Average Error: 10.5 → 1.6
Time: 13.9s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, 1, y \cdot \frac{x}{z}\right) - x\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, 1, y \cdot \frac{x}{z}\right) - x
double f(double x, double y, double z) {
        double r27358651 = x;
        double r27358652 = y;
        double r27358653 = z;
        double r27358654 = r27358652 - r27358653;
        double r27358655 = 1.0;
        double r27358656 = r27358654 + r27358655;
        double r27358657 = r27358651 * r27358656;
        double r27358658 = r27358657 / r27358653;
        return r27358658;
}
↓
double f(double x, double y, double z) {
        double r27358659 = x;
        double r27358660 = z;
        double r27358661 = r27358659 / r27358660;
        double r27358662 = 1.0;
        double r27358663 = y;
        double r27358664 = r27358663 * r27358661;
        double r27358665 = fma(r27358661, r27358662, r27358664);
        double r27358666 = r27358665 - r27358659;
        return r27358666;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.5
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.5

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

  2. Taylor expanded around 0 3.9

    \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x}\]

  3. Simplified1.6

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

  4. Final simplification1.6

    \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1, y \cdot \frac{x}{z}\right) - x\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))