Average Error: 46.2 → 45.1
Time: 16.1s
Precision: binary64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\]
\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)
↓
x \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)
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) {
	return ((double) (x * ((double) cos(((double) (t * ((double) (b * 0.0625))))))));
}

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

Original46.2
Target44.8
Herbie45.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.2

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

  2. Simplified46.0

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

  3. Taylor expanded around 0 45.6

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

  4. Taylor expanded around inf 45.4

    \[\leadsto x \cdot \left(1 \cdot \cos \color{blue}{\left(0.0625 \cdot \left(t \cdot b\right) + 0.125 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)}\right)\]

  5. Simplified45.4

    \[\leadsto x \cdot \left(1 \cdot \cos \color{blue}{\left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)}\right)\]

  6. Taylor expanded around 0 45.1

    \[\leadsto x \cdot \left(1 \cdot \cos \color{blue}{\left(0.0625 \cdot \left(t \cdot b\right)\right)}\right)\]

  7. Simplified45.1

    \[\leadsto x \cdot \left(1 \cdot \cos \color{blue}{\left(t \cdot \left(b \cdot 0.0625\right)\right)}\right)\]

  8. Final simplification45.1

    \[\leadsto x \cdot \cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\]

Reproduce

herbie shell --seed 2020184 
(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))))