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

Average Error: 46.6 → 43.8

Time: 21.4s

Precision: binary64

Cost: 48964

Math

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

↓

\[\begin{array}{l} t_1 := 0.0625 \cdot \left(t \cdot b\right)\\ t_2 := t \cdot \left(b \cdot \left(a \cdot 0.125\right)\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+206}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \left(\cos t_2 \cdot \cos t_1 - \sin t_2 \cdot \sin t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{expm1}\left(\mathsf{fma}\left(-0.0009765625, t \cdot \left(t \cdot \left(b \cdot b\right)\right), \log 2\right)\right)\\ \end{array} \]

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
 (let* ((t_1 (* 0.0625 (* t b))) (t_2 (* t (* b (* a 0.125)))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        4e+206)
     (*
      x
      (*
       (cos (* (* z t) (+ 0.0625 (/ y 8.0))))
       (- (* (cos t_2) (cos t_1)) (* (sin t_2) (sin t_1)))))
     (* x (expm1 (fma -0.0009765625 (* t (* t (* b b))) (log 2.0)))))))

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 t_1 = 0.0625 * (t * b);
	double t_2 = t * (b * (a * 0.125));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 4e+206) {
		tmp = x * (cos(((z * t) * (0.0625 + (y / 8.0)))) * ((cos(t_2) * cos(t_1)) - (sin(t_2) * sin(t_1))));
	} else {
		tmp = x * expm1(fma(-0.0009765625, (t * (t * (b * b))), log(2.0)));
	}
	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)
	t_1 = Float64(0.0625 * Float64(t * b))
	t_2 = Float64(t * Float64(b * Float64(a * 0.125)))
	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))) <= 4e+206)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * t) * Float64(0.0625 + Float64(y / 8.0)))) * Float64(Float64(cos(t_2) * cos(t_1)) - Float64(sin(t_2) * sin(t_1)))));
	else
		tmp = Float64(x * expm1(fma(-0.0009765625, Float64(t * Float64(t * Float64(b * b))), log(2.0))));
	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_] := Block[{t$95$1 = N[(0.0625 * N[(t * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(b * N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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], 4e+206], N[(x * N[(N[Cos[N[(N[(z * t), $MachinePrecision] * N[(0.0625 + N[(y / 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(Exp[N[(-0.0009765625 * N[(t * N[(t * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	46.6
Target	44.7
Herbie	43.8

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

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))) < 4.0000000000000002e206

   1. Initial program 34.1

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

   2. Simplified 34.0

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{8} + 0.0625\right)\right)\right)} \]
      Proof

      [Start] 34.1	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 34.1	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. Applied egg-rr 34.0

      \[\leadsto x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \color{blue}{\left(\cos \left(t \cdot \left(b \cdot \left(a \cdot 0.125\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot b\right)\right) - \sin \left(t \cdot \left(b \cdot \left(a \cdot 0.125\right)\right)\right) \cdot \sin \left(0.0625 \cdot \left(t \cdot b\right)\right)\right)}\right) \]

   if 4.0000000000000002e206 < (*.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 61.0

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

   2. Simplified 59.8

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{8} + 0.0625\right)\right)\right)} \]
      Proof

      [Start] 61.0	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 61.0	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. Taylor expanded in z around 0 57.8

      \[\leadsto x \cdot \left(\color{blue}{1} \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\frac{a}{8} + 0.0625\right)\right)\right) \]

   4. Taylor expanded in a around 0 56.6

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

   5. Applied egg-rr 56.6

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(0.0625 \cdot \left(t \cdot b\right)\right)\right)\right)} \cdot x \]

   6. Taylor expanded in t around 0 56.1

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 + -0.0009765625 \cdot \left({t}^{2} \cdot {b}^{2}\right)}\right) \cdot x \]

   7. Simplified 55.1

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.0009765625, t \cdot \left(t \cdot \left(b \cdot b\right)\right), \log 2\right)}\right) \cdot x \]
      Proof

      [Start] 56.1	\[ \mathsf{expm1}\left(\log 2 + -0.0009765625 \cdot \left({t}^{2} \cdot {b}^{2}\right)\right) \cdot x \]
      +-commutative [=>] 56.1	\[ \mathsf{expm1}\left(\color{blue}{-0.0009765625 \cdot \left({t}^{2} \cdot {b}^{2}\right) + \log 2}\right) \cdot x \]
      fma-def [=>] 56.1	\[ \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(-0.0009765625, {t}^{2} \cdot {b}^{2}, \log 2\right)}\right) \cdot x \]
      unpow2 [=>] 56.1	\[ \mathsf{expm1}\left(\mathsf{fma}\left(-0.0009765625, \color{blue}{\left(t \cdot t\right)} \cdot {b}^{2}, \log 2\right)\right) \cdot x \]
      associate-*l* [=>] 55.1	\[ \mathsf{expm1}\left(\mathsf{fma}\left(-0.0009765625, \color{blue}{t \cdot \left(t \cdot {b}^{2}\right)}, \log 2\right)\right) \cdot x \]
      unpow2 [=>] 55.1	\[ \mathsf{expm1}\left(\mathsf{fma}\left(-0.0009765625, t \cdot \left(t \cdot \color{blue}{\left(b \cdot b\right)}\right), \log 2\right)\right) \cdot x \]

3. Recombined 2 regimes into one program.

4. Final simplification 43.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{+206}:\\ \;\;\;\;x \cdot \left(\cos \left(\left(z \cdot t\right) \cdot \left(0.0625 + \frac{y}{8}\right)\right) \cdot \left(\cos \left(t \cdot \left(b \cdot \left(a \cdot 0.125\right)\right)\right) \cdot \cos \left(0.0625 \cdot \left(t \cdot b\right)\right) - \sin \left(t \cdot \left(b \cdot \left(a \cdot 0.125\right)\right)\right) \cdot \sin \left(0.0625 \cdot \left(t \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{expm1}\left(\mathsf{fma}\left(-0.0009765625, t \cdot \left(t \cdot \left(b \cdot b\right)\right), \log 2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	44.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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))))