Average Error: 10.3 → 0.1
Time: 15.7s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5596220615.762752532958984375:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x \cdot 1}{z} - x\right)\\ \mathbf{elif}\;x \le 1201898336259836928:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5596220615.762752532958984375:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x \cdot 1}{z} - x\right)\\

\mathbf{elif}\;x \le 1201898336259836928:\\
\;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r389084 = x;
        double r389085 = y;
        double r389086 = z;
        double r389087 = r389085 - r389086;
        double r389088 = 1.0;
        double r389089 = r389087 + r389088;
        double r389090 = r389084 * r389089;
        double r389091 = r389090 / r389086;
        return r389091;
}
↓
double f(double x, double y, double z) {
        double r389092 = x;
        double r389093 = -5596220615.762753;
        bool r389094 = r389092 <= r389093;
        double r389095 = z;
        double r389096 = r389092 / r389095;
        double r389097 = y;
        double r389098 = 1.0;
        double r389099 = r389092 * r389098;
        double r389100 = r389099 / r389095;
        double r389101 = r389100 - r389092;
        double r389102 = fma(r389096, r389097, r389101);
        double r389103 = 1.201898336259837e+18;
        bool r389104 = r389092 <= r389103;
        double r389105 = r389097 - r389095;
        double r389106 = r389098 + r389105;
        double r389107 = r389092 * r389106;
        double r389108 = r389107 / r389095;
        double r389109 = r389097 + r389098;
        double r389110 = r389109 - r389095;
        double r389111 = r389095 / r389110;
        double r389112 = r389092 / r389111;
        double r389113 = r389104 ? r389108 : r389112;
        double r389114 = r389094 ? r389102 : r389113;
        return r389114;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.3
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -5596220615.762753

    1. Initial program 27.6

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

    2. Taylor expanded around 0 8.9

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

    3. Simplified0.1

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

    if -5596220615.762753 < x < 1.201898336259837e+18

    1. Initial program 0.2

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

    if 1.201898336259837e+18 < x

    1. Initial program 28.3

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

    3. Applied associate-/l*0.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5596220615.762752532958984375:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x \cdot 1}{z} - x\right)\\ \mathbf{elif}\;x \le 1201898336259836928:\\ \;\;\;\;\frac{x \cdot \left(1 + \left(y - z\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))