1\right)\]
  4. Using strategy rm

  5. Applied associate-*l*0.4

    \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(\frac{0.1111111111111111}{x} + y\right) - 1\right)\right)}\]

  6. Simplified0.4

    \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right)\right)}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \left(y - 1\right)\right)\right)\]

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

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\frac{\color{blue}{1 \cdot 0.1111111111111111}}{\sqrt{x} \cdot \sqrt{x}} + \left(y - 1\right)\right)\right)\]

  10. Applied times-frac0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{0.1111111111111111}{\sqrt{x}}} + \left(y - 1\right)\right)\right)\]

  11. Applied fma-def0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{0.1111111111111111}{\sqrt{x}}, y - 1\right)}\right)\]

  12. Final simplification0.4

    \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{x}}, \frac{0.1111111111111111}{\sqrt{x}}, y - 1\right)\right)\]

Reproduce

herbie shell --seed 2020042 +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)))