Average Error: 46.0 → 43.9
Time: 9.6s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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 \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r1061585 = x;
        double r1061586 = y;
        double r1061587 = 2.0;
        double r1061588 = r1061586 * r1061587;
        double r1061589 = 1.0;
        double r1061590 = r1061588 + r1061589;
        double r1061591 = z;
        double r1061592 = r1061590 * r1061591;
        double r1061593 = t;
        double r1061594 = r1061592 * r1061593;
        double r1061595 = 16.0;
        double r1061596 = r1061594 / r1061595;
        double r1061597 = cos(r1061596);
        double r1061598 = r1061585 * r1061597;
        double r1061599 = a;
        double r1061600 = r1061599 * r1061587;
        double r1061601 = r1061600 + r1061589;
        double r1061602 = b;
        double r1061603 = r1061601 * r1061602;
        double r1061604 = r1061603 * r1061593;
        double r1061605 = r1061604 / r1061595;
        double r1061606 = cos(r1061605);
        double r1061607 = r1061598 * r1061606;
        return r1061607;
}
↓
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 r1061608 = x;
        double r1061609 = 0.0;
        double r1061610 = 16.0;
        double r1061611 = r1061609 / r1061610;
        double r1061612 = cos(r1061611);
        double r1061613 = r1061608 * r1061612;
        return r1061613;
}

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
Target44.2
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 46.0

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

  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 43.9

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

  4. Final simplification43.9

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

Reproduce

herbie shell --seed 2019362 
(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))))