Average Error: 46.4 → 44.4
Time: 33.5s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\]
\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)
↓
\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r5021 = x;
        double r5022 = y;
        double r5023 = 2.0;
        double r5024 = r5022 * r5023;
        double r5025 = 1.0;
        double r5026 = r5024 + r5025;
        double r5027 = z;
        double r5028 = r5026 * r5027;
        double r5029 = t;
        double r5030 = r5028 * r5029;
        double r5031 = 16.0;
        double r5032 = r5030 / r5031;
        double r5033 = cos(r5032);
        double r5034 = r5021 * r5033;
        double r5035 = a;
        double r5036 = r5035 * r5023;
        double r5037 = r5036 + r5025;
        double r5038 = b;
        double r5039 = r5037 * r5038;
        double r5040 = r5039 * r5029;
        double r5041 = r5040 / r5031;
        double r5042 = cos(r5041);
        double r5043 = r5034 * r5042;
        return r5043;
}
↓
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 r5044 = x;
        double r5045 = 0.0;
        double r5046 = 16.0;
        double r5047 = r5045 / r5046;
        double r5048 = cos(r5047);
        double r5049 = r5044 * r5048;
        double r5050 = r5049 * r5048;
        return r5050;
}

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

Original46.4
Target44.7
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.4

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

  4. Final simplification44.4

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

Reproduce

herbie shell --seed 2020025 
(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))))