Average Error: 46.7 → 44.5
Time: 20.1s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r1067939 = x;
        double r1067940 = y;
        double r1067941 = 2.0;
        double r1067942 = r1067940 * r1067941;
        double r1067943 = 1.0;
        double r1067944 = r1067942 + r1067943;
        double r1067945 = z;
        double r1067946 = r1067944 * r1067945;
        double r1067947 = t;
        double r1067948 = r1067946 * r1067947;
        double r1067949 = 16.0;
        double r1067950 = r1067948 / r1067949;
        double r1067951 = cos(r1067950);
        double r1067952 = r1067939 * r1067951;
        double r1067953 = a;
        double r1067954 = r1067953 * r1067941;
        double r1067955 = r1067954 + r1067943;
        double r1067956 = b;
        double r1067957 = r1067955 * r1067956;
        double r1067958 = r1067957 * r1067947;
        double r1067959 = r1067958 / r1067949;
        double r1067960 = cos(r1067959);
        double r1067961 = r1067952 * r1067960;
        return r1067961;
}
↓
double f(double x, double __attribute__((unused)) y, double __attribute__((unused)) z, double __attribute__((unused)) t, double __attribute__((unused)) a, double __attribute__((unused)) b) {
        double r1067962 = x;
        double r1067963 = 0.0;
        double r1067964 = 16.0;
        double r1067965 = r1067963 / r1067964;
        double r1067966 = cos(r1067965);
        double r1067967 = r1067962 * r1067966;
        return r1067967;
}

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
Target44.8
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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. Simplified46.7

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

  3. Taylor expanded around 0 45.8

    \[\leadsto \cos \left(\frac{\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right)\]

  4. Taylor expanded around 0 44.5

    \[\leadsto \color{blue}{1} \cdot \left(x \cdot \cos \left(\frac{0}{16}\right)\right)\]

  5. Final simplification44.5

    \[\leadsto x \cdot \cos \left(\frac{0}{16}\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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))))