Average Error: 0.1 → 0.1
Time: 17.0s
Precision: 64
\[\left(1.0 - x\right) + y \cdot \sqrt{x}\]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1.0 - x\right) + \mathsf{fma}\left(x, -1, x\right)\]
\left(1.0 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1.0 - x\right) + \mathsf{fma}\left(x, -1, x\right)
double f(double x, double y) {
        double r3513617 = 1.0;
        double r3513618 = x;
        double r3513619 = r3513617 - r3513618;
        double r3513620 = y;
        double r3513621 = sqrt(r3513618);
        double r3513622 = r3513620 * r3513621;
        double r3513623 = r3513619 + r3513622;
        return r3513623;
}
↓
double f(double x, double y) {
        double r3513624 = y;
        double r3513625 = x;
        double r3513626 = sqrt(r3513625);
        double r3513627 = 1.0;
        double r3513628 = r3513627 - r3513625;
        double r3513629 = fma(r3513624, r3513626, r3513628);
        double r3513630 = -1.0;
        double r3513631 = fma(r3513625, r3513630, r3513625);
        double r3513632 = r3513629 + r3513631;
        return r3513632;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1.0 - x\right) + y \cdot \sqrt{x}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - x}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.5

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1.0\right) - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\]

  5. Applied add-cube-cbrt0.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, \sqrt{x}, 1.0\right)}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\]

  6. Applied prod-diff0.9

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

  7. Simplified0.1

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

  8. Simplified0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1.0 - x\right) + \color{blue}{\mathsf{fma}\left(x, -1, x\right)}\]

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1.0 - x\right) + \mathsf{fma}\left(x, -1, x\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  (+ (- 1.0 x) (* y (sqrt x))))