Average Error: 10.0 → 0.2
Time: 11.1s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le -3.045531662173799354349276567581092360629 \cdot 10^{53} \lor \neg \left(\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le 8.613111255585373997556790002011210210152 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{1 \cdot x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le -3.045531662173799354349276567581092360629 \cdot 10^{53} \lor \neg \left(\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le 8.613111255585373997556790002011210210152 \cdot 10^{-42}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{1 \cdot x}{z} - x\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r433122 = x;
        double r433123 = y;
        double r433124 = z;
        double r433125 = r433123 - r433124;
        double r433126 = 1.0;
        double r433127 = r433125 + r433126;
        double r433128 = r433122 * r433127;
        double r433129 = r433128 / r433124;
        return r433129;
}
↓
double f(double x, double y, double z) {
        double r433130 = y;
        double r433131 = z;
        double r433132 = r433130 - r433131;
        double r433133 = 1.0;
        double r433134 = r433132 + r433133;
        double r433135 = x;
        double r433136 = r433134 * r433135;
        double r433137 = r433136 / r433131;
        double r433138 = -3.0455316621737994e+53;
        bool r433139 = r433137 <= r433138;
        double r433140 = 8.613111255585374e-42;
        bool r433141 = r433137 <= r433140;
        double r433142 = !r433141;
        bool r433143 = r433139 || r433142;
        double r433144 = r433135 / r433131;
        double r433145 = r433133 * r433135;
        double r433146 = r433145 / r433131;
        double r433147 = r433146 - r433135;
        double r433148 = fma(r433144, r433130, r433147);
        double r433149 = r433130 + r433133;
        double r433150 = r433149 - r433131;
        double r433151 = r433131 / r433150;
        double r433152 = r433135 / r433151;
        double r433153 = r433143 ? r433148 : r433152;
        return r433153;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.0
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -3.0455316621737994e+53 or 8.613111255585374e-42 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 16.9

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

    2. Taylor expanded around 0 5.8

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

    3. Simplified0.1

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

    if -3.0455316621737994e+53 < (/ (* x (+ (- y z) 1.0)) z) < 8.613111255585374e-42

    1. Initial program 0.1

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

    3. Applied associate-/l*0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le -3.045531662173799354349276567581092360629 \cdot 10^{53} \lor \neg \left(\frac{\left(\left(y - z\right) + 1\right) \cdot x}{z} \le 8.613111255585373997556790002011210210152 \cdot 10^{-42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{1 \cdot x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))