Average Error: 46.0 → 44.5
Time: 56.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r43373126 = x;
        double r43373127 = y;
        double r43373128 = 2.0;
        double r43373129 = r43373127 * r43373128;
        double r43373130 = 1.0;
        double r43373131 = r43373129 + r43373130;
        double r43373132 = z;
        double r43373133 = r43373131 * r43373132;
        double r43373134 = t;
        double r43373135 = r43373133 * r43373134;
        double r43373136 = 16.0;
        double r43373137 = r43373135 / r43373136;
        double r43373138 = cos(r43373137);
        double r43373139 = r43373126 * r43373138;
        double r43373140 = a;
        double r43373141 = r43373140 * r43373128;
        double r43373142 = r43373141 + r43373130;
        double r43373143 = b;
        double r43373144 = r43373142 * r43373143;
        double r43373145 = r43373144 * r43373134;
        double r43373146 = r43373145 / r43373136;
        double r43373147 = cos(r43373146);
        double r43373148 = r43373139 * r43373147;
        return r43373148;
}
↓
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 r43373149 = x;
        return r43373149;
}

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

Original46.0
Target44.7
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Taylor expanded around 0 45.4

    \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  3. Taylor expanded around 0 44.5

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

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019163 
(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))))))

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