Average Error: 0.2 → 0.0
Time: 1.4s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)
double f(double x, double y, double z) {
        double r760998 = 4.0;
        double r760999 = x;
        double r761000 = y;
        double r761001 = r760999 - r761000;
        double r761002 = z;
        double r761003 = 0.5;
        double r761004 = r761002 * r761003;
        double r761005 = r761001 - r761004;
        double r761006 = r760998 * r761005;
        double r761007 = r761006 / r761002;
        return r761007;
}
↓
double f(double x, double y, double z) {
        double r761008 = 4.0;
        double r761009 = x;
        double r761010 = z;
        double r761011 = r761009 / r761010;
        double r761012 = y;
        double r761013 = r761012 / r761010;
        double r761014 = 2.0;
        double r761015 = fma(r761008, r761013, r761014);
        double r761016 = -r761015;
        double r761017 = fma(r761008, r761011, r761016);
        return r761017;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.2

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

  2. Simplified0.3

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))