Average Error: 0.2 → 0.0
Time: 3.5s
Precision: binary64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
↓
\[4 \cdot \frac{x}{y} + \mathsf{fma}\left(\frac{z}{y}, -4, 4\right) \]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
4 \cdot \frac{x}{y} + \mathsf{fma}\left(\frac{z}{y}, -4, 4\right)

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
↓
(FPCore (x y z) :precision binary64 (+ (* 4.0 (/ x y)) (fma (/ z y) -4.0 4.0)))
double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}
↓
double code(double x, double y, double z) {
	return (4.0 * (x / y)) + fma((z / y), -4.0, 4.0);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]

  2. Taylor expanded in x around 0 0.0

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

  3. Applied associate--l+_binary640.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))