Average Error: 0.0 → 0.2
Time: 45.6s
Precision: 64
\[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
↓
\[x - \frac{\frac{y}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0} \cdot \sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}\]
x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}
↓
x - \frac{\frac{y}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0} \cdot \sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}
double f(double x, double y) {
        double r11292681 = x;
        double r11292682 = y;
        double r11292683 = 1.0;
        double r11292684 = r11292681 * r11292682;
        double r11292685 = 2.0;
        double r11292686 = r11292684 / r11292685;
        double r11292687 = r11292683 + r11292686;
        double r11292688 = r11292682 / r11292687;
        double r11292689 = r11292681 - r11292688;
        return r11292689;
}
↓
double f(double x, double y) {
        double r11292690 = x;
        double r11292691 = y;
        double r11292692 = r11292690 * r11292691;
        double r11292693 = 2.0;
        double r11292694 = r11292692 / r11292693;
        double r11292695 = 1.0;
        double r11292696 = r11292694 + r11292695;
        double r11292697 = cbrt(r11292696);
        double r11292698 = r11292697 * r11292697;
        double r11292699 = r11292691 / r11292698;
        double r11292700 = r11292699 / r11292697;
        double r11292701 = r11292690 - r11292700;
        return r11292701;
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[x - \frac{y}{1.0 + \frac{x \cdot y}{2.0}}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.2

    \[\leadsto x - \frac{y}{\color{blue}{\left(\sqrt[3]{1.0 + \frac{x \cdot y}{2.0}} \cdot \sqrt[3]{1.0 + \frac{x \cdot y}{2.0}}\right) \cdot \sqrt[3]{1.0 + \frac{x \cdot y}{2.0}}}}\]

  4. Applied associate-/r*0.2

    \[\leadsto x - \color{blue}{\frac{\frac{y}{\sqrt[3]{1.0 + \frac{x \cdot y}{2.0}} \cdot \sqrt[3]{1.0 + \frac{x \cdot y}{2.0}}}}{\sqrt[3]{1.0 + \frac{x \cdot y}{2.0}}}}\]

  5. Final simplification0.2

    \[\leadsto x - \frac{\frac{y}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0} \cdot \sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}}{\sqrt[3]{\frac{x \cdot y}{2.0} + 1.0}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, B"
  (- x (/ y (+ 1.0 (/ (* x y) 2.0)))))