Average Error: 10.5 → 1.6
Time: 15.0s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)
double f(double x, double y, double z) {
        double r40021599 = x;
        double r40021600 = y;
        double r40021601 = z;
        double r40021602 = r40021600 - r40021601;
        double r40021603 = 1.0;
        double r40021604 = r40021602 + r40021603;
        double r40021605 = r40021599 * r40021604;
        double r40021606 = r40021605 / r40021601;
        return r40021606;
}
↓
double f(double x, double y, double z) {
        double r40021607 = x;
        double r40021608 = z;
        double r40021609 = r40021607 / r40021608;
        double r40021610 = y;
        double r40021611 = 1.0;
        double r40021612 = r40021611 * r40021609;
        double r40021613 = r40021612 - r40021607;
        double r40021614 = fma(r40021609, r40021610, r40021613);
        return r40021614;
}

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

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

  4. Final simplification1.6

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

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))