Average Error: 10.5 → 0.7
Time: 3.9s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -4.54809909365855772791856768185597692522 \cdot 10^{-9} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 1.231384797345861064411732315459653816138 \cdot 10^{161}\right):\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\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):\\
\;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r594820 = x;
        double r594821 = y;
        double r594822 = z;
        double r594823 = r594821 - r594822;
        double r594824 = 1.0;
        double r594825 = r594823 + r594824;
        double r594826 = r594820 * r594825;
        double r594827 = r594826 / r594822;
        return r594827;
}
↓
double f(double x, double y, double z) {
        double r594828 = x;
        double r594829 = y;
        double r594830 = z;
        double r594831 = r594829 - r594830;
        double r594832 = 1.0;
        double r594833 = r594831 + r594832;
        double r594834 = r594828 * r594833;
        double r594835 = r594834 / r594830;
        double r594836 = -4.548099093658558e-09;
        bool r594837 = r594835 <= r594836;
        double r594838 = 1.231384797345861e+161;
        bool r594839 = r594835 <= r594838;
        double r594840 = !r594839;
        bool r594841 = r594837 || r594840;
        double r594842 = r594828 / r594830;
        double r594843 = r594833 * r594842;
        double r594844 = r594830 / r594833;
        double r594845 = r594828 / r594844;
        double r594846 = r594841 ? r594843 : r594845;
        return r594846;
}

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 clear-num6.2

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

    7. Applied div-inv6.3

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

    8. Applied add-sqr-sqrt6.3

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

    9. Applied times-frac5.7

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

    10. Applied *-un-lft-identity5.7

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

    11. Applied times-frac0.3

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

    12. Simplified0.3

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

    13. Simplified0.2

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

    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

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

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))