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

Percentage Accurate: 28.1% → 32.2%

Time: 24.6s

Alternatives: 3

Speedup: 225.0×

Specification

Math

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

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))))

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));
}

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

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));
}

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))

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

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

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]

Initial Program: 28.1% accurate, 1.0× speedup

Math

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

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))))

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));
}

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

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));
}

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))

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

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

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]

Alternative 1: 32.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} z = |z|\\ t = |t|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{\frac{16}{t}}{z}}\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{t_1} \cdot \frac{\mathsf{fma}\left(y, 2, 1\right)}{t_1}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (sqrt (/ (/ 16.0 t) z))))
   (if (<=
        (*
         (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0)))
         (cos (/ (* t (* (+ 1.0 (* 2.0 a)) b)) 16.0)))
        2e-25)
     (*
      x
      (*
       (cos (* (/ 1.0 t_1) (/ (fma y 2.0 1.0) t_1)))
       (cos (/ (* b (fma a 2.0 1.0)) (/ 16.0 t)))))
     x)))

C

z = abs(z);
t = abs(t);
double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = sqrt(((16.0 / t) / z));
	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-25) {
		tmp = x * (cos(((1.0 / t_1) * (fma(y, 2.0, 1.0) / t_1))) * cos(((b * fma(a, 2.0, 1.0)) / (16.0 / t))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

z = abs(z)
t = abs(t)
function code(x, y, z, t, a, b)
	t_1 = sqrt(Float64(Float64(16.0 / t) / z))
	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-25)
		tmp = Float64(x * Float64(cos(Float64(Float64(1.0 / t_1) * Float64(fma(y, 2.0, 1.0) / t_1))) * cos(Float64(Float64(b * fma(a, 2.0, 1.0)) / Float64(16.0 / t)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Sqrt[N[(N[(16.0 / t), $MachinePrecision] / z), $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], 2e-25], N[(x * N[(N[Cos[N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(y * 2.0 + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $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]]

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))) < 2.00000000000000008e-25

   1. Initial program 46.6%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   2. Step-by-step derivation

      1. *-commutative 46.6%

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

      2. associate-*l* 46.6%

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

      3. cos-neg 46.6%

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

      4. distribute-frac-neg 46.6%

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

      5. distribute-lft-neg-in 46.6%

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

      6. distribute-rgt-neg-out 46.6%

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

      7. associate-*l* 46.6%

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

      8. *-commutative 46.6%

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

      9. associate-*l* 46.6%

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

   3. Simplified 47.0%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right) \cdot z}{\frac{16}{t}}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(a, 2, 1\right) \cdot b}{\frac{16}{t}}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-/l* 47.0%

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

      2. *-un-lft-identity 47.0%

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

      3. add-sqr-sqrt 18.7%

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

      4. times-frac 19.0%

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

   5. Applied egg-rr 19.0%

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

   if 2.00000000000000008e-25 < (*.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 14.9%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in z around 0 19.3%

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

   5. Taylor expanded in t around 0 24.0%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 2 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(\cos \left(\frac{1}{\sqrt{\frac{\frac{16}{t}}{z}}} \cdot \frac{\mathsf{fma}\left(y, 2, 1\right)}{\sqrt{\frac{\frac{16}{t}}{z}}}\right) \cdot \cos \left(\frac{b \cdot \mathsf{fma}\left(a, 2, 1\right)}{\frac{16}{t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 32.2% accurate, 0.3× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
(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-17)
   (*
    x
    (*
     (cos (/ (fma y 2.0 1.0) (/ (/ 16.0 t) z)))
     (cos (* (/ t 16.0) (* b (fma 2.0 a 1.0))))))
   x))

C

z = abs(z);
t = abs(t);
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-17) {
		tmp = x * (cos((fma(y, 2.0, 1.0) / ((16.0 / t) / z))) * cos(((t / 16.0) * (b * fma(2.0, a, 1.0)))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

z = abs(z)
t = abs(t)
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-17)
		tmp = Float64(x * Float64(cos(Float64(fma(y, 2.0, 1.0) / Float64(Float64(16.0 / t) / z))) * cos(Float64(Float64(t / 16.0) * Float64(b * fma(2.0, a, 1.0))))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
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-17], N[(x * N[(N[Cos[N[(N[(y * 2.0 + 1.0), $MachinePrecision] / N[(N[(16.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t / 16.0), $MachinePrecision] * N[(b * N[(2.0 * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]

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))) < 1.00000000000000007e-17

   1. Initial program 46.8%

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

   3. Simplified 47.2%

      \[\leadsto \color{blue}{x \cdot \left(\cos \left(\frac{\mathsf{fma}\left(y, 2, 1\right)}{16} \cdot \left(z \cdot t\right)\right) \cdot \cos \left(\frac{t}{16} \cdot \left(\mathsf{fma}\left(2, a, 1\right) \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*l/ 47.2%

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

      2. associate-*l* 46.8%

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

      3. fma-def 46.8%

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

      4. associate-/l* 46.9%

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

      5. fma-def 46.9%

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

      6. associate-/l* 47.8%

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

   5. Applied egg-rr 47.8%

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

   if 1.00000000000000007e-17 < (*.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 14.3%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in z around 0 18.9%

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

   5. Taylor expanded in t around 0 23.5%

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

4. Final simplification 33.8%

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

Alternative 3: 31.2% accurate, 225.0× speedup

Math

\[\begin{array}{l} z = |z|\\ t = |t|\\ \\ x \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
(FPCore (x y z t a b) :precision binary64 x)

C

z = abs(z);
t = abs(t);
double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this 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
    code = x
end function

Java

z = Math.abs(z);
t = Math.abs(t);
public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

z = abs(z)
t = abs(t)
def code(x, y, z, t, a, b):
	return x

Julia

z = abs(z)
t = abs(t)
function code(x, y, z, t, a, b)
	return x
end

MATLAB

z = abs(z)
t = abs(t)
function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: t should be positive before calling this function
code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 28.0%

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

3. Simplified 28.3%

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

4. Taylor expanded in z around 0 29.5%

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

5. Taylor expanded in t around 0 32.2%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 32.2%

   \[\leadsto x \]

Developer target: 30.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * cos(((b / 16.0) * (t / ((1.0 - (a * 2.0)) + pow((a * 2.0), 2.0)))));
}

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(((b / 16.0d0) * (t / ((1.0d0 - (a * 2.0d0)) + ((a * 2.0d0) ** 2.0d0)))))
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a, b)
	return Float64(x * cos(Float64(Float64(b / 16.0) * Float64(t / Float64(Float64(1.0 - Float64(a * 2.0)) + (Float64(a * 2.0) ^ 2.0))))))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * N[Cos[N[(N[(b / 16.0), $MachinePrecision] * N[(t / N[(N[(1.0 - N[(a * 2.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(a * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(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))))