Average Error: 46.3 → 45.2
Time: 30.4s
Precision: binary64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot t\right)\right)\]
\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(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot t\right)\right)
(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (* x (cos (* (+ 1.0 (* 2.0 a)) (* (/ b 16.0) t)))))
double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos(((((y * 2.0) + 1.0) * z) * t) / 16.0)) * cos(((((a * 2.0) + 1.0) * b) * t) / 16.0);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return x * cos((1.0 + (2.0 * a)) * ((b / 16.0) * t));
}

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

Original46.3
Target44.5
Herbie45.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.3

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

  2. Simplified45.7

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

  3. Taylor expanded around 0 45.2

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

  4. Final simplification45.2

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

Reproduce

herbie shell --seed 2021176 
(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))))