Average Error: 3.5 → 0.9
Time: 4.5s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;x \leq -2.114852865568445 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{elif}\;x \leq 7.834655119261467 \cdot 10^{-123}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(1, 1, z \cdot \left(y + -1\right)\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, t_0\right)\\ \end{array} \]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\begin{array}{l}
t_0 := \left(1 - y\right) \cdot z\\
\mathbf{if}\;x \leq -2.114852865568445 \cdot 10^{+126}:\\
\;\;\;\;x \cdot \left(1 - t_0\right)\\

\mathbf{elif}\;x \leq 7.834655119261467 \cdot 10^{-123}:\\
\;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \mathsf{fma}\left(1, 1, z \cdot \left(y + -1\right)\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, t_0\right)\\


\end{array}

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= x -2.114852865568445e+126)
     (* x (- 1.0 t_0))
     (if (<= x 7.834655119261467e-123)
       (+ x (* z (- (* x y) x)))
       (+
        (* x (fma 1.0 1.0 (* z (+ y -1.0))))
        (* x (fma (- z) (- 1.0 y) t_0)))))))
double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}
↓
double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (x <= -2.114852865568445e+126) {
		tmp = x * (1.0 - t_0);
	} else if (x <= 7.834655119261467e-123) {
		tmp = x + (z * ((x * y) - x));
	} else {
		tmp = (x * fma(1.0, 1.0, (z * (y + -1.0)))) + (x * fma(-z, (1.0 - y), t_0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.5
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.114852865568445e126

    1. Initial program 0.1

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

    if -2.114852865568445e126 < x < 7.8346551192614673e-123

    1. Initial program 5.2

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

    2. Applied sub-neg_binary645.2

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

    3. Applied distribute-rgt-in_binary645.2

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

    4. Simplified5.2

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

    5. Simplified3.0

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

    6. Taylor expanded in z around 0 0.8

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

    if 7.8346551192614673e-123 < x

    1. Initial program 1.3

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

    2. Applied *-un-lft-identity_binary641.3

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

    3. Applied prod-diff_binary641.3

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

    4. Applied distribute-rgt-in_binary641.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, 1, -z \cdot \left(1 - y\right)\right) \cdot x + \mathsf{fma}\left(-z, 1 - y, z \cdot \left(1 - y\right)\right) \cdot x} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.114852865568445 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{elif}\;x \leq 7.834655119261467 \cdot 10^{-123}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(1, 1, z \cdot \left(y + -1\right)\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, \left(1 - y\right) \cdot z\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))