Average Error: 10.2 → 0.1
Time: 2.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

\mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\

\end{array}
double f(double x, double y, double z) {
        double r607720 = x;
        double r607721 = y;
        double r607722 = z;
        double r607723 = r607721 - r607722;
        double r607724 = 1.0;
        double r607725 = r607723 + r607724;
        double r607726 = r607720 * r607725;
        double r607727 = r607726 / r607722;
        return r607727;
}
↓
double f(double x, double y, double z) {
        double r607728 = x;
        double r607729 = -3.4374981009777875e+27;
        bool r607730 = r607728 <= r607729;
        double r607731 = z;
        double r607732 = r607728 / r607731;
        double r607733 = 1.0;
        double r607734 = y;
        double r607735 = r607733 + r607734;
        double r607736 = r607732 * r607735;
        double r607737 = r607736 - r607728;
        double r607738 = 8.649219799147649e-36;
        bool r607739 = r607728 <= r607738;
        double r607740 = r607728 * r607734;
        double r607741 = 1.0;
        double r607742 = r607741 / r607731;
        double r607743 = r607740 * r607742;
        double r607744 = fma(r607733, r607732, r607743);
        double r607745 = r607744 - r607728;
        double r607746 = r607734 - r607731;
        double r607747 = r607746 + r607733;
        double r607748 = r607731 / r607747;
        double r607749 = r607728 / r607748;
        double r607750 = r607739 ? r607745 : r607749;
        double r607751 = r607730 ? r607737 : r607750;
        return r607751;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.4374981009777875e+27

    1. Initial program 28.2

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

    2. Taylor expanded around 0 9.1

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

    3. Simplified9.1

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

    4. Taylor expanded around 0 9.1

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

    5. Simplified0.1

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

    if -3.4374981009777875e+27 < x < 8.649219799147649e-36

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.1

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

    if 8.649219799147649e-36 < x

    1. Initial program 21.3

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

    3. Applied associate-/l*0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))