Average Error: 9.6 → 0.2
Time: 12.7s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\ \mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{x}{z} \cdot 1.0 + \frac{y \cdot x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y + \frac{x}{z} \cdot 1.0\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1.0\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\

\mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\
\;\;\;\;\left(\frac{x}{z} \cdot 1.0 + \frac{y \cdot x}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r33018558 = x;
        double r33018559 = y;
        double r33018560 = z;
        double r33018561 = r33018559 - r33018560;
        double r33018562 = 1.0;
        double r33018563 = r33018561 + r33018562;
        double r33018564 = r33018558 * r33018563;
        double r33018565 = r33018564 / r33018560;
        return r33018565;
}
↓
double f(double x, double y, double z) {
        double r33018566 = x;
        double r33018567 = -1.3418778201574405e-43;
        bool r33018568 = r33018566 <= r33018567;
        double r33018569 = z;
        double r33018570 = y;
        double r33018571 = r33018570 - r33018569;
        double r33018572 = 1.0;
        double r33018573 = r33018571 + r33018572;
        double r33018574 = r33018569 / r33018573;
        double r33018575 = r33018566 / r33018574;
        double r33018576 = 1.2340705215670025e-123;
        bool r33018577 = r33018566 <= r33018576;
        double r33018578 = r33018566 / r33018569;
        double r33018579 = r33018578 * r33018572;
        double r33018580 = r33018570 * r33018566;
        double r33018581 = r33018580 / r33018569;
        double r33018582 = r33018579 + r33018581;
        double r33018583 = r33018582 - r33018566;
        double r33018584 = r33018578 * r33018570;
        double r33018585 = r33018584 + r33018579;
        double r33018586 = r33018585 - r33018566;
        double r33018587 = r33018577 ? r33018583 : r33018586;
        double r33018588 = r33018568 ? r33018575 : r33018587;
        return r33018588;
}

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

Original9.6
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1.0\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.3418778201574405e-43

    1. Initial program 20.1

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

    3. Applied associate-/l*0.2

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

    if -1.3418778201574405e-43 < x < 1.2340705215670025e-123

    1. Initial program 0.1

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

    3. Applied associate-/l*4.9

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

    4. Taylor expanded around 0 0.1

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

    5. Simplified3.5

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

    7. Applied associate-*l/0.1

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

    if 1.2340705215670025e-123 < x

    1. Initial program 15.9

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

    3. Applied associate-/l*1.0

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

    4. Taylor expanded around 0 5.5

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

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3418778201574405 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1.0}}\\ \mathbf{elif}\;x \le 1.2340705215670025 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{x}{z} \cdot 1.0 + \frac{y \cdot x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} \cdot y + \frac{x}{z} \cdot 1.0\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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))