Average Error: 10.4 → 1.7
Time: 2.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, 1 + y, -x\right)
double f(double x, double y, double z) {
        double r543525 = x;
        double r543526 = y;
        double r543527 = z;
        double r543528 = r543526 - r543527;
        double r543529 = 1.0;
        double r543530 = r543528 + r543529;
        double r543531 = r543525 * r543530;
        double r543532 = r543531 / r543527;
        return r543532;
}
↓
double f(double x, double y, double z) {
        double r543533 = x;
        double r543534 = z;
        double r543535 = r543533 / r543534;
        double r543536 = 1.0;
        double r543537 = y;
        double r543538 = r543536 + r543537;
        double r543539 = -r543533;
        double r543540 = fma(r543535, r543538, r543539);
        return r543540;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.4
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.4

    \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity10.4

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

  4. Applied times-frac3.6

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

  5. Simplified3.6

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

  6. Taylor expanded around 0 3.6

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

  7. Simplified1.7

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

  8. Final simplification1.7

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))