Average Error: 19.4 → 0.2
Time: 8.9s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
↓
\[\frac{\frac{y}{\left(x + y\right) + 1.0} \cdot \left(\frac{1}{x + y} \cdot x\right)}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}
↓
\frac{\frac{y}{\left(x + y\right) + 1.0} \cdot \left(\frac{1}{x + y} \cdot x\right)}{x + y}
double f(double x, double y) {
        double r7392555 = x;
        double r7392556 = y;
        double r7392557 = r7392555 * r7392556;
        double r7392558 = r7392555 + r7392556;
        double r7392559 = r7392558 * r7392558;
        double r7392560 = 1.0;
        double r7392561 = r7392558 + r7392560;
        double r7392562 = r7392559 * r7392561;
        double r7392563 = r7392557 / r7392562;
        return r7392563;
}
↓
double f(double x, double y) {
        double r7392564 = y;
        double r7392565 = x;
        double r7392566 = r7392565 + r7392564;
        double r7392567 = 1.0;
        double r7392568 = r7392566 + r7392567;
        double r7392569 = r7392564 / r7392568;
        double r7392570 = 1.0;
        double r7392571 = r7392570 / r7392566;
        double r7392572 = r7392571 * r7392565;
        double r7392573 = r7392569 * r7392572;
        double r7392574 = r7392573 / r7392566;
        return r7392574;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.4
Target0.1
Herbie0.2
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.4

    \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac7.8

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity7.8

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}\]

  6. Applied times-frac0.2

    \[\leadsto \color{blue}{\left(\frac{1}{x + y} \cdot \frac{x}{x + y}\right)} \cdot \frac{y}{\left(x + y\right) + 1.0}\]

  7. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \left(\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1.0}\right)}\]
  8. Using strategy rm

  9. Applied associate-*l/7.7

    \[\leadsto \frac{1}{x + y} \cdot \color{blue}{\frac{x \cdot \frac{y}{\left(x + y\right) + 1.0}}{x + y}}\]

  10. Applied associate-*r/7.7

    \[\leadsto \color{blue}{\frac{\frac{1}{x + y} \cdot \left(x \cdot \frac{y}{\left(x + y\right) + 1.0}\right)}{x + y}}\]

  11. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) + 1.0}}}{x + y}\]
  12. Using strategy rm

  13. Applied div-inv0.2

    \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{1}{y + x}\right)} \cdot \frac{y}{\left(y + x\right) + 1.0}}{x + y}\]

  14. Final simplification0.2

    \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1.0} \cdot \left(\frac{1}{x + y} \cdot x\right)}{x + y}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))