Average Error: 9.4 → 0.9
Time: 12.3s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 1.9565246455280066 \cdot 10^{-10}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le 1.9565246455280066 \cdot 10^{-10}:\\
\;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r28613383 = x;
        double r28613384 = y;
        double r28613385 = z;
        double r28613386 = r28613384 - r28613385;
        double r28613387 = 1.0;
        double r28613388 = r28613386 + r28613387;
        double r28613389 = r28613383 * r28613388;
        double r28613390 = r28613389 / r28613385;
        return r28613390;
}
↓
double f(double x, double y, double z) {
        double r28613391 = z;
        double r28613392 = 1.9565246455280066e-10;
        bool r28613393 = r28613391 <= r28613392;
        double r28613394 = 1.0;
        double r28613395 = x;
        double r28613396 = r28613395 / r28613391;
        double r28613397 = r28613394 * r28613396;
        double r28613398 = y;
        double r28613399 = r28613396 * r28613398;
        double r28613400 = r28613397 + r28613399;
        double r28613401 = r28613400 - r28613395;
        double r28613402 = r28613398 - r28613391;
        double r28613403 = r28613402 + r28613394;
        double r28613404 = r28613403 / r28613391;
        double r28613405 = r28613395 * r28613404;
        double r28613406 = r28613393 ? r28613401 : r28613405;
        return r28613406;
}

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

Original9.4
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < 1.9565246455280066e-10

    1. Initial program 6.6

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*4.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}}\]

    4. Taylor expanded around 0 2.2

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + 1.0 \cdot \frac{x}{z}\right) - x}\]

    5. Simplified1.3

      \[\leadsto \color{blue}{\left(\frac{x}{z} \cdot y + \frac{x}{z} \cdot 1.0\right) - x}\]

    if 1.9565246455280066e-10 < z

    1. Initial program 15.4

      \[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}}\]
    4. Using strategy rm

    5. Applied div-inv0.1

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{z}{\left(y - z\right) + 1.0}}}\]

    6. Simplified0.1

      \[\leadsto x \cdot \color{blue}{\frac{\left(y - z\right) + 1.0}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 1.9565246455280066 \cdot 10^{-10}:\\ \;\;\;\;\left(1.0 \cdot \frac{x}{z} + \frac{x}{z} \cdot y\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1.0}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))