Average Error: 46.1 → 44.1
Time: 2.2m
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 r46177090 = x;
        double r46177091 = y;
        double r46177092 = 2.0;
        double r46177093 = r46177091 * r46177092;
        double r46177094 = 1.0;
        double r46177095 = r46177093 + r46177094;
        double r46177096 = z;
        double r46177097 = r46177095 * r46177096;
        double r46177098 = t;
        double r46177099 = r46177097 * r46177098;
        double r46177100 = 16.0;
        double r46177101 = r46177099 / r46177100;
        double r46177102 = cos(r46177101);
        double r46177103 = r46177090 * r46177102;
        double r46177104 = a;
        double r46177105 = r46177104 * r46177092;
        double r46177106 = r46177105 + r46177094;
        double r46177107 = b;
        double r46177108 = r46177106 * r46177107;
        double r46177109 = r46177108 * r46177098;
        double r46177110 = r46177109 / r46177100;
        double r46177111 = cos(r46177110);
        double r46177112 = r46177103 * r46177111;
        return r46177112;
}
↓
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 r46177113 = x;
        return r46177113;
}

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

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

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.4

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))