Average Error: 0.1 → 0.1
Time: 18.2s
Precision: 64
\[\left(1 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) + \left(x - x\right)\]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) + \left(x - x\right)
double f(double x, double y) {
        double r3083827 = 1.0;
        double r3083828 = x;
        double r3083829 = r3083827 - r3083828;
        double r3083830 = y;
        double r3083831 = sqrt(r3083828);
        double r3083832 = r3083830 * r3083831;
        double r3083833 = r3083829 + r3083832;
        return r3083833;
}
↓
double f(double x, double y) {
        double r3083834 = y;
        double r3083835 = x;
        double r3083836 = sqrt(r3083835);
        double r3083837 = 1.0;
        double r3083838 = r3083837 - r3083835;
        double r3083839 = fma(r3083834, r3083836, r3083838);
        double r3083840 = r3083835 - r3083835;
        double r3083841 = r3083839 + r3083840;
        return r3083841;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.5

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

  5. Applied add-sqr-sqrt15.2

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

  6. Applied prod-diff15.2

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) + \left(x - x\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))