Average Error: 45.8 → 43.9
Time: 27.3s
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 r656123 = x;
        double r656124 = y;
        double r656125 = 2.0;
        double r656126 = r656124 * r656125;
        double r656127 = 1.0;
        double r656128 = r656126 + r656127;
        double r656129 = z;
        double r656130 = r656128 * r656129;
        double r656131 = t;
        double r656132 = r656130 * r656131;
        double r656133 = 16.0;
        double r656134 = r656132 / r656133;
        double r656135 = cos(r656134);
        double r656136 = r656123 * r656135;
        double r656137 = a;
        double r656138 = r656137 * r656125;
        double r656139 = r656138 + r656127;
        double r656140 = b;
        double r656141 = r656139 * r656140;
        double r656142 = r656141 * r656131;
        double r656143 = r656142 / r656133;
        double r656144 = cos(r656143);
        double r656145 = r656136 * r656144;
        return r656145;
}
↓
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 r656146 = x;
        return r656146;
}

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

Original45.8
Target44.2
Herbie43.9
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.8

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 43.9

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

  4. Final simplification43.9

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019303 +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))))