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

Average Error: 46.7 → 44.0

Time: 12.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\cos \left(0.0625 \cdot \left(t \cdot b\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(z \cdot \mathsf{fma}\left(y, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\mathsf{fma}\left(t \cdot \left(t \cdot \left(b \cdot b\right)\right), -0.0009765625, \log 2\right)} + -1\right)\\ \end{array} \]

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)))
      2e+282)
   (* (cos (* 0.0625 (* t b))) (* x (cos (* t (* z (fma y 0.125 0.0625))))))
   (* x (+ (exp (fma (* t (* t (* b b))) -0.0009765625 (log 2.0))) -1.0))))

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))) <= 2e+282) {
		tmp = cos((0.0625 * (t * b))) * (x * cos((t * (z * fma(y, 0.125, 0.0625)))));
	} else {
		tmp = x * (exp(fma((t * (t * (b * b))), -0.0009765625, log(2.0))) + -1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(Float64(Float64(Float64(a * 2.0) + 1.0) * b) * t) / 16.0)))
end

↓

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(Float64(Float64(t * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))) <= 2e+282)
		tmp = Float64(cos(Float64(0.0625 * Float64(t * b))) * Float64(x * cos(Float64(t * Float64(z * fma(y, 0.125, 0.0625))))));
	else
		tmp = Float64(x * Float64(exp(fma(Float64(t * Float64(t * Float64(b * b))), -0.0009765625, log(2.0))) + -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(N[(N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+282], N[(N[Cos[N[(0.0625 * N[(t * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[Cos[N[(t * N[(z * N[(y * 0.125 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Exp[N[(N[(t * N[(t * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0009765625 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

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	46.7
Target	44.8
Herbie	44.0

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

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))) < 2.00000000000000007e282

   1. Initial program 34.7

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

   2. Simplified 35.0

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

   3. Taylor expanded in a around 0 35.1

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

   4. Taylor expanded in z around -inf 34.9

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

   if 2.00000000000000007e282 < (*.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 63.0

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

   2. Simplified 62.4

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

   3. Taylor expanded in a around 0 61.0

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

   4. Taylor expanded in z around 0 58.3

      \[\leadsto \color{blue}{\cos \left(0.0625 \cdot \left(t \cdot b\right)\right) \cdot x} \]

   5. Applied egg-rr 58.3

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(t \cdot \left(b \cdot 0.0625\right)\right)\right)} - 1\right)} \cdot x \]

   6. Taylor expanded in t around 0 57.5

      \[\leadsto \left(e^{\color{blue}{\log 2 - 0.0009765625 \cdot \left({t}^{2} \cdot {b}^{2}\right)}} - 1\right) \cdot x \]

   7. Simplified 56.5

      \[\leadsto \left(e^{\color{blue}{\mathsf{fma}\left(t \cdot \left(t \cdot \left(b \cdot b\right)\right), -0.0009765625, \log 2\right)}} - 1\right) \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\cos \left(0.0625 \cdot \left(t \cdot b\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(z \cdot \mathsf{fma}\left(y, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(e^{\mathsf{fma}\left(t \cdot \left(t \cdot \left(b \cdot b\right)\right), -0.0009765625, \log 2\right)} + -1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))