Average Error: 10.2 → 0.4
Time: 2.5s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.180654286668062195300281725905696886021 \cdot 10^{104} \lor \neg \left(z \le 2.201016357696081879363280295698727206869 \cdot 10^{-60}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x}{\frac{z}{y}}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.180654286668062195300281725905696886021 \cdot 10^{104} \lor \neg \left(z \le 2.201016357696081879363280295698727206869 \cdot 10^{-60}\right):\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x}{\frac{z}{y}}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r853459 = x;
        double r853460 = y;
        double r853461 = z;
        double r853462 = r853460 - r853461;
        double r853463 = 1.0;
        double r853464 = r853462 + r853463;
        double r853465 = r853459 * r853464;
        double r853466 = r853465 / r853461;
        return r853466;
}
↓
double f(double x, double y, double z) {
        double r853467 = z;
        double r853468 = -8.180654286668062e+104;
        bool r853469 = r853467 <= r853468;
        double r853470 = 2.201016357696082e-60;
        bool r853471 = r853467 <= r853470;
        double r853472 = !r853471;
        bool r853473 = r853469 || r853472;
        double r853474 = 1.0;
        double r853475 = x;
        double r853476 = r853475 / r853467;
        double r853477 = y;
        double r853478 = r853467 / r853477;
        double r853479 = r853475 / r853478;
        double r853480 = fma(r853474, r853476, r853479);
        double r853481 = r853480 - r853475;
        double r853482 = r853475 * r853477;
        double r853483 = r853482 / r853467;
        double r853484 = fma(r853474, r853476, r853483);
        double r853485 = r853484 - r853475;
        double r853486 = r853473 ? r853481 : r853485;
        return r853486;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -8.180654286668062e+104 or 2.201016357696082e-60 < z

    1. Initial program 17.4

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

    3. Applied associate-/l*0.1

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

    4. Taylor expanded around 0 5.5

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

    5. Simplified5.5

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

    7. Applied clear-num5.5

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

    9. Applied *-un-lft-identity5.5

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

    10. Applied times-frac0.2

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

    11. Applied associate-/r*0.1

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

    12. Simplified0.1

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

    if -8.180654286668062e+104 < z < 2.201016357696082e-60

    1. Initial program 1.4

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

    3. Applied associate-/l*6.8

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

    4. Taylor expanded around 0 0.7

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

    5. Simplified0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.180654286668062195300281725905696886021 \cdot 10^{104} \lor \neg \left(z \le 2.201016357696081879363280295698727206869 \cdot 10^{-60}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x}{\frac{z}{y}}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))