Average Error: 10.0 → 1.7
Time: 2.8s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r668560 = x;
        double r668561 = y;
        double r668562 = z;
        double r668563 = r668561 - r668562;
        double r668564 = 1.0;
        double r668565 = r668563 + r668564;
        double r668566 = r668560 * r668565;
        double r668567 = r668566 / r668562;
        return r668567;
}
↓
double f(double x, double y, double z) {
        double r668568 = x;
        double r668569 = 3.5216806508699143e-102;
        bool r668570 = r668568 <= r668569;
        double r668571 = 1.0;
        double r668572 = z;
        double r668573 = r668568 / r668572;
        double r668574 = y;
        double r668575 = r668568 * r668574;
        double r668576 = 1.0;
        double r668577 = r668576 / r668572;
        double r668578 = r668575 * r668577;
        double r668579 = fma(r668571, r668573, r668578);
        double r668580 = r668579 - r668568;
        double r668581 = r668571 + r668574;
        double r668582 = r668573 * r668581;
        double r668583 = r668582 - r668568;
        double r668584 = r668570 ? r668580 : r668583;
        return r668584;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.0
Target0.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.5216806508699143e-102

    1. Initial program 7.1

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

    2. Taylor expanded around 0 2.3

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

    3. Simplified2.3

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

    5. Applied div-inv2.3

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

    if 3.5216806508699143e-102 < x

    1. Initial program 17.1

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

    2. Taylor expanded around 0 5.6

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

    3. Simplified5.6

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

    4. Taylor expanded around 0 5.6

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

    5. Simplified0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(1 + y\right)} - x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))