Average Error: 3.6 → 1.7
Time: 30.7s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\mathsf{fma}\left(x \cdot z, y, x\right) - x \cdot z \]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\mathsf{fma}\left(x \cdot z, y, x\right) - x \cdot z

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.6
Target0.2
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Initial program 3.6

    \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

  2. Simplified3.6

    \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)} \]

  3. Taylor expanded in y around 0 1.7

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

  4. Taylor expanded in y around inf 1.7

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

  5. Simplified1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))