Average Error: 0.1 → 0.1
Time: 18.1s
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 r4046331 = 1.0;
        double r4046332 = x;
        double r4046333 = r4046331 - r4046332;
        double r4046334 = y;
        double r4046335 = sqrt(r4046332);
        double r4046336 = r4046334 * r4046335;
        double r4046337 = r4046333 + r4046336;
        return r4046337;
}
↓
double f(double x, double y) {
        double r4046338 = y;
        double r4046339 = x;
        double r4046340 = sqrt(r4046339);
        double r4046341 = 1.0;
        double r4046342 = r4046341 - r4046339;
        double r4046343 = fma(r4046338, r4046340, r4046342);
        double r4046344 = r4046339 - r4046339;
        double r4046345 = r4046343 + r4046344;
        return r4046345;
}

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-sqrt15.0

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

  6. Applied prod-diff15.0

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

  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 2019192 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))