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

Average Error: 46.6 → 44.0

Time: 11.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(y, 0.125, 0.0625\right) \cdot \left(z \cdot t\right)\right)\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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)))
      2e+17)
   (*
    (log (+ 1.0 (expm1 (cos (* (fma y 0.125 0.0625) (* z t))))))
    (* x (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))) <= 2e+17) {
		tmp = log((1.0 + expm1(cos((fma(y, 0.125, 0.0625) * (z * t)))))) * (x * cos((t * (b * fma(a, 0.125, 0.0625)))));
	} else {
		tmp = x;
	}
	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+17)
		tmp = Float64(log(Float64(1.0 + expm1(cos(Float64(fma(y, 0.125, 0.0625) * Float64(z * t)))))) * Float64(x * cos(Float64(t * Float64(b * fma(a, 0.125, 0.0625))))));
	else
		tmp = x;
	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+17], N[(N[Log[N[(1.0 + N[(Exp[N[Cos[N[(N[(y * 0.125 + 0.0625), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[Cos[N[(t * N[(b * N[(a * 0.125 + 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

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	46.6
Target	44.9
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))) < 2e17

   1. Initial program 34.5

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

   2. Simplified 34.8

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

   3. Applied egg-rr 34.5

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(y, 0.125, 0.0625\right) \cdot \left(z \cdot t\right)\right)\right)\right)} \cdot \left(x \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right) \]

   if 2e17 < (*.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 56.6

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

   2. Simplified 56.2

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

   3. Applied egg-rr 55.8

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

   4. Taylor expanded in z around 0 54.2

      \[\leadsto \color{blue}{1} \cdot \left(x \cdot \sqrt[3]{{\cos \left(\mathsf{fma}\left(a, 0.125, 0.0625\right) \cdot \left(t \cdot b\right)\right)}^{3}}\right) \]

   5. Taylor expanded in t around 0 51.8

      \[\leadsto 1 \cdot \left(x \cdot \sqrt[3]{\color{blue}{1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(y, 0.125, 0.0625\right) \cdot \left(z \cdot t\right)\right)\right)\right) \cdot \left(x \cdot \cos \left(t \cdot \left(b \cdot \mathsf{fma}\left(a, 0.125, 0.0625\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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))))