Average Error: 45.9 → 44.0
Time: 24.9s
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 r576143 = x;
        double r576144 = y;
        double r576145 = 2.0;
        double r576146 = r576144 * r576145;
        double r576147 = 1.0;
        double r576148 = r576146 + r576147;
        double r576149 = z;
        double r576150 = r576148 * r576149;
        double r576151 = t;
        double r576152 = r576150 * r576151;
        double r576153 = 16.0;
        double r576154 = r576152 / r576153;
        double r576155 = cos(r576154);
        double r576156 = r576143 * r576155;
        double r576157 = a;
        double r576158 = r576157 * r576145;
        double r576159 = r576158 + r576147;
        double r576160 = b;
        double r576161 = r576159 * r576160;
        double r576162 = r576161 * r576151;
        double r576163 = r576162 / r576153;
        double r576164 = cos(r576163);
        double r576165 = r576156 * r576164;
        return r576165;
}
↓
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 r576166 = x;
        return r576166;
}

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

Original45.9
Target44.3
Herbie44.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.9

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 44.0

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot 1\]

  4. Final simplification44.0

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019325 
(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))))