Average Error: 19.7 → 0.1
Time: 4.0s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}
double f(double x, double y) {
        double r396110 = x;
        double r396111 = y;
        double r396112 = r396110 * r396111;
        double r396113 = r396110 + r396111;
        double r396114 = r396113 * r396113;
        double r396115 = 1.0;
        double r396116 = r396113 + r396115;
        double r396117 = r396114 * r396116;
        double r396118 = r396112 / r396117;
        return r396118;
}
↓
double f(double x, double y) {
        double r396119 = x;
        double r396120 = y;
        double r396121 = r396119 + r396120;
        double r396122 = r396119 / r396121;
        double r396123 = 1.0;
        double r396124 = r396121 + r396123;
        double r396125 = r396120 / r396124;
        double r396126 = r396125 / r396121;
        double r396127 = r396122 * r396126;
        return r396127;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original19.7
Target0.1
Herbie0.1
\[\frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}}\]

Derivation

  1. Initial program 19.7

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

  3. Applied times-frac8.0

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

  5. Applied associate-/r*0.2

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

  7. Applied associate-*l/0.1

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

  9. Applied *-un-lft-identity0.1

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

  10. Applied times-frac0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

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

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1))))