Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1

Average Error: 46.3 → 43.6

Time: 14.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5.7150111089301706 \cdot 10^{+144}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \cos \left(\sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)} \cdot \left(\sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)} \cdot \sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (*
  (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
  (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
       (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
      5.7150111089301706e+144)
   (*
    (* x (cos (* (* z t) (+ 0.0625 (* y 0.125)))))
    (cos
     (*
      (cbrt (* (* t b) (+ 0.0625 (* a 0.125))))
      (*
       (cbrt (* (* t b) (+ 0.0625 (* a 0.125))))
       (cbrt (* (* t b) (+ 0.0625 (* a 0.125))))))))
   x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * cos(((((y * 2.0) + 1.0) * z) * t) / 16.0)) * cos(((((a * 2.0) + 1.0) * b) * t) / 16.0);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * cos(((((y * 2.0) + 1.0) * z) * t) / 16.0)) * cos((t * ((1.0 + (2.0 * a)) * b)) / 16.0)) <= 5.7150111089301706e+144) {
		tmp = (x * cos((z * t) * (0.0625 + (y * 0.125)))) * cos(cbrt((t * b) * (0.0625 + (a * 0.125))) * (cbrt((t * b) * (0.0625 + (a * 0.125))) * cbrt((t * b) * (0.0625 + (a * 0.125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

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

Target

Original	46.3
Target	44.5
Herbie	43.6

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16))) < 5.71501110893017056e144

   1. Initial program 34.4

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

   2. Simplified 34.2

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt_binary64_21255 34.4

      \[\leadsto \left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)} \cdot \sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)}\right) \cdot \sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)}\right)}\]

   if 5.71501110893017056e144 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y 2) 1) z) t) 16))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a 2) 1) b) t) 16)))

   1. Initial program 58.7

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

   2. Simplified 57.6

      \[\leadsto \color{blue}{\left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)}\]

   3. Taylor expanded around 0 55.9

      \[\leadsto \left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \color{blue}{1}\]

   4. Taylor expanded around 0 53.1

      \[\leadsto \color{blue}{x} \cdot 1\]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5.7150111089301706 \cdot 10^{+144}:\\ \;\;\;\;\left(x \cdot \cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + y \cdot 0.125\right)\right)\right) \cdot \cos \left(\sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)} \cdot \left(\sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)} \cdot \sqrt[3]{\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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 16.0) (/ 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))))