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

Average Error: 72.63% → 68.3%

Time: 25.6s

Precision: binary64

Cost: 87044

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+273}:\\ \;\;\;\;x \cdot \left(\cos \left({\left({\left({t_1}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{t_1}\right)}^{3}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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
 (let* ((t_1 (cbrt (* (fma y 2.0 1.0) (* z (* t 0.0625))))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        2e+273)
     (*
      x
      (*
       (cos (pow (* (pow (pow t_1 2.0) 0.3333333333333333) (cbrt t_1)) 3.0))
       (cos (/ (* b (fma a 2.0 1.0)) (/ 16.0 t)))))
     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 t_1 = cbrt((fma(y, 2.0, 1.0) * (z * (t * 0.0625))));
	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+273) {
		tmp = x * (cos(pow((pow(pow(t_1, 2.0), 0.3333333333333333) * cbrt(t_1)), 3.0)) * cos(((b * fma(a, 2.0, 1.0)) / (16.0 / t))));
	} 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)
	t_1 = cbrt(Float64(fma(y, 2.0, 1.0) * Float64(z * Float64(t * 0.0625))))
	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+273)
		tmp = Float64(x * Float64(cos((Float64(((t_1 ^ 2.0) ^ 0.3333333333333333) * cbrt(t_1)) ^ 3.0)) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) / Float64(16.0 / t)))));
	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_] := Block[{t$95$1 = N[Power[N[(N[(y * 2.0 + 1.0), $MachinePrecision] * N[(z * N[(t * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $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], 2e+273], N[(x * N[(N[Cos[N[Power[N[(N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(b * N[(a * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Target

Original	72.63%
Target	69.88%
Herbie	68.3%

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

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))) < 1.99999999999999989e273

   1. Initial program 53.06

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

   2. Simplified 53.13

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right)} \]
      Proof

      [Start] 53.06	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 53.06	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]
      associate-/l* [=>] 53.07	\[ x \cdot \left(\cos \color{blue}{\left(\frac{\left(y \cdot 2 + 1\right) \cdot z}{\frac{16}{t}}\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      fma-def [=>] 53.07	\[ x \cdot \left(\cos \left(\frac{\color{blue}{\mathsf{fma}\left(y, 2, 1\right)} \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      associate-/l* [=>] 53.13	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{\frac{16}{t}}\right)}\right) \]
      fma-def [=>] 53.13	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b}{\frac{16}{t}}\right)\right) \]

   3. Applied egg-rr 53.14

      \[\leadsto x \cdot \left(\cos \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{3}\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   4. Applied egg-rr 53.24

      \[\leadsto x \cdot \left(\cos \left({\color{blue}{\left({\left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}}\right)}}^{3}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right) \]

   if 1.99999999999999989e273 < (*.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 98.11

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

   2. Simplified 96.14

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

      [Start] 98.11	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 98.11	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. Taylor expanded in z around 0 92.81

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

   4. Taylor expanded in t around 0 87.91

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

4. Final simplification 68.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{+273}:\\ \;\;\;\;x \cdot \left(\cos \left({\left({\left({\left(\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, 2, 1\right) \cdot \left(z \cdot \left(t \cdot 0.0625\right)\right)}}\right)}^{3}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	69.36%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))