Average Error: 9.6 → 0.2
Time: 20.5s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\ \mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{y \cdot x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\

\mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{y \cdot x}{z}\right) - x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\

\end{array}
double f(double x, double y, double z) {
        double r29190193 = x;
        double r29190194 = y;
        double r29190195 = z;
        double r29190196 = r29190194 - r29190195;
        double r29190197 = 1.0;
        double r29190198 = r29190196 + r29190197;
        double r29190199 = r29190193 * r29190198;
        double r29190200 = r29190199 / r29190195;
        return r29190200;
}
↓
double f(double x, double y, double z) {
        double r29190201 = x;
        double r29190202 = -1.3418778201574405e-43;
        bool r29190203 = r29190201 <= r29190202;
        double r29190204 = z;
        double r29190205 = y;
        double r29190206 = r29190205 - r29190204;
        double r29190207 = 1.0;
        double r29190208 = r29190206 + r29190207;
        double r29190209 = r29190204 / r29190208;
        double r29190210 = r29190201 / r29190209;
        double r29190211 = 1.2340705215670025e-123;
        bool r29190212 = r29190201 <= r29190211;
        double r29190213 = r29190201 / r29190204;
        double r29190214 = r29190205 * r29190201;
        double r29190215 = r29190214 / r29190204;
        double r29190216 = fma(r29190213, r29190207, r29190215);
        double r29190217 = r29190216 - r29190201;
        double r29190218 = r29190205 * r29190213;
        double r29190219 = fma(r29190213, r29190207, r29190218);
        double r29190220 = r29190219 - r29190201;
        double r29190221 = r29190212 ? r29190217 : r29190220;
        double r29190222 = r29190203 ? r29190210 : r29190221;
        return r29190222;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original9.6
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3418778201574405e-43

    1. Initial program 20.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}}\]

    if -1.3418778201574405e-43 < x < 1.2340705215670025e-123

    1. Initial program 0.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]

    2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]

    3. Simplified3.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x}\]
    4. Using strategy rm

    5. Applied associate-*r/0.1

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, 1.0, \color{blue}{\frac{y \cdot x}{z}}\right) - x\]

    if 1.2340705215670025e-123 < x

    1. Initial program 15.9

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]

    2. Taylor expanded around 0 5.5

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\ \mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, \frac{y \cdot x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1.0, y \cdot \frac{x}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))