Average Error: 0.2 → 0.0
Time: 2.9s
Precision: binary64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y} \]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y}

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

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. Simplified0.2

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

  3. Taylor expanded in y around 0 0.0

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

  4. Taylor expanded in x around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(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)))