Average Error: 0.1 → 0.0
Time: 1.1s
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 code(double x, double y, double z) {
	return ((4.0 * ((x - y) - (z * 0.5))) / z);
}
↓
double code(double x, double y, double z) {
	return fma(4.0, (x / z), -fma(4.0, (y / z), 2.0));
}

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. Simplified0.3

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

  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 2020105 +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))