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

Percentage Accurate: 27.9% → 31.9%

Time: 22.1s

Alternatives: 6

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: 27.9% 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: 31.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \cos \left(\frac{t\_m \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot t\_1 \leq 10^{+271}:\\ \;\;\;\;t\_1 \cdot \left(x \cdot \cos \left(\frac{{\left({\left(z\_m \cdot t\_m\right)}^{0.3333333333333333}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* t_m (* (+ 1.0 (* 2.0 a)) b)) 16.0))))
   (if (<=
        (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0))) t_1)
        1e+271)
     (*
      t_1
      (* x (cos (/ (pow (pow (* z_m t_m) 0.3333333333333333) 3.0) 16.0))))
     x)))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	double t_1 = cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 1e+271) {
		tmp = t_1 * (x * cos((pow(pow((z_m * t_m), 0.3333333333333333), 3.0) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
t_m = abs(t)
real(8) function code(x, y, z_m, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(((t_m * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (((x * cos((((((y * 2.0d0) + 1.0d0) * z_m) * t_m) / 16.0d0))) * t_1) <= 1d+271) then
        tmp = t_1 * (x * cos(((((z_m * t_m) ** 0.3333333333333333d0) ** 3.0d0) / 16.0d0)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	double t_1 = Math.cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (((x * Math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 1e+271) {
		tmp = t_1 * (x * Math.cos((Math.pow(Math.pow((z_m * t_m), 0.3333333333333333), 3.0) / 16.0)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	t_1 = math.cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if ((x * math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 1e+271:
		tmp = t_1 * (x * math.cos((math.pow(math.pow((z_m * t_m), 0.3333333333333333), 3.0) / 16.0)))
	else:
		tmp = x
	return tmp

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	t_1 = cos(Float64(Float64(t_m * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0))
	tmp = 0.0
	if (Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 1e+271)
		tmp = Float64(t_1 * Float64(x * cos(Float64(((Float64(z_m * t_m) ^ 0.3333333333333333) ^ 3.0) / 16.0))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x, y, z_m, t_m, a, b)
	t_1 = cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (((x * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * t_1) <= 1e+271)
		tmp = t_1 * (x * cos(((((z_m * t_m) ^ 0.3333333333333333) ^ 3.0) / 16.0)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(t$95$m * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 1e+271], N[(t$95$1 * N[(x * N[Cos[N[(N[Power[N[Power[N[(z$95$m * t$95$m), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision] / 16.0), $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))) < 9.99999999999999953e270

   1. Initial program 49.7%

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

      1. add-cube-cbrt 50.2%

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

      2. pow3 50.0%

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

      3. *-commutative 50.0%

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

      4. fma-define 50.0%

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

      5. *-commutative 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in y around 0 33.3%

      \[\leadsto \left(x \cdot \cos \left(\frac{{\color{blue}{\left({\left(t \cdot z\right)}^{0.3333333333333333}\right)}}^{3}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 9.99999999999999953e270 < (*.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 3.7%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in z around 0 7.5%

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

   6. Taylor expanded in t around 0 13.6%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 10^{+271}:\\ \;\;\;\;\cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \cdot \left(x \cdot \cos \left(\frac{{\left({\left(z \cdot t\right)}^{0.3333333333333333}\right)}^{3}}{16}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\_m\right) \cdot t\_m}{16}\right)\right) \cdot \cos \left(\frac{t\_m \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (let* ((t_1
         (*
          (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z_m) t_m) 16.0)))
          (cos (/ (* t_m (* (+ 1.0 (* 2.0 a)) b)) 16.0)))))
   (if (<= t_1 5e+199) t_1 x)))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	double t_1 = (x * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 5e+199) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
t_m = abs(t)
real(8) function code(x, y, z_m, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * cos((((((y * 2.0d0) + 1.0d0) * z_m) * t_m) / 16.0d0))) * cos(((t_m * ((1.0d0 + (2.0d0 * a)) * b)) / 16.0d0))
    if (t_1 <= 5d+199) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	double t_1 = (x * Math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * Math.cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	double tmp;
	if (t_1 <= 5e+199) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	t_1 = (x * math.cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * math.cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0))
	tmp = 0
	if t_1 <= 5e+199:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	t_1 = Float64(Float64(x * cos(Float64(Float64(Float64(Float64(Float64(y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(Float64(Float64(t_m * Float64(Float64(1.0 + Float64(2.0 * a)) * b)) / 16.0)))
	tmp = 0.0
	if (t_1 <= 5e+199)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x, y, z_m, t_m, a, b)
	t_1 = (x * cos((((((y * 2.0) + 1.0) * z_m) * t_m) / 16.0))) * cos(((t_m * ((1.0 + (2.0 * a)) * b)) / 16.0));
	tmp = 0.0;
	if (t_1 <= 5e+199)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Cos[N[(N[(N[(N[(N[(y * 2.0), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t$95$m * N[(N[(1.0 + N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+199], t$95$1, 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))) < 4.9999999999999998e199

