Average Error: 46.7 → 44.6
Time: 11.4s
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 r1044081 = x;
        double r1044082 = y;
        double r1044083 = 2.0;
        double r1044084 = r1044082 * r1044083;
        double r1044085 = 1.0;
        double r1044086 = r1044084 + r1044085;
        double r1044087 = z;
        double r1044088 = r1044086 * r1044087;
        double r1044089 = t;
        double r1044090 = r1044088 * r1044089;
        double r1044091 = 16.0;
        double r1044092 = r1044090 / r1044091;
        double r1044093 = cos(r1044092);
        double r1044094 = r1044081 * r1044093;
        double r1044095 = a;
        double r1044096 = r1044095 * r1044083;
        double r1044097 = r1044096 + r1044085;
        double r1044098 = b;
        double r1044099 = r1044097 * r1044098;
        double r1044100 = r1044099 * r1044089;
        double r1044101 = r1044100 / r1044091;
        double r1044102 = cos(r1044101);
        double r1044103 = r1044094 * r1044102;
        return r1044103;
}
↓
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 r1044104 = x;
        double r1044105 = 0.0;
        double r1044106 = 16.0;
        double r1044107 = r1044105 / r1044106;
        double r1044108 = cos(r1044107);
        double r1044109 = r1044104 * r1044108;
        return r1044109;
}

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
Target45.0
Herbie44.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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. Taylor expanded around 0 45.9

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

  3. Taylor expanded around 0 44.6

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

  4. Final simplification44.6

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

Reproduce

herbie shell --seed 2020057 +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))))