Average Error: 10.0 → 0.5
Time: 32.0s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.10171392839965858826786137925009047894 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{elif}\;x \le 3.928280010097489435879619358180988729078 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(y + 1\right) \cdot x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.10171392839965858826786137925009047894 \cdot 10^{-54}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\

\mathbf{elif}\;x \le 3.928280010097489435879619358180988729078 \cdot 10^{-146}:\\
\;\;\;\;\frac{\left(y + 1\right) \cdot x}{z} - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r31336565 = x;
        double r31336566 = y;
        double r31336567 = z;
        double r31336568 = r31336566 - r31336567;
        double r31336569 = 1.0;
        double r31336570 = r31336568 + r31336569;
        double r31336571 = r31336565 * r31336570;
        double r31336572 = r31336571 / r31336567;
        return r31336572;
}
↓
double f(double x, double y, double z) {
        double r31336573 = x;
        double r31336574 = -3.1017139283996586e-54;
        bool r31336575 = r31336573 <= r31336574;
        double r31336576 = z;
        double r31336577 = r31336573 / r31336576;
        double r31336578 = y;
        double r31336579 = 1.0;
        double r31336580 = r31336579 * r31336577;
        double r31336581 = r31336580 - r31336573;
        double r31336582 = fma(r31336577, r31336578, r31336581);
        double r31336583 = 3.9282800100974894e-146;
        bool r31336584 = r31336573 <= r31336583;
        double r31336585 = r31336578 + r31336579;
        double r31336586 = r31336585 * r31336573;
        double r31336587 = r31336586 / r31336576;
        double r31336588 = r31336587 - r31336573;
        double r31336589 = r31336578 - r31336576;
        double r31336590 = r31336579 + r31336589;
        double r31336591 = r31336576 / r31336590;
        double r31336592 = r31336573 / r31336591;
        double r31336593 = r31336584 ? r31336588 : r31336592;
        double r31336594 = r31336575 ? r31336582 : r31336593;
        return r31336594;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.0
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.1017139283996586e-54

    1. Initial program 20.8

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

    3. Applied associate-/l*0.4

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

    4. Taylor expanded around 0 7.1

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

    5. Simplified0.1

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

    if -3.1017139283996586e-54 < x < 3.9282800100974894e-146

    1. Initial program 0.2

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

    3. Applied associate-/l*6.9

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

    4. Taylor expanded around 0 0.1

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

    5. Simplified3.8

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

    6. Taylor expanded around 0 0.1

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

    7. Simplified0.1

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

    if 3.9282800100974894e-146 < x

    1. Initial program 14.7

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

    3. Applied associate-/l*1.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.10171392839965858826786137925009047894 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{elif}\;x \le 3.928280010097489435879619358180988729078 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(y + 1\right) \cdot x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))