Average Error: 46.7 → 45.9
Time: 19.3s
Precision: binary64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \left(\sqrt[3]{\sqrt[3]{t}} \cdot {\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{2}\right)\right)\right)\right)\]
\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(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \left(\sqrt[3]{\sqrt[3]{t}} \cdot {\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{2}\right)\right)\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) (((double) (((double) cbrt(t)) * ((double) cbrt(t)))) * ((double) (((double) (1.0 + ((double) (2.0 * a)))) * ((double) (((double) (b / 16.0)) * ((double) (((double) cbrt(((double) cbrt(t)))) * ((double) pow(((double) cbrt(((double) cbrt(t)))), 2.0))))))))))))));
}

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

Original46.7
Target45.0
Herbie45.9
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.7

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

  2. Simplified46.4

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

  3. Taylor expanded around 0 46.0

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

  5. Applied add-cube-cbrt46.0

    \[\leadsto x \cdot \left(1 \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \cdot \left(\left(1 + 2 \cdot a\right) \cdot \frac{b}{16}\right)\right)\right)\]

  6. Applied associate-*l*46.0

    \[\leadsto x \cdot \left(1 \cdot \cos \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(\left(1 + 2 \cdot a\right) \cdot \frac{b}{16}\right)\right)\right)}\right)\]

  7. Simplified45.9

    \[\leadsto x \cdot \left(1 \cdot \cos \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \color{blue}{\left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \sqrt[3]{t}\right)\right)}\right)\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrt45.9

    \[\leadsto x \cdot \left(1 \cdot \cos \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t}}\right)}\right)\right)\right)\right)\]

  10. Simplified45.9

    \[\leadsto x \cdot \left(1 \cdot \cos \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)\right)\right)\right)\right)\]

  11. Final simplification45.9

    \[\leadsto x \cdot \cos \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{b}{16} \cdot \left(\sqrt[3]{\sqrt[3]{t}} \cdot {\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{2}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020181 
(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))))