Average Error: 0.1 → 0.0
Time: 2.1s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)
double f(double x, double y, double z) {
        double r1176084 = 4.0;
        double r1176085 = x;
        double r1176086 = y;
        double r1176087 = r1176085 - r1176086;
        double r1176088 = z;
        double r1176089 = 0.5;
        double r1176090 = r1176088 * r1176089;
        double r1176091 = r1176087 - r1176090;
        double r1176092 = r1176084 * r1176091;
        double r1176093 = r1176092 / r1176088;
        return r1176093;
}
↓
double f(double x, double y, double z) {
        double r1176094 = 4.0;
        double r1176095 = x;
        double r1176096 = y;
        double r1176097 = r1176095 - r1176096;
        double r1176098 = z;
        double r1176099 = r1176097 / r1176098;
        double r1176100 = 2.0;
        double r1176101 = -r1176100;
        double r1176102 = fma(r1176094, r1176099, r1176101);
        return r1176102;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 +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))