Average Error: 19.4 → 0.1
Time: 12.6s
Precision: 64
\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{y}{\frac{x + y}{x}}}{x + y}}{\left(x + y\right) + 1}\]
\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}
↓
\frac{\frac{\frac{y}{\frac{x + y}{x}}}{x + y}}{\left(x + y\right) + 1}
double f(double x, double y) {
        double r513189 = x;
        double r513190 = y;
        double r513191 = r513189 * r513190;
        double r513192 = r513189 + r513190;
        double r513193 = r513192 * r513192;
        double r513194 = 1.0;
        double r513195 = r513192 + r513194;
        double r513196 = r513193 * r513195;
        double r513197 = r513191 / r513196;
        return r513197;
}
↓
double f(double x, double y) {
        double r513198 = y;
        double r513199 = x;
        double r513200 = r513199 + r513198;
        double r513201 = r513200 / r513199;
        double r513202 = r513198 / r513201;
        double r513203 = r513202 / r513200;
        double r513204 = 1.0;
        double r513205 = r513200 + r513204;
        double r513206 = r513203 / r513205;
        return r513206;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

Original19.4
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 19.4

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

  3. Applied times-frac7.6

    \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}}\]
  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-*r/0.2

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

  9. Applied associate-*l/0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +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))))