Average Error: 0.1 → 0.0
Time: 8.9s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[4 \cdot \left(\frac{x}{z} - \frac{y}{z}\right) - 2\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
4 \cdot \left(\frac{x}{z} - \frac{y}{z}\right) - 2
double f(double x, double y, double z) {
        double r616829 = 4.0;
        double r616830 = x;
        double r616831 = y;
        double r616832 = r616830 - r616831;
        double r616833 = z;
        double r616834 = 0.5;
        double r616835 = r616833 * r616834;
        double r616836 = r616832 - r616835;
        double r616837 = r616829 * r616836;
        double r616838 = r616837 / r616833;
        return r616838;
}
↓
double f(double x, double y, double z) {
        double r616839 = 4.0;
        double r616840 = x;
        double r616841 = z;
        double r616842 = r616840 / r616841;
        double r616843 = y;
        double r616844 = r616843 / r616841;
        double r616845 = r616842 - r616844;
        double r616846 = r616839 * r616845;
        double r616847 = 2.0;
        double r616848 = r616846 - r616847;
        return r616848;
}

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

Original0.1
Target0.0
Herbie0.0
\[4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - \mathsf{fma}\left(0.5, z, y\right)}{\frac{z}{4}}}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - \left(4 \cdot \frac{y}{z} + 2\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{4 \cdot \left(\frac{x}{z} - \frac{y}{z}\right) - 2}\]

  5. Final simplification0.0

    \[\leadsto 4 \cdot \left(\frac{x}{z} - \frac{y}{z}\right) - 2\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z))))

  (/ (* 4 (- (- x y) (* z 0.5))) z))