Average Error: 10.5 → 0.3
Time: 3.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.61943798913748523 \cdot 10^{55}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{\left(y - z\right) + 1}{z}}}\\ \mathbf{elif}\;z \le 2.2731223542198846 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.61943798913748523 \cdot 10^{55}:\\
\;\;\;\;\frac{x}{\frac{1}{\frac{\left(y - z\right) + 1}{z}}}\\

\mathbf{elif}\;z \le 2.2731223542198846 \cdot 10^{49}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{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 r743396 = x;
        double r743397 = y;
        double r743398 = z;
        double r743399 = r743397 - r743398;
        double r743400 = 1.0;
        double r743401 = r743399 + r743400;
        double r743402 = r743396 * r743401;
        double r743403 = r743402 / r743398;
        return r743403;
}
↓
double f(double x, double y, double z) {
        double r743404 = z;
        double r743405 = -2.619437989137485e+55;
        bool r743406 = r743404 <= r743405;
        double r743407 = x;
        double r743408 = 1.0;
        double r743409 = y;
        double r743410 = r743409 - r743404;
        double r743411 = 1.0;
        double r743412 = r743410 + r743411;
        double r743413 = r743412 / r743404;
        double r743414 = r743408 / r743413;
        double r743415 = r743407 / r743414;
        double r743416 = 2.2731223542198846e+49;
        bool r743417 = r743404 <= r743416;
        double r743418 = r743407 / r743404;
        double r743419 = r743407 * r743409;
        double r743420 = r743419 / r743404;
        double r743421 = fma(r743411, r743418, r743420);
        double r743422 = r743421 - r743407;
        double r743423 = r743404 / r743412;
        double r743424 = r743407 / r743423;
        double r743425 = r743417 ? r743422 : r743424;
        double r743426 = r743406 ? r743415 : r743425;
        return r743426;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.619437989137485e+55

    1. Initial program 20.9

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

    3. Applied associate-/l*0.1

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

    5. Applied clear-num0.1

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

    if -2.619437989137485e+55 < z < 2.2731223542198846e+49

    1. Initial program 0.8

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

    if 2.2731223542198846e+49 < z

    1. Initial program 18.2

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

    3. Applied associate-/l*0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.61943798913748523 \cdot 10^{55}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{\left(y - z\right) + 1}{z}}}\\ \mathbf{elif}\;z \le 2.2731223542198846 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))