Average Error: 10.2 → 1.1
Time: 2.2s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.12244133704874589 \cdot 10^{104}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{elif}\;z \le 2.007427933209735 \cdot 10^{67}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot 1 + \left(y \cdot \frac{x}{z} - x\right)\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.12244133704874589 \cdot 10^{104}:\\
\;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\

\mathbf{elif}\;z \le 2.007427933209735 \cdot 10^{67}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r487387 = x;
        double r487388 = y;
        double r487389 = z;
        double r487390 = r487388 - r487389;
        double r487391 = 1.0;
        double r487392 = r487390 + r487391;
        double r487393 = r487387 * r487392;
        double r487394 = r487393 / r487389;
        return r487394;
}
↓
double f(double x, double y, double z) {
        double r487395 = z;
        double r487396 = -1.1224413370487459e+104;
        bool r487397 = r487395 <= r487396;
        double r487398 = x;
        double r487399 = y;
        double r487400 = r487399 - r487395;
        double r487401 = 1.0;
        double r487402 = r487400 + r487401;
        double r487403 = r487402 / r487395;
        double r487404 = r487398 * r487403;
        double r487405 = 2.007427933209735e+67;
        bool r487406 = r487395 <= r487405;
        double r487407 = r487398 * r487399;
        double r487408 = r487407 / r487395;
        double r487409 = r487398 / r487395;
        double r487410 = r487401 * r487409;
        double r487411 = r487408 + r487410;
        double r487412 = r487411 - r487398;
        double r487413 = r487409 * r487401;
        double r487414 = r487399 * r487409;
        double r487415 = r487414 - r487398;
        double r487416 = r487413 + r487415;
        double r487417 = r487406 ? r487412 : r487416;
        double r487418 = r487397 ? r487404 : r487417;
        return r487418;
}

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

Original10.2
Target0.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1224413370487459e+104

    1. Initial program 21.4

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

    3. Applied *-un-lft-identity21.4

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

    4. Applied times-frac0.1

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

    5. Simplified0.1

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

    if -1.1224413370487459e+104 < z < 2.007427933209735e+67

    1. Initial program 1.6

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

    2. Taylor expanded around 0 0.8

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

    if 2.007427933209735e+67 < z

    1. Initial program 20.8

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

    2. Taylor expanded around 0 6.5

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

    3. Taylor expanded around 0 6.5

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

    4. Simplified2.5

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

    6. Applied distribute-lft-in2.5

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

    7. Applied associate--l+2.5

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

    8. Simplified2.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.12244133704874589 \cdot 10^{104}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \mathbf{elif}\;z \le 2.007427933209735 \cdot 10^{67}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot 1 + \left(y \cdot \frac{x}{z} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))