Average Error: 46.3 → 45.3
Time: 22.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\]
\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{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r491639 = x;
        double r491640 = y;
        double r491641 = 2.0;
        double r491642 = r491640 * r491641;
        double r491643 = 1.0;
        double r491644 = r491642 + r491643;
        double r491645 = z;
        double r491646 = r491644 * r491645;
        double r491647 = t;
        double r491648 = r491646 * r491647;
        double r491649 = 16.0;
        double r491650 = r491648 / r491649;
        double r491651 = cos(r491650);
        double r491652 = r491639 * r491651;
        double r491653 = a;
        double r491654 = r491653 * r491641;
        double r491655 = r491654 + r491643;
        double r491656 = b;
        double r491657 = r491655 * r491656;
        double r491658 = r491657 * r491647;
        double r491659 = r491658 / r491649;
        double r491660 = cos(r491659);
        double r491661 = r491652 * r491660;
        return r491661;
}
↓
double f(double x, double __attribute__((unused)) y, double __attribute__((unused)) z, double t, double a, double b) {
        double r491662 = x;
        double r491663 = a;
        double r491664 = 2.0;
        double r491665 = r491663 * r491664;
        double r491666 = 1.0;
        double r491667 = r491665 + r491666;
        double r491668 = b;
        double r491669 = t;
        double r491670 = r491668 * r491669;
        double r491671 = r491667 * r491670;
        double r491672 = 16.0;
        double r491673 = r491671 / r491672;
        double r491674 = cos(r491673);
        double r491675 = r491662 * r491674;
        return r491675;
}

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

Original46.3
Target44.6
Herbie45.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.3

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

  2. Taylor expanded around 0 45.6

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
  3. Using strategy rm

  4. Applied associate-*l*45.3

    \[\leadsto \left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}}{16}\right)\]
  5. Using strategy rm

  6. Applied add-cbrt-cube45.3

    \[\leadsto \left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right) \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)}}\]

  7. Simplified45.3

    \[\leadsto \left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(\cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\right)}^{3}}}\]
  8. Using strategy rm

  9. Applied add-log-exp45.3

    \[\leadsto \left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{{\left(\cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\right)}^{3}}\right)}}\]

  10. Final simplification45.3

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

Reproduce

herbie shell --seed 2019298 
(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))))