Average Error: 10.4 → 0.5
Time: 7.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.6583684505465714 \cdot 10^{34} \lor \neg \left(x \le 5.3429144278201139 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.6583684505465714 \cdot 10^{34} \lor \neg \left(x \le 5.3429144278201139 \cdot 10^{-89}\right):\\
\;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\

\end{array}
double f(double x, double y, double z) {
        double r3155 = x;
        double r3156 = y;
        double r3157 = z;
        double r3158 = r3156 - r3157;
        double r3159 = 1.0;
        double r3160 = r3158 + r3159;
        double r3161 = r3155 * r3160;
        double r3162 = r3161 / r3157;
        return r3162;
}
↓
double f(double x, double y, double z) {
        double r3163 = x;
        double r3164 = -2.6583684505465714e+34;
        bool r3165 = r3163 <= r3164;
        double r3166 = 5.342914427820114e-89;
        bool r3167 = r3163 <= r3166;
        double r3168 = !r3167;
        bool r3169 = r3165 || r3168;
        double r3170 = z;
        double r3171 = r3163 / r3170;
        double r3172 = y;
        double r3173 = r3172 - r3170;
        double r3174 = 1.0;
        double r3175 = r3173 + r3174;
        double r3176 = r3171 * r3175;
        double r3177 = r3163 * r3172;
        double r3178 = r3177 / r3170;
        double r3179 = fma(r3174, r3171, r3178);
        double r3180 = r3179 - r3163;
        double r3181 = r3169 ? r3176 : r3180;
        return r3181;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.4
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.6583684505465714e+34 or 5.342914427820114e-89 < x

    1. Initial program 22.7

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

    3. Applied associate-/l*0.4

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

    5. Applied associate-/r/0.9

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

    if -2.6583684505465714e+34 < x < 5.342914427820114e-89

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.6583684505465714 \cdot 10^{34} \lor \neg \left(x \le 5.3429144278201139 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))