Average Error: 46.4 → 44.4
Time: 1.0m
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 r44589028 = x;
        double r44589029 = y;
        double r44589030 = 2.0;
        double r44589031 = r44589029 * r44589030;
        double r44589032 = 1.0;
        double r44589033 = r44589031 + r44589032;
        double r44589034 = z;
        double r44589035 = r44589033 * r44589034;
        double r44589036 = t;
        double r44589037 = r44589035 * r44589036;
        double r44589038 = 16.0;
        double r44589039 = r44589037 / r44589038;
        double r44589040 = cos(r44589039);
        double r44589041 = r44589028 * r44589040;
        double r44589042 = a;
        double r44589043 = r44589042 * r44589030;
        double r44589044 = r44589043 + r44589032;
        double r44589045 = b;
        double r44589046 = r44589044 * r44589045;
        double r44589047 = r44589046 * r44589036;
        double r44589048 = r44589047 / r44589038;
        double r44589049 = cos(r44589048);
        double r44589050 = r44589041 * r44589049;
        return r44589050;
}
↓
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 r44589051 = x;
        return r44589051;
}

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. Simplified46.1

    \[\leadsto \color{blue}{\left(\cos \left(\frac{t}{\frac{\frac{16}{z}}{\mathsf{fma}\left(2, y, 1\right)}}\right) \cdot x\right) \cdot \cos \left(\frac{b}{\frac{\frac{16}{t}}{\mathsf{fma}\left(a, 2, 1\right)}}\right)}\]

  3. Taylor expanded around 0 45.5

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

  4. Taylor expanded around 0 44.4

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

  5. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))))