Average Error: 9.8 → 0.8
Time: 19.8s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.6266794377330075 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \mathbf{elif}\;x \le 1.3597213738421031 \cdot 10^{-288}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x + 1.0 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.6266794377330075 \cdot 10^{-145}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\

\mathbf{elif}\;x \le 1.3597213738421031 \cdot 10^{-288}:\\
\;\;\;\;\frac{\left(y - z\right) \cdot x + 1.0 \cdot x}{z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\

\end{array}
double f(double x, double y, double z) {
        double r26343909 = x;
        double r26343910 = y;
        double r26343911 = z;
        double r26343912 = r26343910 - r26343911;
        double r26343913 = 1.0;
        double r26343914 = r26343912 + r26343913;
        double r26343915 = r26343909 * r26343914;
        double r26343916 = r26343915 / r26343911;
        return r26343916;
}
↓
double f(double x, double y, double z) {
        double r26343917 = x;
        double r26343918 = -4.6266794377330075e-145;
        bool r26343919 = r26343917 <= r26343918;
        double r26343920 = z;
        double r26343921 = r26343917 / r26343920;
        double r26343922 = 1.0;
        double r26343923 = y;
        double r26343924 = r26343923 * r26343921;
        double r26343925 = fma(r26343921, r26343922, r26343924);
        double r26343926 = r26343925 - r26343917;
        double r26343927 = 1.3597213738421031e-288;
        bool r26343928 = r26343917 <= r26343927;
        double r26343929 = r26343923 - r26343920;
        double r26343930 = r26343929 * r26343917;
        double r26343931 = r26343922 * r26343917;
        double r26343932 = r26343930 + r26343931;
        double r26343933 = r26343932 / r26343920;
        double r26343934 = r26343928 ? r26343933 : r26343926;
        double r26343935 = r26343919 ? r26343926 : r26343934;
        return r26343935;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.8
Target0.5
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4.6266794377330075e-145 or 1.3597213738421031e-288 < x

    1. Initial program 11.9

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]

    2. Taylor expanded around 0 4.1

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]

    3. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x}\]

    if -4.6266794377330075e-145 < x < 1.3597213738421031e-288

    1. Initial program 0.2

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
    2. Using strategy rm

    3. Applied distribute-lft-in0.2

      \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1.0}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.6266794377330075 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \mathbf{elif}\;x \le 1.3597213738421031 \cdot 10^{-288}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x + 1.0 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))