Average Error: 0.1 → 0.1
Time: 16.9s
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 r4280253 = 1.0;
        double r4280254 = x;
        double r4280255 = r4280253 - r4280254;
        double r4280256 = y;
        double r4280257 = sqrt(r4280254);
        double r4280258 = r4280256 * r4280257;
        double r4280259 = r4280255 + r4280258;
        return r4280259;
}
↓
double f(double x, double y) {
        double r4280260 = y;
        double r4280261 = x;
        double r4280262 = sqrt(r4280261);
        double r4280263 = 1.0;
        double r4280264 = r4280263 - r4280261;
        double r4280265 = fma(r4280260, r4280262, r4280264);
        double r4280266 = r4280261 - r4280261;
        double r4280267 = r4280265 + r4280266;
        return r4280267;
}

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-sqrt15.3

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

  6. Applied prod-diff15.3

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

  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 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))