Average Error: 9.8 → 0.4
Time: 11.4s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -9.2290174913998737 \cdot 10^{129} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.0292350933407683 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -9.2290174913998737 \cdot 10^{129} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.0292350933407683 \cdot 10^{-45}\right):\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r560122 = x;
        double r560123 = y;
        double r560124 = z;
        double r560125 = r560123 - r560124;
        double r560126 = 1.0;
        double r560127 = r560125 + r560126;
        double r560128 = r560122 * r560127;
        double r560129 = r560128 / r560124;
        return r560129;
}
↓
double f(double x, double y, double z) {
        double r560130 = x;
        double r560131 = y;
        double r560132 = z;
        double r560133 = r560131 - r560132;
        double r560134 = 1.0;
        double r560135 = r560133 + r560134;
        double r560136 = r560130 * r560135;
        double r560137 = r560136 / r560132;
        double r560138 = -9.229017491399874e+129;
        bool r560139 = r560137 <= r560138;
        double r560140 = 3.029235093340768e-45;
        bool r560141 = r560137 <= r560140;
        double r560142 = !r560141;
        bool r560143 = r560139 || r560142;
        double r560144 = r560130 / r560132;
        double r560145 = r560134 + r560131;
        double r560146 = r560144 * r560145;
        double r560147 = r560146 - r560130;
        double r560148 = r560132 / r560135;
        double r560149 = r560130 / r560148;
        double r560150 = r560143 ? r560147 : r560149;
        return r560150;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -9.229017491399874e+129 or 3.029235093340768e-45 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 18.0

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

    2. Taylor expanded around 0 6.4

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

    3. Simplified0.1

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

    if -9.229017491399874e+129 < (/ (* x (+ (- y z) 1.0)) z) < 3.029235093340768e-45

    1. Initial program 0.1

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

    3. Applied associate-/l*0.8

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -9.2290174913998737 \cdot 10^{129} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.0292350933407683 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))