Average Error: 46.1 → 44.0
Time: 17.6s
Precision: binary64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.07481359844060788 \cdot 10^{-243}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 2.1761479401547903 \cdot 10^{-75}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.07481359844060788 \cdot 10^{-243}:\\
\;\;\;\;x\\

\mathbf{elif}\;t \le 2.1761479401547903 \cdot 10^{-75}:\\
\;\;\;\;\left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (x * ((double) cos(((double) (((double) (((double) (((double) (((double) (y * 2.0)) + 1.0)) * z)) * t)) / 16.0)))))) * ((double) cos(((double) (((double) (((double) (((double) (((double) (a * 2.0)) + 1.0)) * b)) * t)) / 16.0))))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((t <= -2.074813598440608e-243)) {
		VAR = x;
	} else {
		double VAR_1;
		if ((t <= 2.1761479401547903e-75)) {
			VAR_1 = ((double) (((double) (x * ((double) cos(((double) (((double) (((double) (((double) (y * 2.0)) + 1.0)) * ((double) (z * t)))) / 16.0)))))) * ((double) cos(((double) (((double) (((double) (((double) (((double) (a * 2.0)) + 1.0)) * b)) * t)) / 16.0))))));
		} else {
			VAR_1 = x;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original46.1
Target44.3
Herbie44.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.07481359844060788e-243 or 2.1761479401547903e-75 < t

    1. Initial program 52.0

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    2. Taylor expanded around 0 51.0

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    3. Taylor expanded around 0 49.4

      \[\leadsto \color{blue}{x}\]

    if -2.07481359844060788e-243 < t < 2.1761479401547903e-75

    1. Initial program 27.0

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
    2. Using strategy rm

    3. Applied associate-*l*26.2

      \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification44.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.07481359844060788 \cdot 10^{-243}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 2.1761479401547903 \cdot 10^{-75}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(y \cdot 2 + 1\right) \cdot \left(z \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))