Average Error: 10.4 → 0.5
Time: 18.1s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.955374476152934635601992853388084850774 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{elif}\;x \le 8.101941502747141806933018865526614192325 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\left(1 + \left(y - z\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.955374476152934635601992853388084850774 \cdot 10^{-168}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\

\mathbf{elif}\;x \le 8.101941502747141806933018865526614192325 \cdot 10^{-99}:\\
\;\;\;\;\frac{1}{z} \cdot \left(\left(1 + \left(y - z\right)\right) \cdot x\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r26523149 = x;
        double r26523150 = y;
        double r26523151 = z;
        double r26523152 = r26523150 - r26523151;
        double r26523153 = 1.0;
        double r26523154 = r26523152 + r26523153;
        double r26523155 = r26523149 * r26523154;
        double r26523156 = r26523155 / r26523151;
        return r26523156;
}
↓
double f(double x, double y, double z) {
        double r26523157 = x;
        double r26523158 = -1.9553744761529346e-168;
        bool r26523159 = r26523157 <= r26523158;
        double r26523160 = z;
        double r26523161 = r26523157 / r26523160;
        double r26523162 = y;
        double r26523163 = 1.0;
        double r26523164 = r26523163 * r26523161;
        double r26523165 = r26523164 - r26523157;
        double r26523166 = fma(r26523161, r26523162, r26523165);
        double r26523167 = 8.101941502747142e-99;
        bool r26523168 = r26523157 <= r26523167;
        double r26523169 = 1.0;
        double r26523170 = r26523169 / r26523160;
        double r26523171 = r26523162 - r26523160;
        double r26523172 = r26523163 + r26523171;
        double r26523173 = r26523172 * r26523157;
        double r26523174 = r26523170 * r26523173;
        double r26523175 = r26523160 / r26523172;
        double r26523176 = r26523157 / r26523175;
        double r26523177 = r26523168 ? r26523174 : r26523176;
        double r26523178 = r26523159 ? r26523166 : r26523177;
        return r26523178;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
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 < -1.9553744761529346e-168

    1. Initial program 13.9

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

    2. Taylor expanded around 0 4.5

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

    3. Simplified0.5

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

    if -1.9553744761529346e-168 < x < 8.101941502747142e-99

    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. Using strategy rm

    5. Applied div-inv7.0

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

    6. Applied *-un-lft-identity7.0

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

    7. Applied times-frac0.3

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

    8. Simplified0.3

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

    if 8.101941502747142e-99 < x

    1. Initial program 18.2

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

    3. Applied associate-/l*0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.955374476152934635601992853388084850774 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, 1 \cdot \frac{x}{z} - x\right)\\ \mathbf{elif}\;x \le 8.101941502747141806933018865526614192325 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{z} \cdot \left(\left(1 + \left(y - z\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + \left(y - z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))