Average Error: 10.2 → 0.1
Time: 7.7s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\left(1 \cdot \frac{x}{z} + \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\
\;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\

\mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\
\;\;\;\;\left(1 \cdot \frac{x}{z} + \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r563042 = x;
        double r563043 = y;
        double r563044 = z;
        double r563045 = r563043 - r563044;
        double r563046 = 1.0;
        double r563047 = r563045 + r563046;
        double r563048 = r563042 * r563047;
        double r563049 = r563048 / r563044;
        return r563049;
}
↓
double f(double x, double y, double z) {
        double r563050 = x;
        double r563051 = -3.4374981009777875e+27;
        bool r563052 = r563050 <= r563051;
        double r563053 = z;
        double r563054 = r563050 / r563053;
        double r563055 = 1.0;
        double r563056 = y;
        double r563057 = r563055 + r563056;
        double r563058 = r563054 * r563057;
        double r563059 = r563058 - r563050;
        double r563060 = 8.649219799147649e-36;
        bool r563061 = r563050 <= r563060;
        double r563062 = r563055 * r563054;
        double r563063 = r563050 * r563056;
        double r563064 = 1.0;
        double r563065 = r563064 / r563053;
        double r563066 = r563063 * r563065;
        double r563067 = r563062 + r563066;
        double r563068 = r563067 - r563050;
        double r563069 = r563056 - r563053;
        double r563070 = r563069 + r563055;
        double r563071 = r563053 / r563070;
        double r563072 = r563050 / r563071;
        double r563073 = r563061 ? r563068 : r563072;
        double r563074 = r563052 ? r563059 : r563073;
        return r563074;
}

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
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -3.4374981009777875e+27

    1. Initial program 28.2

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

    2. Taylor expanded around 0 9.1

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

    3. Simplified0.1

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

    if -3.4374981009777875e+27 < x < 8.649219799147649e-36

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified3.0

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

    5. Applied distribute-lft-in3.0

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

    6. Simplified3.0

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

    7. Simplified0.1

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

    9. Applied div-inv0.1

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

    if 8.649219799147649e-36 < x

    1. Initial program 21.3

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

    3. Applied associate-/l*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.4374981009777875 \cdot 10^{27}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 8.649219799147649 \cdot 10^{-36}:\\ \;\;\;\;\left(1 \cdot \frac{x}{z} + \left(x \cdot y\right) \cdot \frac{1}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))