Average Error: 0.0 → 0.6
Time: 2.9s
Precision: 64
\[x + \frac{y - x}{z}\]
↓
\[\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{y - x}{z}\right)\]
x + \frac{y - x}{z}
↓
\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{y - x}{z}\right)
double f(double x, double y, double z) {
        double r16514 = x;
        double r16515 = y;
        double r16516 = r16515 - r16514;
        double r16517 = z;
        double r16518 = r16516 / r16517;
        double r16519 = r16514 + r16518;
        return r16519;
}
↓
double f(double x, double y, double z) {
        double r16520 = x;
        double r16521 = cbrt(r16520);
        double r16522 = r16521 * r16521;
        double r16523 = y;
        double r16524 = r16523 - r16520;
        double r16525 = z;
        double r16526 = r16524 / r16525;
        double r16527 = fma(r16522, r16521, r16526);
        return r16527;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[x + \frac{y - x}{z}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.6

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

  4. Applied fma-def0.6

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

  5. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{y - x}{z}\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Sample:$swelfordMean from math-functions-0.1.5.2"
  :precision binary64
  (+ x (/ (- y x) z)))