Average Error: 0.1 → 0.0
Time: 1.3s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)
double f(double x, double y, double z) {
        double r990567 = 4.0;
        double r990568 = x;
        double r990569 = y;
        double r990570 = r990568 - r990569;
        double r990571 = z;
        double r990572 = 0.5;
        double r990573 = r990571 * r990572;
        double r990574 = r990570 - r990573;
        double r990575 = r990567 * r990574;
        double r990576 = r990575 / r990571;
        return r990576;
}
↓
double f(double x, double y, double z) {
        double r990577 = 4.0;
        double r990578 = x;
        double r990579 = z;
        double r990580 = r990578 / r990579;
        double r990581 = y;
        double r990582 = r990581 / r990579;
        double r990583 = r990580 - r990582;
        double r990584 = 2.0;
        double r990585 = -r990584;
        double r990586 = fma(r990577, r990583, r990585);
        return r990586;
}

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. Using strategy rm

  5. Applied div-sub0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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))