Average Error: 46.4 → 44.4
Time: 13.8s
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 r1500078 = x;
        double r1500079 = y;
        double r1500080 = 2.0;
        double r1500081 = r1500079 * r1500080;
        double r1500082 = 1.0;
        double r1500083 = r1500081 + r1500082;
        double r1500084 = z;
        double r1500085 = r1500083 * r1500084;
        double r1500086 = t;
        double r1500087 = r1500085 * r1500086;
        double r1500088 = 16.0;
        double r1500089 = r1500087 / r1500088;
        double r1500090 = cos(r1500089);
        double r1500091 = r1500078 * r1500090;
        double r1500092 = a;
        double r1500093 = r1500092 * r1500080;
        double r1500094 = r1500093 + r1500082;
        double r1500095 = b;
        double r1500096 = r1500094 * r1500095;
        double r1500097 = r1500096 * r1500086;
        double r1500098 = r1500097 / r1500088;
        double r1500099 = cos(r1500098);
        double r1500100 = r1500091 * r1500099;
        return r1500100;
}
↓
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 r1500101 = x;
        return r1500101;
}

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

    \[\leadsto \color{blue}{x}\]

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020025 +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))))