Average Error: 10.7 → 1.5
Time: 22.0s
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 r31067553 = x;
        double r31067554 = y;
        double r31067555 = z;
        double r31067556 = r31067554 - r31067555;
        double r31067557 = 1.0;
        double r31067558 = r31067556 + r31067557;
        double r31067559 = r31067553 * r31067558;
        double r31067560 = r31067559 / r31067555;
        return r31067560;
}
↓
double f(double x, double y, double z) {
        double r31067561 = x;
        double r31067562 = z;
        double r31067563 = r31067561 / r31067562;
        double r31067564 = y;
        double r31067565 = 1.0;
        double r31067566 = r31067565 * r31067563;
        double r31067567 = r31067566 - r31067561;
        double r31067568 = fma(r31067563, r31067564, r31067567);
        return r31067568;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.7
Target0.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.7

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

  2. Taylor expanded around 0 3.5

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

  3. Simplified1.5

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

  4. Final simplification1.5

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

Reproduce

herbie shell --seed 2019169 +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))