Average Error: 46.3 → 44.3
Time: 10.0s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot x\]
\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)
↓
\cos \left(\frac{0}{16}\right) \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r874043 = x;
        double r874044 = y;
        double r874045 = 2.0;
        double r874046 = r874044 * r874045;
        double r874047 = 1.0;
        double r874048 = r874046 + r874047;
        double r874049 = z;
        double r874050 = r874048 * r874049;
        double r874051 = t;
        double r874052 = r874050 * r874051;
        double r874053 = 16.0;
        double r874054 = r874052 / r874053;
        double r874055 = cos(r874054);
        double r874056 = r874043 * r874055;
        double r874057 = a;
        double r874058 = r874057 * r874045;
        double r874059 = r874058 + r874047;
        double r874060 = b;
        double r874061 = r874059 * r874060;
        double r874062 = r874061 * r874051;
        double r874063 = r874062 / r874053;
        double r874064 = cos(r874063);
        double r874065 = r874056 * r874064;
        return r874065;
}
↓
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 r874066 = 0.0;
        double r874067 = 16.0;
        double r874068 = r874066 / r874067;
        double r874069 = cos(r874068);
        double r874070 = x;
        double r874071 = r874069 * r874070;
        return r874071;
}

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
Herbie44.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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

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

  3. Taylor expanded around 0 44.3

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

  4. Final simplification44.3

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

Reproduce

herbie shell --seed 2020003 
(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))))