Average Error: 10.2 → 0.6
Time: 11.6s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.863929268284692307003518013897528112257 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{y}{z} \cdot x\right) - x\\ \mathbf{elif}\;x \le 1.076984167266228777802393156484664836043 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{y}{z} \cdot x\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.863929268284692307003518013897528112257 \cdot 10^{-90}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{y}{z} \cdot x\right) - x\\

\mathbf{elif}\;x \le 1.076984167266228777802393156484664836043 \cdot 10^{-131}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r25804363 = x;
        double r25804364 = y;
        double r25804365 = z;
        double r25804366 = r25804364 - r25804365;
        double r25804367 = 1.0;
        double r25804368 = r25804366 + r25804367;
        double r25804369 = r25804363 * r25804368;
        double r25804370 = r25804369 / r25804365;
        return r25804370;
}
↓
double f(double x, double y, double z) {
        double r25804371 = x;
        double r25804372 = -1.8639292682846923e-90;
        bool r25804373 = r25804371 <= r25804372;
        double r25804374 = z;
        double r25804375 = r25804371 / r25804374;
        double r25804376 = 1.0;
        double r25804377 = y;
        double r25804378 = r25804377 / r25804374;
        double r25804379 = r25804378 * r25804371;
        double r25804380 = fma(r25804375, r25804376, r25804379);
        double r25804381 = r25804380 - r25804371;
        double r25804382 = 1.0769841672662288e-131;
        bool r25804383 = r25804371 <= r25804382;
        double r25804384 = r25804371 * r25804377;
        double r25804385 = 1.0;
        double r25804386 = r25804385 / r25804374;
        double r25804387 = r25804384 * r25804386;
        double r25804388 = fma(r25804375, r25804376, r25804387);
        double r25804389 = r25804388 - r25804371;
        double r25804390 = r25804383 ? r25804389 : r25804381;
        double r25804391 = r25804373 ? r25804381 : r25804390;
        return r25804391;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.8639292682846923e-90 or 1.0769841672662288e-131 < x

    1. Initial program 17.0

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

    2. Taylor expanded around 0 5.9

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

    3. Simplified0.2

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

    5. Applied div-inv0.2

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

    6. Applied associate-*l*1.0

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

    7. Simplified1.0

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

    if -1.8639292682846923e-90 < x < 1.0769841672662288e-131

    1. Initial program 0.2

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified4.0

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

    5. Applied div-inv4.0

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

    6. Applied associate-*l*7.2

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

    7. Simplified7.2

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

    9. Applied div-inv7.2

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

    10. Applied associate-*r*0.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.863929268284692307003518013897528112257 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{y}{z} \cdot x\right) - x\\ \mathbf{elif}\;x \le 1.076984167266228777802393156484664836043 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{y}{z} \cdot x\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))