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

Average Accuracy: 28.7% → 32.4%

Time: 23.8s

Precision: binary64

Cost: 34884

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\left(a \cdot \left(t \cdot b\right)\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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)))
      4e-142)
   (*
    x
    (*
     (cos (/ (* z (fma y 2.0 1.0)) (/ 16.0 t)))
     (cos (* (* a (* t b)) 0.125))))
   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))) <= 4e-142) {
		tmp = x * (cos(((z * fma(y, 2.0, 1.0)) / (16.0 / t))) * cos(((a * (t * b)) * 0.125)));
	} 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))) <= 4e-142)
		tmp = Float64(x * Float64(cos(Float64(Float64(z * fma(y, 2.0, 1.0)) / Float64(16.0 / t))) * cos(Float64(Float64(a * Float64(t * b)) * 0.125))));
	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], 4e-142], N[(x * N[(N[Cos[N[(N[(z * N[(y * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[(16.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(a * N[(t * b), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Target

Original	28.7%
Target	31.3%
Herbie	32.4%

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

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))) < 4.0000000000000002e-142

   1. Initial program 48.3%

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

   2. Simplified 48.4%

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

      [Start] 48.3	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 48.3	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]
      associate-/l* [=>] 48.3	\[ x \cdot \left(\cos \color{blue}{\left(\frac{\left(y \cdot 2 + 1\right) \cdot z}{\frac{16}{t}}\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      fma-def [=>] 48.3	\[ x \cdot \left(\cos \left(\frac{\color{blue}{\mathsf{fma}\left(y, 2, 1\right)} \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      associate-/l* [=>] 48.4	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{\frac{16}{t}}\right)}\right) \]
      fma-def [=>] 48.4	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(a, 2, 1\right)} \cdot b}{\frac{16}{t}}\right)\right) \]

   3. Taylor expanded in a around inf 47.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)}\right) \]

   4. Simplified 47.7%

      \[\leadsto x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(\left(\left(t \cdot b\right) \cdot a\right) \cdot 0.125\right)}\right) \]
      Proof

      [Start] 47.7	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(0.125 \cdot \left(a \cdot \left(t \cdot b\right)\right)\right)\right) \]
      *-commutative [=>] 47.7	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \color{blue}{\left(\left(a \cdot \left(t \cdot b\right)\right) \cdot 0.125\right)}\right) \]
      *-commutative [=>] 47.7	\[ x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\color{blue}{\left(\left(t \cdot b\right) \cdot a\right)} \cdot 0.125\right)\right) \]

   if 4.0000000000000002e-142 < (*.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 16.5%

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

   2. Simplified 17.2%

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

      [Start] 16.5	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      *-commutative [=>] 16.5	\[ \color{blue}{\left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot x\right)} \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 16.5	\[ \color{blue}{\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]
      associate-/l* [=>] 16.6	\[ \cos \color{blue}{\left(\frac{\left(y \cdot 2 + 1\right) \cdot z}{\frac{16}{t}}\right)} \cdot \left(x \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      associate-/l* [=>] 17.2	\[ \cos \color{blue}{\left(\frac{y \cdot 2 + 1}{\frac{\frac{16}{t}}{z}}\right)} \cdot \left(x \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      fma-def [=>] 17.2	\[ \cos \left(\frac{\color{blue}{\mathsf{fma}\left(y, 2, 1\right)}}{\frac{\frac{16}{t}}{z}}\right) \cdot \left(x \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right) \]
      associate-*l/ [<=] 17.2	\[ \cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \left(x \cdot \cos \color{blue}{\left(\frac{\left(a \cdot 2 + 1\right) \cdot b}{16} \cdot t\right)}\right) \]
      *-commutative [=>] 17.2	\[ \cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \left(x \cdot \cos \left(\frac{\left(\color{blue}{2 \cdot a} + 1\right) \cdot b}{16} \cdot t\right)\right) \]
      fma-def [=>] 17.2	\[ \cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{\frac{\frac{16}{t}}{z}}\right) \cdot \left(x \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(2, a, 1\right)} \cdot b}{16} \cdot t\right)\right) \]

   3. Taylor expanded in b around 0 19.8%

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

   4. Taylor expanded in t around 0 22.9%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{-142}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{z \cdot \mathsf{fma}\left(y, 2, 1\right)}{\frac{16}{t}}\right) \cdot \cos \left(\left(a \cdot \left(t \cdot b\right)\right) \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	32.4%
Cost	28740

\[\begin{array}{l} t_1 := x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\\ \mathbf{if}\;t_1 \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 4 \cdot 10^{-142}:\\ \;\;\;\;t_1 \cdot \cos \left(\frac{2 \cdot \left(t \cdot \left(a \cdot b\right)\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	31.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023142 
(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))))