Average Error: 0.1 → 0.0
Time: 5.0s
Precision: binary64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
↓
\[\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right) \]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))
↓
(FPCore (x y z) :precision binary64 (fma 4.0 (/ (- x y) z) -2.0))
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 - y) / z), -2.0);
}

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. Simplified0.0

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

    Proof

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022004 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))