   1. Initial program 51.9%

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

   if 4.9999999999999998e199 < (*.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 5.3%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in z around 0 8.6%

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

   6. Taylor expanded in t around 0 13.7%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right) \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{t \cdot \left(\left(1 + 2 \cdot a\right) \cdot b\right)}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 30.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x \cdot \cos \left({\left(\sqrt[3]{0.0625 \cdot \left(t\_m \cdot b\right)}\right)}^{3}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (* x (cos (pow (cbrt (* 0.0625 (* t_m b))) 3.0))))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	return x * cos(pow(cbrt((0.0625 * (t_m * b))), 3.0));
}

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	return x * Math.cos(Math.pow(Math.cbrt((0.0625 * (t_m * b))), 3.0));
}

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	return Float64(x * cos((cbrt(Float64(0.0625 * Float64(t_m * b))) ^ 3.0)))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x * N[Cos[N[Power[N[Power[N[(0.0625 * N[(t$95$m * b), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

3. Simplified 33.4%

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

5. Taylor expanded in z around 0 34.3%

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

   1. add-cube-cbrt 34.2%

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

   2. pow3 34.3%

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

   3. associate-*r* 34.7%

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

   4. div-inv 34.7%

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

   5. metadata-eval 34.7%

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

7. Applied egg-rr 34.7%

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

8. Taylor expanded in a around 0 35.2%

   \[\leadsto \left(x \cdot 1\right) \cdot \cos \left({\left(\sqrt[3]{\color{blue}{0.0625 \cdot \left(b \cdot t\right)}}\right)}^{3}\right) \]

9. Final simplification 35.2%

   \[\leadsto x \cdot \cos \left({\left(\sqrt[3]{0.0625 \cdot \left(t \cdot b\right)}\right)}^{3}\right) \]
10. Add Preprocessing

Alternative 4: 31.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{+34}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{z\_m \cdot t\_m}{16}\right)\right) \cdot \cos \left(\left(t\_m \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (if (<= t_m 1.7e+34)
   (*
    (* x (cos (/ (* z_m t_m) 16.0)))
    (cos (* (* t_m b) (+ 0.0625 (* a 0.125)))))
   x))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 1.7e+34) {
		tmp = (x * cos(((z_m * t_m) / 16.0))) * cos(((t_m * b) * (0.0625 + (a * 0.125))));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

z_m = abs(z)
t_m = abs(t)
real(8) function code(x, y, z_m, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t_m <= 1.7d+34) then
        tmp = (x * cos(((z_m * t_m) / 16.0d0))) * cos(((t_m * b) * (0.0625d0 + (a * 0.125d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	double tmp;
	if (t_m <= 1.7e+34) {
		tmp = (x * Math.cos(((z_m * t_m) / 16.0))) * Math.cos(((t_m * b) * (0.0625 + (a * 0.125))));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	tmp = 0
	if t_m <= 1.7e+34:
		tmp = (x * math.cos(((z_m * t_m) / 16.0))) * math.cos(((t_m * b) * (0.0625 + (a * 0.125))))
	else:
		tmp = x
	return tmp

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	tmp = 0.0
	if (t_m <= 1.7e+34)
		tmp = Float64(Float64(x * cos(Float64(Float64(z_m * t_m) / 16.0))) * cos(Float64(Float64(t_m * b) * Float64(0.0625 + Float64(a * 0.125)))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

z_m = abs(z);
t_m = abs(t);
function tmp_2 = code(x, y, z_m, t_m, a, b)
	tmp = 0.0;
	if (t_m <= 1.7e+34)
		tmp = (x * cos(((z_m * t_m) / 16.0))) * cos(((t_m * b) * (0.0625 + (a * 0.125))));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := If[LessEqual[t$95$m, 1.7e+34], N[(N[(x * N[Cos[N[(N[(z$95$m * t$95$m), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(t$95$m * b), $MachinePrecision] * N[(0.0625 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e34

   1. Initial program 38.8%

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

   3. Taylor expanded in y around 0 39.9%

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

   4. Taylor expanded in a around 0 40.3%

      \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \color{blue}{\left(0.0625 \cdot \left(b \cdot t\right) + 0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative 40.3%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right) + 0.0625 \cdot \left(b \cdot t\right)\right)} \]

      2. associate-*r* 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(0.125 \cdot a\right) \cdot \left(b \cdot t\right)} + 0.0625 \cdot \left(b \cdot t\right)\right) \]

      3. *-commutative 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \left(\left(0.125 \cdot a\right) \cdot \color{blue}{\left(t \cdot b\right)} + 0.0625 \cdot \left(b \cdot t\right)\right) \]

      4. *-commutative 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \left(\left(0.125 \cdot a\right) \cdot \left(t \cdot b\right) + 0.0625 \cdot \color{blue}{\left(t \cdot b\right)}\right) \]

