Average Error: 46.3 → 45.3
Time: 20.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \log \left(e^{\cos \left(\frac{\left(\sqrt[3]{y \cdot 2 + 1} \cdot \sqrt[3]{y \cdot 2 + 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot \sqrt[3]{\sqrt[3]{y \cdot 2 + 1}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot z\right) \cdot t\right)\right)}{16}\right)}\right)\]
\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 \log \left(e^{\cos \left(\frac{\left(\sqrt[3]{y \cdot 2 + 1} \cdot \sqrt[3]{y \cdot 2 + 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot \sqrt[3]{\sqrt[3]{y \cdot 2 + 1}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot z\right) \cdot t\right)\right)}{16}\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) log(((double) exp(((double) cos(((double) (((double) (((double) (((double) cbrt(((double) (((double) (y * 2.0)) + 1.0)))) * ((double) cbrt(((double) (((double) (y * 2.0)) + 1.0)))))) * ((double) (((double) (((double) cbrt(((double) cbrt(((double) (((double) (y * 2.0)) + 1.0)))))) * ((double) cbrt(((double) cbrt(((double) (((double) (y * 2.0)) + 1.0)))))))) * ((double) (((double) (((double) cbrt(((double) cbrt(((double) (((double) (y * 2.0)) + 1.0)))))) * z)) * t)))))) / 16.0))))))))));
}

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

Original46.3
Target44.6
Herbie45.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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. Taylor expanded around 0 45.6

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

  4. Applied add-cube-cbrt45.5

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

  5. Applied associate-*l*45.5

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

  6. Applied associate-*l*45.4

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

  8. Applied add-cube-cbrt45.3

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

  9. Applied associate-*l*45.3

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

  10. Applied associate-*l*45.3

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

  12. Applied add-log-exp45.3

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

  13. Final simplification45.3

    \[\leadsto x \cdot \log \left(e^{\cos \left(\frac{\left(\sqrt[3]{y \cdot 2 + 1} \cdot \sqrt[3]{y \cdot 2 + 1}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot \sqrt[3]{\sqrt[3]{y \cdot 2 + 1}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y \cdot 2 + 1}} \cdot z\right) \cdot t\right)\right)}{16}\right)}\right)\]

Reproduce

herbie shell --seed 2020114 
(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) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2))))))

  (* (* x (cos (/ (* (* (+ (* y 2) 1) z) t) 16))) (cos (/ (* (* (+ (* a 2) 1) b) t) 16))))