Average Error: 3.6 → 0.1
Time: 4.6s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := 1 - \left(1 - y\right) \cdot z\\ t_0 \leq -\infty \lor \neg \left(t_0 \leq 4.9109081016256914 \cdot 10^{+188}\right) \end{array}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \end{array} \]
x \cdot \left(1 - \left(1 - y\right) \cdot z\right)
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_0 := 1 - \left(1 - y\right) \cdot z\\
t_0 \leq -\infty \lor \neg \left(t_0 \leq 4.9109081016256914 \cdot 10^{+188}\right)
\end{array}:\\
\;\;\;\;z \cdot \left(y \cdot x - x\right)\\

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


\end{array}

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (if (let* ((t_0 (- 1.0 (* (- 1.0 y) z))))
       (or (<= t_0 (- INFINITY)) (not (<= t_0 4.9109081016256914e+188))))
   (* z (- (* y x) x))
   (* x (- (fma y z 1.0) 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) {
	double t_0 = 1.0 - ((1.0 - y) * z);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 4.9109081016256914e+188)) {
		tmp = z * ((y * x) - x);
	} else {
		tmp = x * (fma(y, z, 1.0) - z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.6
Target0.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 1 (*.f64 (-.f64 1 y) z)) < -inf.0 or 4.9109081016256914e188 < (-.f64 1 (*.f64 (-.f64 1 y) z))

    1. Initial program 27.8

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

    2. Simplified27.8

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

    3. Taylor expanded in y around 0 0.5

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

    4. Taylor expanded in z around inf 0.4

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

    if -inf.0 < (-.f64 1 (*.f64 (-.f64 1 y) z)) < 4.9109081016256914e188

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(1 - y\right) \cdot z \leq -\infty \lor \neg \left(1 - \left(1 - y\right) \cdot z \leq 4.9109081016256914 \cdot 10^{+188}\right):\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))