Average Error: 46.2 → 45.2
Time: 54.3s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{\sqrt[3]{\sqrt[3]{t}} \cdot \left(\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot z\right) \cdot \left(2 \cdot y + 1\right)\right) \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}{16}\right)\]
\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{\sqrt[3]{\sqrt[3]{t}} \cdot \left(\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot z\right) \cdot \left(2 \cdot y + 1\right)\right) \cdot \sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r41591255 = x;
        double r41591256 = y;
        double r41591257 = 2.0;
        double r41591258 = r41591256 * r41591257;
        double r41591259 = 1.0;
        double r41591260 = r41591258 + r41591259;
        double r41591261 = z;
        double r41591262 = r41591260 * r41591261;
        double r41591263 = t;
        double r41591264 = r41591262 * r41591263;
        double r41591265 = 16.0;
        double r41591266 = r41591264 / r41591265;
        double r41591267 = cos(r41591266);
        double r41591268 = r41591255 * r41591267;
        double r41591269 = a;
        double r41591270 = r41591269 * r41591257;
        double r41591271 = r41591270 + r41591259;
        double r41591272 = b;
        double r41591273 = r41591271 * r41591272;
        double r41591274 = r41591273 * r41591263;
        double r41591275 = r41591274 / r41591265;
        double r41591276 = cos(r41591275);
        double r41591277 = r41591268 * r41591276;
        return r41591277;
}
↓
double f(double x, double y, double z, double t, double __attribute__((unused)) a, double __attribute__((unused)) b) {
        double r41591278 = x;
        double r41591279 = t;
        double r41591280 = cbrt(r41591279);
        double r41591281 = cbrt(r41591280);
        double r41591282 = r41591280 * r41591280;
        double r41591283 = z;
        double r41591284 = r41591282 * r41591283;
        double r41591285 = 2.0;
        double r41591286 = y;
        double r41591287 = r41591285 * r41591286;
        double r41591288 = 1.0;
        double r41591289 = r41591287 + r41591288;
        double r41591290 = r41591284 * r41591289;
        double r41591291 = cbrt(r41591282);
        double r41591292 = r41591290 * r41591291;
        double r41591293 = r41591281 * r41591292;
        double r41591294 = 16.0;
        double r41591295 = r41591293 / r41591294;
        double r41591296 = cos(r41591295);
        double r41591297 = r41591278 * r41591296;
        return r41591297;
}

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

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

Derivation

  1. Initial program 46.2

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

  2. Taylor expanded around 0 45.5

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

  4. Applied add-cube-cbrt45.4

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

  5. Applied associate-*r*45.4

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right) \cdot \sqrt[3]{t}}}{16}\right)\right) \cdot 1\]
  6. Using strategy rm

  7. Applied *-un-lft-identity45.4

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

  8. Applied cbrt-prod45.4

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

  9. Applied associate-*r*45.4

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

  10. Simplified45.2

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\left(\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot z\right) \cdot \left(y \cdot 2 + 1\right)\right)} \cdot \sqrt[3]{t}}{16}\right)\right) \cdot 1\]
  11. Using strategy rm

  12. Applied add-cube-cbrt45.2

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

  13. Applied cbrt-prod45.2

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

  14. Applied associate-*r*45.2

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

  15. Final simplification45.2

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))