Average Error: 46.5 → 44.4
Time: 11.2s
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 r916454 = x;
        double r916455 = y;
        double r916456 = 2.0;
        double r916457 = r916455 * r916456;
        double r916458 = 1.0;
        double r916459 = r916457 + r916458;
        double r916460 = z;
        double r916461 = r916459 * r916460;
        double r916462 = t;
        double r916463 = r916461 * r916462;
        double r916464 = 16.0;
        double r916465 = r916463 / r916464;
        double r916466 = cos(r916465);
        double r916467 = r916454 * r916466;
        double r916468 = a;
        double r916469 = r916468 * r916456;
        double r916470 = r916469 + r916458;
        double r916471 = b;
        double r916472 = r916470 * r916471;
        double r916473 = r916472 * r916462;
        double r916474 = r916473 / r916464;
        double r916475 = cos(r916474);
        double r916476 = r916467 * r916475;
        return r916476;
}
↓
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 r916477 = x;
        return r916477;
}

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
Target44.7
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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

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

  3. Taylor expanded around 0 44.4

    \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot 1\]

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020036 
(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))))