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

Average Error: 45.9 → 43.5

Time: 19.8s

Precision: binary64

Cost: 28484

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+270}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \left(y \cdot \left(z \cdot 0.125\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

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)))
      1e+270)
   (*
    x
    (* (cos (* t (* y (* z 0.125)))) (cos (* (* t b) (+ 0.0625 (* a 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))) <= 1e+270) {
		tmp = x * (cos((t * (y * (z * 0.125)))) * cos(((t * b) * (0.0625 + (a * 0.125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos((((((a * 2.0d0) + 1.0d0) * b) * t) / 16.0d0))
end function

↓

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z) * t) / 16.0d0))) * cos(((t * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))) <= 1d+270) then
        tmp = x * (cos((t * (y * (z * 0.125d0)))) * cos(((t * b) * (0.0625d0 + (a * 0.125d0)))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * Math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+270) {
		tmp = x * (Math.cos((t * (y * (z * 0.125)))) * Math.cos(((t * b) * (0.0625 + (a * 0.125)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos((((((a * 2.0) + 1.0) * b) * t) / 16.0))

↓

def code(x, y, z, t, a, b):
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * math.cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+270:
		tmp = x * (math.cos((t * (y * (z * 0.125)))) * math.cos(((t * b) * (0.0625 + (a * 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))) <= 1e+270)
		tmp = Float64(x * Float64(cos(Float64(t * Float64(y * Float64(z * 0.125)))) * cos(Float64(Float64(t * b) * Float64(0.0625 + Float64(a * 0.125))))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos((((((a * 2.0) + 1.0) * b) * t) / 16.0));
end

↓

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z) * t) / 16.0))) * cos(((t * ((1.0 + (2.0 * a)) * b)) / 16.0))) <= 1e+270)
		tmp = x * (cos((t * (y * (z * 0.125)))) * cos(((t * b) * (0.0625 + (a * 0.125)))));
	else
		tmp = x;
	end
	tmp_2 = 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], 1e+270], N[(x * N[(N[Cos[N[(t * N[(y * N[(z * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t * b), $MachinePrecision] * N[(0.0625 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

Target

Original	45.9
Target	44.3
Herbie	43.5

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

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))) < 1e270

   1. Initial program 33.9

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

   2. Simplified 34.2

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

      [Start] 33.9	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 33.9	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. Taylor expanded in y around inf 34.4

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

   4. Simplified 34.4

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

      [Start] 34.4	\[ x \cdot \left(\cos \left(0.125 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      *-commutative [=>] 34.4	\[ x \cdot \left(\cos \left(0.125 \cdot \left(y \cdot \color{blue}{\left(z \cdot t\right)}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      associate-*r* [=>] 34.4	\[ x \cdot \left(\cos \color{blue}{\left(\left(0.125 \cdot y\right) \cdot \left(z \cdot t\right)\right)} \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      associate-*r* [=>] 34.4	\[ x \cdot \left(\cos \color{blue}{\left(\left(\left(0.125 \cdot y\right) \cdot z\right) \cdot t\right)} \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      *-commutative [<=] 34.4	\[ x \cdot \left(\cos \color{blue}{\left(t \cdot \left(\left(0.125 \cdot y\right) \cdot z\right)\right)} \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      *-commutative [=>] 34.4	\[ x \cdot \left(\cos \left(t \cdot \left(\color{blue}{\left(y \cdot 0.125\right)} \cdot z\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]
      associate-*l* [=>] 34.4	\[ x \cdot \left(\cos \left(t \cdot \color{blue}{\left(y \cdot \left(0.125 \cdot z\right)\right)}\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + 0.125 \cdot a\right)\right)\right) \]

   if 1e270 < (*.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 62.8

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

   2. Simplified 61.5

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

      [Start] 62.8	\[ \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
      associate-*l* [=>] 62.8	\[ \color{blue}{x \cdot \left(\cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\right)} \]

   3. Taylor expanded in z around 0 59.5

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

   4. Taylor expanded in t around 0 56.3

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

4. Final simplification 43.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+270}:\\ \;\;\;\;x \cdot \left(\cos \left(t \cdot \left(y \cdot \left(z \cdot 0.125\right)\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	44.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023066 
(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))))