Average Error: 10.0 → 1.7
Time: 2.5s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\
\;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{z}\right) - x\\

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

\end{array}
double f(double x, double y, double z) {
        double r909194 = x;
        double r909195 = y;
        double r909196 = z;
        double r909197 = r909195 - r909196;
        double r909198 = 1.0;
        double r909199 = r909197 + r909198;
        double r909200 = r909194 * r909199;
        double r909201 = r909200 / r909196;
        return r909201;
}
↓
double f(double x, double y, double z) {
        double r909202 = x;
        double r909203 = 3.5216806508699143e-102;
        bool r909204 = r909202 <= r909203;
        double r909205 = y;
        double r909206 = r909202 * r909205;
        double r909207 = 1.0;
        double r909208 = z;
        double r909209 = r909207 / r909208;
        double r909210 = r909206 * r909209;
        double r909211 = 1.0;
        double r909212 = r909202 / r909208;
        double r909213 = r909211 * r909212;
        double r909214 = r909210 + r909213;
        double r909215 = r909214 - r909202;
        double r909216 = r909211 + r909205;
        double r909217 = r909212 * r909216;
        double r909218 = r909217 - r909202;
        double r909219 = r909204 ? r909215 : r909218;
        return r909219;
}

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

Original10.0
Target0.5
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.5216806508699143e-102

    1. Initial program 7.1

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

    2. Taylor expanded around 0 2.3

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

    4. Applied div-inv2.3

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

    if 3.5216806508699143e-102 < x

    1. Initial program 17.1

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

    2. Taylor expanded around 0 5.6

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

    3. Taylor expanded around 0 5.6

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

    4. Simplified0.3

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(1 + y\right)} - x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.5216806508699143 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(x \cdot y\right) \cdot \frac{1}{z} + 1 \cdot \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))