Average Error: 45.9 → 44.0
Time: 11.1s
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 r804535 = x;
        double r804536 = y;
        double r804537 = 2.0;
        double r804538 = r804536 * r804537;
        double r804539 = 1.0;
        double r804540 = r804538 + r804539;
        double r804541 = z;
        double r804542 = r804540 * r804541;
        double r804543 = t;
        double r804544 = r804542 * r804543;
        double r804545 = 16.0;
        double r804546 = r804544 / r804545;
        double r804547 = cos(r804546);
        double r804548 = r804535 * r804547;
        double r804549 = a;
        double r804550 = r804549 * r804537;
        double r804551 = r804550 + r804539;
        double r804552 = b;
        double r804553 = r804551 * r804552;
        double r804554 = r804553 * r804543;
        double r804555 = r804554 / r804545;
        double r804556 = cos(r804555);
        double r804557 = r804548 * r804556;
        return r804557;
}
↓
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 r804558 = 0.0;
        double r804559 = 16.0;
        double r804560 = r804558 / r804559;
        double r804561 = cos(r804560);
        double r804562 = x;
        double r804563 = r804561 * r804562;
        return r804563;
}

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
Target44.2
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 45.3

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

  3. Taylor expanded around 0 44.0

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

  4. Final simplification44.0

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

Reproduce

herbie shell --seed 2020024 
(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))))