Average Error: 46.0 → 45.5
Time: 53.6s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[\left(\left(\left(\sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)}\right)\right)\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
\left(\left(\left(\sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)}\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r45103904 = x;
        double r45103905 = y;
        double r45103906 = 2.0;
        double r45103907 = r45103905 * r45103906;
        double r45103908 = 1.0;
        double r45103909 = r45103907 + r45103908;
        double r45103910 = z;
        double r45103911 = r45103909 * r45103910;
        double r45103912 = t;
        double r45103913 = r45103911 * r45103912;
        double r45103914 = 16.0;
        double r45103915 = r45103913 / r45103914;
        double r45103916 = cos(r45103915);
        double r45103917 = r45103904 * r45103916;
        double r45103918 = a;
        double r45103919 = r45103918 * r45103906;
        double r45103920 = r45103919 + r45103908;
        double r45103921 = b;
        double r45103922 = r45103920 * r45103921;
        double r45103923 = r45103922 * r45103912;
        double r45103924 = r45103923 / r45103914;
        double r45103925 = cos(r45103924);
        double r45103926 = r45103917 * r45103925;
        return r45103926;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r45103927 = y;
        double r45103928 = 2.0;
        double r45103929 = r45103927 * r45103928;
        double r45103930 = 1.0;
        double r45103931 = r45103929 + r45103930;
        double r45103932 = z;
        double r45103933 = t;
        double r45103934 = r45103932 * r45103933;
        double r45103935 = r45103931 * r45103934;
        double r45103936 = 16.0;
        double r45103937 = r45103935 / r45103936;
        double r45103938 = cos(r45103937);
        double r45103939 = cbrt(r45103938);
        double r45103940 = r45103939 * r45103939;
        double r45103941 = r45103940 * r45103939;
        double r45103942 = x;
        double r45103943 = r45103941 * r45103942;
        double r45103944 = b;
        double r45103945 = r45103933 * r45103944;
        double r45103946 = a;
        double r45103947 = r45103928 * r45103946;
        double r45103948 = r45103930 + r45103947;
        double r45103949 = r45103945 * r45103948;
        double r45103950 = r45103949 / r45103936;
        double r45103951 = cos(r45103950);
        double r45103952 = cbrt(r45103951);
        double r45103953 = r45103952 * r45103952;
        double r45103954 = r45103952 * r45103953;
        double r45103955 = r45103943 * r45103954;
        return r45103955;
}

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

Original46.0
Target44.7
Herbie45.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
  2. Using strategy rm

  3. Applied associate-*l*45.8

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}}{16.0}\right)\]

  4. Taylor expanded around inf 45.8

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{2.0 \cdot \left(t \cdot \left(z \cdot y\right)\right) + 1.0 \cdot \left(t \cdot z\right)}}{16.0}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)\]

  5. Simplified45.5

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}}{16.0}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)\]
  6. Using strategy rm

  7. Applied add-cube-cbrt45.5

    \[\leadsto \left(x \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)}\right)}\right) \cdot \cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrt45.5

    \[\leadsto \left(x \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot z\right) \cdot \left(1.0 + y \cdot 2.0\right)}{16.0}\right)}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(a \cdot 2.0 + 1.0\right) \cdot \left(b \cdot t\right)}{16.0}\right)}\right)}\]

  10. Final simplification45.5

    \[\leadsto \left(\left(\left(\sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\left(y \cdot 2.0 + 1.0\right) \cdot \left(z \cdot t\right)}{16.0}\right)}\right) \cdot x\right) \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)} \cdot \sqrt[3]{\cos \left(\frac{\left(t \cdot b\right) \cdot \left(1.0 + 2.0 \cdot a\right)}{16.0}\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))