Average Error: 10.5 → 1.6
Time: 11.9s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\mathsf{fma}\left(\frac{x}{z}, y, \frac{1 \cdot x}{z} - x\right)\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\mathsf{fma}\left(\frac{x}{z}, y, \frac{1 \cdot x}{z} - x\right)
double f(double x, double y, double z) {
        double r452889 = x;
        double r452890 = y;
        double r452891 = z;
        double r452892 = r452890 - r452891;
        double r452893 = 1.0;
        double r452894 = r452892 + r452893;
        double r452895 = r452889 * r452894;
        double r452896 = r452895 / r452891;
        return r452896;
}
↓
double f(double x, double y, double z) {
        double r452897 = x;
        double r452898 = z;
        double r452899 = r452897 / r452898;
        double r452900 = y;
        double r452901 = 1.0;
        double r452902 = r452901 * r452897;
        double r452903 = r452902 / r452898;
        double r452904 = r452903 - r452897;
        double r452905 = fma(r452899, r452900, r452904);
        return r452905;
}

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

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

  4. Final simplification1.6

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

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))