Average Error: 46.5 → 44.3
Time: 26.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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 r1042090 = x;
        double r1042091 = y;
        double r1042092 = 2.0;
        double r1042093 = r1042091 * r1042092;
        double r1042094 = 1.0;
        double r1042095 = r1042093 + r1042094;
        double r1042096 = z;
        double r1042097 = r1042095 * r1042096;
        double r1042098 = t;
        double r1042099 = r1042097 * r1042098;
        double r1042100 = 16.0;
        double r1042101 = r1042099 / r1042100;
        double r1042102 = cos(r1042101);
        double r1042103 = r1042090 * r1042102;
        double r1042104 = a;
        double r1042105 = r1042104 * r1042092;
        double r1042106 = r1042105 + r1042094;
        double r1042107 = b;
        double r1042108 = r1042106 * r1042107;
        double r1042109 = r1042108 * r1042098;
        double r1042110 = r1042109 / r1042100;
        double r1042111 = cos(r1042110);
        double r1042112 = r1042103 * r1042111;
        return r1042112;
}
↓
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 r1042113 = x;
        return r1042113;
}

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

Original46.5
Target44.6
Herbie44.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.5

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

  2. Taylor expanded around 0 45.7

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\]

  3. Taylor expanded around 0 44.3

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\]

  4. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019305 +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))))