Average Error: 46.9 → 45.8
Time: 18.4s
Precision: binary64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\]
\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(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\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 (* (* t b) (+ 0.0625 (* a 0.125))))))
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((t * b) * (0.0625 + (a * 0.125)));
}

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

Original46.9
Target45.1
Herbie45.8
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.9

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

  2. Simplified46.3

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

  3. Taylor expanded around 0 45.8

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

  4. Final simplification45.8

    \[\leadsto x \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))