Average Error: 46.7 → 44.6
Time: 10.2s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot x\]
\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)
↓
\cos \left(\frac{0}{16}\right) \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r1098554 = x;
        double r1098555 = y;
        double r1098556 = 2.0;
        double r1098557 = r1098555 * r1098556;
        double r1098558 = 1.0;
        double r1098559 = r1098557 + r1098558;
        double r1098560 = z;
        double r1098561 = r1098559 * r1098560;
        double r1098562 = t;
        double r1098563 = r1098561 * r1098562;
        double r1098564 = 16.0;
        double r1098565 = r1098563 / r1098564;
        double r1098566 = cos(r1098565);
        double r1098567 = r1098554 * r1098566;
        double r1098568 = a;
        double r1098569 = r1098568 * r1098556;
        double r1098570 = r1098569 + r1098558;
        double r1098571 = b;
        double r1098572 = r1098570 * r1098571;
        double r1098573 = r1098572 * r1098562;
        double r1098574 = r1098573 / r1098564;
        double r1098575 = cos(r1098574);
        double r1098576 = r1098567 * r1098575;
        return r1098576;
}
↓
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 r1098577 = 0.0;
        double r1098578 = 16.0;
        double r1098579 = r1098577 / r1098578;
        double r1098580 = cos(r1098579);
        double r1098581 = x;
        double r1098582 = r1098580 * r1098581;
        return r1098582;
}

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

Original46.7
Target45.0
Herbie44.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.7

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

  2. Taylor expanded around 0 45.9

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

  3. Taylor expanded around 0 44.6

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

  4. Final simplification44.6

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

Reproduce

herbie shell --seed 2020057 
(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))))