Average Error: 46.2 → 44.2
Time: 15.2s
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 r785025 = x;
        double r785026 = y;
        double r785027 = 2.0;
        double r785028 = r785026 * r785027;
        double r785029 = 1.0;
        double r785030 = r785028 + r785029;
        double r785031 = z;
        double r785032 = r785030 * r785031;
        double r785033 = t;
        double r785034 = r785032 * r785033;
        double r785035 = 16.0;
        double r785036 = r785034 / r785035;
        double r785037 = cos(r785036);
        double r785038 = r785025 * r785037;
        double r785039 = a;
        double r785040 = r785039 * r785027;
        double r785041 = r785040 + r785029;
        double r785042 = b;
        double r785043 = r785041 * r785042;
        double r785044 = r785043 * r785033;
        double r785045 = r785044 / r785035;
        double r785046 = cos(r785045);
        double r785047 = r785038 * r785046;
        return r785047;
}
↓
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 r785048 = 0.0;
        double r785049 = 16.0;
        double r785050 = r785048 / r785049;
        double r785051 = cos(r785050);
        double r785052 = x;
        double r785053 = r785051 * r785052;
        return r785053;
}

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

Original46.2
Target44.5
Herbie44.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.2

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

  2. Taylor expanded around 0 45.5

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2020045 
(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))))