Average Error: 10.4 → 1.8
Time: 32.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, 1, y \cdot \frac{x}{z} - x\right)\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, 1, y \cdot \frac{x}{z} - x\right)
double f(double x, double y, double z) {
        double r23861520 = x;
        double r23861521 = y;
        double r23861522 = z;
        double r23861523 = r23861521 - r23861522;
        double r23861524 = 1.0;
        double r23861525 = r23861523 + r23861524;
        double r23861526 = r23861520 * r23861525;
        double r23861527 = r23861526 / r23861522;
        return r23861527;
}
↓
double f(double x, double y, double z) {
        double r23861528 = x;
        double r23861529 = z;
        double r23861530 = r23861528 / r23861529;
        double r23861531 = 1.0;
        double r23861532 = y;
        double r23861533 = r23861532 * r23861530;
        double r23861534 = r23861533 - r23861528;
        double r23861535 = fma(r23861530, r23861531, r23861534);
        return r23861535;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.4
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.4

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

  2. Taylor expanded around 0 3.7

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

  3. Simplified1.8

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

  4. Final simplification1.8

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

Reproduce

herbie shell --seed 2019200 +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))