Average Error: 10.5 → 0.7
Time: 4.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\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 -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\
\;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r713566 = x;
        double r713567 = y;
        double r713568 = z;
        double r713569 = r713567 - r713568;
        double r713570 = 1.0;
        double r713571 = r713569 + r713570;
        double r713572 = r713566 * r713571;
        double r713573 = r713572 / r713568;
        return r713573;
}
↓
double f(double x, double y, double z) {
        double r713574 = x;
        double r713575 = y;
        double r713576 = z;
        double r713577 = r713575 - r713576;
        double r713578 = 1.0;
        double r713579 = r713577 + r713578;
        double r713580 = r713574 * r713579;
        double r713581 = r713580 / r713576;
        double r713582 = -4.548099093658558e-09;
        bool r713583 = r713581 <= r713582;
        double r713584 = 1.231384797345861e+161;
        bool r713585 = r713581 <= r713584;
        double r713586 = !r713585;
        bool r713587 = r713583 || r713586;
        double r713588 = r713574 / r713576;
        double r713589 = r713588 * r713579;
        double r713590 = r713576 / r713579;
        double r713591 = r713574 / r713590;
        double r713592 = r713587 ? r713589 : r713591;
        return r713592;
}

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

Original10.5
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (+ (- y z) 1.0)) z) < -4.548099093658558e-09 or 1.231384797345861e+161 < (/ (* x (+ (- y z) 1.0)) z)

    1. Initial program 21.5

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

    3. Applied associate-/l*5.6

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

    5. Applied associate-/r/0.2

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

    if -4.548099093658558e-09 < (/ (* x (+ (- y z) 1.0)) z) < 1.231384797345861e+161

    1. Initial program 0.1

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

    3. Applied associate-/l*1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))