Average Error: 46.1 → 44.1
Time: 2.1m
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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
double f(double x, double y, double z, double t, double a, double b) {
        double r39558095 = x;
        double r39558096 = y;
        double r39558097 = 2.0;
        double r39558098 = r39558096 * r39558097;
        double r39558099 = 1.0;
        double r39558100 = r39558098 + r39558099;
        double r39558101 = z;
        double r39558102 = r39558100 * r39558101;
        double r39558103 = t;
        double r39558104 = r39558102 * r39558103;
        double r39558105 = 16.0;
        double r39558106 = r39558104 / r39558105;
        double r39558107 = cos(r39558106);
        double r39558108 = r39558095 * r39558107;
        double r39558109 = a;
        double r39558110 = r39558109 * r39558097;
        double r39558111 = r39558110 + r39558099;
        double r39558112 = b;
        double r39558113 = r39558111 * r39558112;
        double r39558114 = r39558113 * r39558103;
        double r39558115 = r39558114 / r39558105;
        double r39558116 = cos(r39558115);
        double r39558117 = r39558108 * r39558116;
        return r39558117;
}
↓
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 r39558118 = x;
        return r39558118;
}

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

Original46.1
Target44.4
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Simplified45.7

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

  3. Taylor expanded around 0 45.0

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

  4. Taylor expanded around 0 44.1

    \[\leadsto 1 \cdot \color{blue}{x}\]

  5. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))