      5. distribute-rgt-out 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(t \cdot b\right) \cdot \left(0.125 \cdot a + 0.0625\right)\right)} \]

      6. *-commutative 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(b \cdot t\right)} \cdot \left(0.125 \cdot a + 0.0625\right)\right) \]

      7. *-commutative 40.4%

         \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \left(\left(b \cdot t\right) \cdot \left(\color{blue}{a \cdot 0.125} + 0.0625\right)\right) \]

   6. Simplified 40.4%

      \[\leadsto \left(x \cdot \cos \left(\frac{t \cdot z}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \left(a \cdot 0.125 + 0.0625\right)\right)} \]

   if 1.7e34 < t

   1. Initial program 9.0%

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

   3. Simplified 8.4%

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

   5. Taylor expanded in z around 0 12.6%

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

   6. Taylor expanded in t around 0 14.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+34}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{z \cdot t}{16}\right)\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x \cdot \cos \left(\left(t\_m \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b)
 :precision binary64
 (* x (cos (* (* t_m b) (+ 0.0625 (* a 0.125))))))

C

z_m = fabs(z);
t_m = fabs(t);
double code(double x, double y, double z_m, double t_m, double a, double b) {
	return x * cos(((t_m * b) * (0.0625 + (a * 0.125))));
}

Fortran

z_m = abs(z)
t_m = abs(t)
real(8) function code(x, y, z_m, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x * cos(((t_m * b) * (0.0625d0 + (a * 0.125d0))))
end function

Java

z_m = Math.abs(z);
t_m = Math.abs(t);
public static double code(double x, double y, double z_m, double t_m, double a, double b) {
	return x * Math.cos(((t_m * b) * (0.0625 + (a * 0.125))));
}

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	return x * math.cos(((t_m * b) * (0.0625 + (a * 0.125))))

Julia

z_m = abs(z)
t_m = abs(t)
function code(x, y, z_m, t_m, a, b)
	return Float64(x * cos(Float64(Float64(t_m * b) * Float64(0.0625 + Float64(a * 0.125)))))
end

MATLAB

z_m = abs(z);
t_m = abs(t);
function tmp = code(x, y, z_m, t_m, a, b)
	tmp = x * cos(((t_m * b) * (0.0625 + (a * 0.125))));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := N[(x * N[Cos[N[(N[(t$95$m * b), $MachinePrecision] * N[(0.0625 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

3. Simplified 33.4%

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

5. Taylor expanded in z around 0 34.3%

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

6. Taylor expanded in a around 0 34.6%

   \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(0.0625 \cdot \left(b \cdot t\right) + 0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative 34.6%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(0.125 \cdot \left(a \cdot \left(b \cdot t\right)\right) + 0.0625 \cdot \left(b \cdot t\right)\right)} \]

   2. associate-*r* 34.7%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\color{blue}{\left(0.125 \cdot a\right) \cdot \left(b \cdot t\right)} + 0.0625 \cdot \left(b \cdot t\right)\right) \]

   3. distribute-rgt-out 34.7%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\left(b \cdot t\right) \cdot \left(0.125 \cdot a + 0.0625\right)\right)} \]

   4. *-commutative 34.7%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\color{blue}{\left(t \cdot b\right)} \cdot \left(0.125 \cdot a + 0.0625\right)\right) \]

   5. *-commutative 34.7%

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\left(t \cdot b\right) \cdot \left(\color{blue}{a \cdot 0.125} + 0.0625\right)\right) \]

8. Simplified 34.7%

   \[\leadsto \left(x \cdot 1\right) \cdot \cos \color{blue}{\left(\left(t \cdot b\right) \cdot \left(a \cdot 0.125 + 0.0625\right)\right)} \]

9. Final simplification 34.7%

   \[\leadsto x \cdot \cos \left(\left(t \cdot b\right) \cdot \left(0.0625 + a \cdot 0.125\right)\right) \]
10. Add Preprocessing

Alternative 6: 31.3% accurate, 225.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ t_m = \left|t\right| \\ x \end{array} \]

FPCore

z_m = (fabs.f64 z)
t_m = (fabs.f64 t)
(FPCore (x y z_m t_m a b) :precision binary64 x)

C

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

Fortran

z_m = abs(z)
t_m = abs(t)
real(8) function code(x, y, z_m, t_m, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t_m
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

z_m = math.fabs(z)
t_m = math.fabs(t)
def code(x, y, z_m, t_m, a, b):
	return x

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[x_, y_, z$95$m_, t$95$m_, a_, b_] := x

Derivation

1. Initial program 33.0%

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

3. Simplified 33.4%

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

5. Taylor expanded in z around 0 34.3%

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

6. Taylor expanded in t around 0 34.7%

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

7. Final simplification 34.7%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 30.8% 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 2024030 
(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))))