Average Error: 3.4 → 0.3
Time: 4.6s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;z \leq -2.426846197592874 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.61320657470593 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(\left(1 + z \cdot y\right) - z\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, z \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\begin{array}{l}
t_0 := z \cdot \left(y \cdot x - x\right)\\
\mathbf{if}\;z \leq -2.426846197592874 \cdot 10^{+27}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;z \leq 5.61320657470593 \cdot 10^{-89}:\\
\;\;\;\;x \cdot \left(\left(1 + z \cdot y\right) - z\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, z \cdot \left(1 - y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + t_0\\


\end{array}

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- (* y x) x))))
   (if (<= z -2.426846197592874e+27)
     t_0
     (if (<= z 5.61320657470593e-89)
       (+
        (* x (- (+ 1.0 (* z y)) z))
        (* x (fma (- z) (- 1.0 y) (* z (- 1.0 y)))))
       (+ x 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 = z * ((y * x) - x);
	double tmp;
	if (z <= -2.426846197592874e+27) {
		tmp = t_0;
	} else if (z <= 5.61320657470593e-89) {
		tmp = (x * ((1.0 + (z * y)) - z)) + (x * fma(-z, (1.0 - y), (z * (1.0 - y))));
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.42684619759287393e27

    1. Initial program 9.7

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

    2. Applied sub-neg_binary649.7

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

    3. Applied distribute-rgt-in_binary649.7

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

    4. Simplified9.7

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

    5. Simplified0.1

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

    6. Taylor expanded in z around 0 0.1

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

    7. Taylor expanded in z around inf 0.1

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

    if -2.42684619759287393e27 < z < 5.61320657470593017e-89

    1. Initial program 0.1

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

    2. Applied *-un-lft-identity_binary640.1

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

    3. Applied prod-diff_binary640.1

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

    4. Applied distribute-rgt-in_binary640.1

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

    5. Taylor expanded in x around 0 0.1

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

    if 5.61320657470593017e-89 < z

    1. Initial program 6.1

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

    2. Applied sub-neg_binary646.1

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

    3. Applied distribute-rgt-in_binary646.1

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

    4. Simplified6.1

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

    5. Simplified0.5

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

    6. Taylor expanded in z around 0 0.9

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.426846197592874 \cdot 10^{+27}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;z \leq 5.61320657470593 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \left(\left(1 + z \cdot y\right) - z\right) + x \cdot \mathsf{fma}\left(-z, 1 - y, z \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))