Average Error: 46.4 → 45.7
Time: 12.7s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\left(x \cdot \cos \left(\frac{{\left(\left(t \cdot z\right) \cdot 1\right)}^{1}}{16}\right)\right) \cdot \cos \left(\frac{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}{16}\right)\]
\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)
↓
\left(x \cdot \cos \left(\frac{{\left(\left(t \cdot z\right) \cdot 1\right)}^{1}}{16}\right)\right) \cdot \cos \left(\frac{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r873568 = x;
        double r873569 = y;
        double r873570 = 2.0;
        double r873571 = r873569 * r873570;
        double r873572 = 1.0;
        double r873573 = r873571 + r873572;
        double r873574 = z;
        double r873575 = r873573 * r873574;
        double r873576 = t;
        double r873577 = r873575 * r873576;
        double r873578 = 16.0;
        double r873579 = r873577 / r873578;
        double r873580 = cos(r873579);
        double r873581 = r873568 * r873580;
        double r873582 = a;
        double r873583 = r873582 * r873570;
        double r873584 = r873583 + r873572;
        double r873585 = b;
        double r873586 = r873584 * r873585;
        double r873587 = r873586 * r873576;
        double r873588 = r873587 / r873578;
        double r873589 = cos(r873588);
        double r873590 = r873581 * r873589;
        return r873590;
}
↓
double f(double x, double __attribute__((unused)) y, double z, double t, double a, double b) {
        double r873591 = x;
        double r873592 = t;
        double r873593 = z;
        double r873594 = r873592 * r873593;
        double r873595 = 1.0;
        double r873596 = r873594 * r873595;
        double r873597 = 1.0;
        double r873598 = pow(r873596, r873597);
        double r873599 = 16.0;
        double r873600 = r873598 / r873599;
        double r873601 = cos(r873600);
        double r873602 = r873591 * r873601;
        double r873603 = a;
        double r873604 = 2.0;
        double r873605 = r873603 * r873604;
        double r873606 = r873605 + r873595;
        double r873607 = cbrt(r873606);
        double r873608 = r873607 * r873607;
        double r873609 = b;
        double r873610 = r873609 * r873592;
        double r873611 = r873607 * r873610;
        double r873612 = r873608 * r873611;
        double r873613 = r873612 / r873599;
        double r873614 = cos(r873613);
        double r873615 = r873602 * r873614;
        return r873615;
}

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

Original46.4
Target44.6
Herbie45.7
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  3. Applied associate-*l*46.1

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

  5. Applied pow146.1

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

  6. Applied pow146.1

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

  7. Applied pow146.1

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

  8. Applied pow-prod-down46.1

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

  9. Applied pow-prod-down46.1

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

  10. Simplified45.8

    \[\leadsto \left(x \cdot \cos \left(\frac{{\color{blue}{\left(\left(t \cdot z\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right)}}^{1}}{16}\right)\right) \cdot \cos \left(\frac{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}{16}\right)\]

  11. Taylor expanded around 0 45.7

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

  12. Simplified45.7

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

  14. Applied add-cube-cbrt45.7

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

  15. Applied associate-*l*45.7

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

  16. Final simplification45.7

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(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))))