Average Error: 10.9 → 0.3
Time: 2.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\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 -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\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 r658210 = x;
        double r658211 = y;
        double r658212 = z;
        double r658213 = r658211 - r658212;
        double r658214 = 1.0;
        double r658215 = r658213 + r658214;
        double r658216 = r658210 * r658215;
        double r658217 = r658216 / r658212;
        return r658217;
}
↓
double f(double x, double y, double z) {
        double r658218 = x;
        double r658219 = y;
        double r658220 = z;
        double r658221 = r658219 - r658220;
        double r658222 = 1.0;
        double r658223 = r658221 + r658222;
        double r658224 = r658218 * r658223;
        double r658225 = r658224 / r658220;
        double r658226 = -6.81649671214237e+110;
        bool r658227 = r658225 <= r658226;
        double r658228 = 4.167561421650247e-52;
        bool r658229 = r658225 <= r658228;
        double r658230 = !r658229;
        bool r658231 = r658227 || r658230;
        double r658232 = r658218 / r658220;
        double r658233 = r658222 + r658219;
        double r658234 = r658232 * r658233;
        double r658235 = r658234 - r658218;
        double r658236 = r658220 / r658223;
        double r658237 = r658218 / r658236;
        double r658238 = r658231 ? r658235 : r658237;
        return r658238;
}

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

Original10.9
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -6.81649671214237e+110 or 4.167561421650247e-52 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 19.3

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

    3. Applied associate-/l*4.9

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

    4. Taylor expanded around 0 6.3

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

    5. Taylor expanded around 0 6.3

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

    6. Simplified0.2

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

    if -6.81649671214237e+110 < (/ (* x (+ (- y z) 1.0)) z) < 4.167561421650247e-52

    1. Initial program 0.2

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

    3. Applied associate-/l*0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.816496712142369533985882229910807371226 \cdot 10^{110} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 4.167561421650247042871141348207441053161 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))