Average Error: 10.3 → 3.6
Time: 21.8s
Precision: 64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\left(\frac{x \cdot y}{z} + 1 \cdot \frac{x}{z}\right) - x
double f(double x, double y, double z) {
        double r481802 = x;
        double r481803 = y;
        double r481804 = z;
        double r481805 = r481803 - r481804;
        double r481806 = 1.0;
        double r481807 = r481805 + r481806;
        double r481808 = r481802 * r481807;
        double r481809 = r481808 / r481804;
        return r481809;
}
↓
double f(double x, double y, double z) {
        double r481810 = x;
        double r481811 = y;
        double r481812 = r481810 * r481811;
        double r481813 = z;
        double r481814 = r481812 / r481813;
        double r481815 = 1.0;
        double r481816 = r481810 / r481813;
        double r481817 = r481815 * r481816;
        double r481818 = r481814 + r481817;
        double r481819 = r481818 - r481810;
        return r481819;
}

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

Original10.3
Target0.4
Herbie3.6
\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.3

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

  2. Taylor expanded around 0 3.6

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

  3. Simplified1.6

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

  5. Applied distribute-lft-in1.6

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

  6. Simplified3.6

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

  7. Simplified3.6

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

  8. Final simplification3.6

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.7148310671343599e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.87410881643954616e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))