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

Average Error: 46.3 → 43.6

Time: 16.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 1.0401117086229122 \cdot 10^{+308}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)}\\ \left(x \cdot \cos \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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)))
      1.0401117086229122e+308)
   (let* ((t_1 (cbrt (* t (* z (fma 0.125 y 0.0625))))))
     (*
      (* x (cos (* t_1 (* t_1 t_1))))
      (cos (* (* t b) (fma a 0.125 0.0625)))))
   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)) <= 1.0401117086229122e+308) {
		double t_1_1 = cbrt(t * (z * fma(0.125, y, 0.0625)));
		tmp = (x * cos(t_1_1 * (t_1_1 * t_1_1))) * cos((t * b) * fma(a, 0.125, 0.0625));
	} 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	44.7
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))) < 1.04011170862291215e308

   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 34.4

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

   3. Applied add-cube-cbrt_binary64 34.5

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

   4. Simplified 34.4

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

   5. Simplified 34.4

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

   if 1.04011170862291215e308 < (*.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 64.0

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

   2. Simplified 62.5

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

   3. Taylor expanded in t around 0 60.5

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

   4. Taylor expanded in z around 0 57.1

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

4. Final simplification 43.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 1.0401117086229122 \cdot 10^{+308}:\\ \;\;\;\;\left(x \cdot \cos \left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)} \cdot \left(\sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)} \cdot \sqrt[3]{t \cdot \left(z \cdot \mathsf{fma}\left(0.125, y, 0.0625\right)\right)}\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))