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

↓

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

\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(t \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot z\right)}{16}\right)

double f(double x, double y, double z, double t, double a, double b) {
        double r560206 = x;
        double r560207 = y;
        double r560208 = 2.0;
        double r560209 = r560207 * r560208;
        double r560210 = 1.0;
        double r560211 = r560209 + r560210;
        double r560212 = z;
        double r560213 = r560211 * r560212;
        double r560214 = t;
        double r560215 = r560213 * r560214;
        double r560216 = 16.0;
        double r560217 = r560215 / r560216;
        double r560218 = cos(r560217);
        double r560219 = r560206 * r560218;
        double r560220 = a;
        double r560221 = r560220 * r560208;
        double r560222 = r560221 + r560210;
        double r560223 = b;
        double r560224 = r560222 * r560223;
        double r560225 = r560224 * r560214;
        double r560226 = r560225 / r560216;
        double r560227 = cos(r560226);
        double r560228 = r560219 * r560227;
        return r560228;
}

↓

double f(double x, double y, double z, double t, double __attribute__((unused)) a, double __attribute__((unused)) b) {
        double r560229 = x;
        double r560230 = t;
        double r560231 = y;
        double r560232 = 2.0;
        double r560233 = 1.0;
        double r560234 = fma(r560231, r560232, r560233);
        double r560235 = cbrt(r560234);
        double r560236 = r560235 * r560235;
        double r560237 = r560230 * r560236;
        double r560238 = z;
        double r560239 = r560235 * r560238;
        double r560240 = r560237 * r560239;
        double r560241 = 16.0;
        double r560242 = r560240 / r560241;
        double r560243 = cos(r560242);
        double r560244 = r560229 * r560243;
        return r560244;
}

Original	46.2
Target	44.4
Herbie	45.4

Derivation

Initial program 46.2
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
Simplified46.2
\[\leadsto \color{blue}{\cos \left(\frac{\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)}{16}\right)\right)}\]
Taylor expanded around 0 45.4
\[\leadsto \cos \left(\frac{\color{blue}{0}}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)}{16}\right)\right)\]
Using strategy rm
Applied add-cube-cbrt45.4
\[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right)} \cdot z\right)}{16}\right)\right)\]
Applied associate-*l*45.4
\[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot z\right)\right)}}{16}\right)\right)\]
Using strategy rm
Applied associate-*r*45.4
\[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\color{blue}{\left(t \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot z\right)}}{16}\right)\right)\]
Final simplification45.4
\[\leadsto x \cdot \cos \left(\frac{\left(t \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right)} \cdot z\right)}{16}\right)\]

Reproduce

herbie shell --seed 2019235 +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))))

Error

Target

Derivation

Reproduce