Average Error: 10.2 → 0.1
Time: 6.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.0032476001413837569:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\ \mathbf{elif}\;z \le 1.5118393353993747 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.0032476001413837569:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\

\mathbf{elif}\;z \le 1.5118393353993747 \cdot 10^{-50}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \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 r618082 = x;
        double r618083 = y;
        double r618084 = z;
        double r618085 = r618083 - r618084;
        double r618086 = 1.0;
        double r618087 = r618085 + r618086;
        double r618088 = r618082 * r618087;
        double r618089 = r618088 / r618084;
        return r618089;
}
↓
double f(double x, double y, double z) {
        double r618090 = z;
        double r618091 = -0.003247600141383757;
        bool r618092 = r618090 <= r618091;
        double r618093 = x;
        double r618094 = r618093 / r618090;
        double r618095 = 1.0;
        double r618096 = y;
        double r618097 = r618096 / r618090;
        double r618098 = r618093 * r618097;
        double r618099 = fma(r618094, r618095, r618098);
        double r618100 = r618099 - r618093;
        double r618101 = 1.5118393353993747e-50;
        bool r618102 = r618090 <= r618101;
        double r618103 = r618093 * r618096;
        double r618104 = r618103 / r618090;
        double r618105 = fma(r618094, r618095, r618104);
        double r618106 = r618105 - r618093;
        double r618107 = r618096 - r618090;
        double r618108 = r618107 + r618095;
        double r618109 = r618090 / r618108;
        double r618110 = r618093 / r618109;
        double r618111 = r618102 ? r618106 : r618110;
        double r618112 = r618092 ? r618100 : r618111;
        return r618112;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.2
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -0.003247600141383757

    1. Initial program 16.7

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

    2. Taylor expanded around 0 5.5

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

    3. Simplified5.5

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

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

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

    6. Applied times-frac0.1

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

    7. Simplified0.1

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

    if -0.003247600141383757 < z < 1.5118393353993747e-50

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 1.5118393353993747e-50 < z

    1. Initial program 14.6

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

    3. Applied associate-/l*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.0032476001413837569:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, x \cdot \frac{y}{z}\right) - x\\ \mathbf{elif}\;z \le 1.5118393353993747 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))