Average Error: 46.8 → 44.7
Time: 26.1s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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
double f(double x, double y, double z, double t, double a, double b) {
        double r564019 = x;
        double r564020 = y;
        double r564021 = 2.0;
        double r564022 = r564020 * r564021;
        double r564023 = 1.0;
        double r564024 = r564022 + r564023;
        double r564025 = z;
        double r564026 = r564024 * r564025;
        double r564027 = t;
        double r564028 = r564026 * r564027;
        double r564029 = 16.0;
        double r564030 = r564028 / r564029;
        double r564031 = cos(r564030);
        double r564032 = r564019 * r564031;
        double r564033 = a;
        double r564034 = r564033 * r564021;
        double r564035 = r564034 + r564023;
        double r564036 = b;
        double r564037 = r564035 * r564036;
        double r564038 = r564037 * r564027;
        double r564039 = r564038 / r564029;
        double r564040 = cos(r564039);
        double r564041 = r564032 * r564040;
        return r564041;
}
↓
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 r564042 = x;
        return r564042;
}

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

Original46.8
Target45.0
Herbie44.7
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.8

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

  2. Taylor expanded around 0 46.0

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

  3. Taylor expanded around 0 44.7

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

  4. Final simplification44.7

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019322 
(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))))