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

Percentage Accurate: 27.7% → 31.8%

Time: 13.3s

Alternatives: 7

Speedup: 24.5×

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: 27.7% 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: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot t\_1 \leq 10^{+283}:\\ \;\;\;\;\cos \left(\left(\left(b \cdot t\right) \cdot 0.125\right) \cdot a\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m)))
   (*
    x_s
    (if (<= (* (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0)) t_1) 1e+283)
      (* (cos (* (* (* b t) 0.125) a)) t_1)
      (* 1.0 (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m;
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+283) {
		tmp = cos((((b * t) * 0.125) * a)) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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) :: t_1
    real(8) :: tmp
    t_1 = cos(((t * (z * (1.0d0 + (2.0d0 * y)))) / 16.0d0)) * x_m
    if ((cos((((b * ((a * 2.0d0) + 1.0d0)) * t) / 16.0d0)) * t_1) <= 1d+283) then
        tmp = cos((((b * t) * 0.125d0) * a)) * t_1
    else
        tmp = 1.0d0 * (1.0d0 * x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m;
	double tmp;
	if ((Math.cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+283) {
		tmp = Math.cos((((b * t) * 0.125) * a)) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = math.cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m
	tmp = 0
	if (math.cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+283:
		tmp = math.cos((((b * t) * 0.125) * a)) * t_1
	else:
		tmp = 1.0 * (1.0 * x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+283)
		tmp = Float64(cos(Float64(Float64(Float64(b * t) * 0.125) * a)) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m;
	tmp = 0.0;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 1e+283)
		tmp = cos((((b * t) * 0.125) * a)) * t_1;
	else
		tmp = 1.0 * (1.0 * x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 1e+283], N[(N[Cos[N[(N[(N[(b * t), $MachinePrecision] * 0.125), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 9.99999999999999955e282

   1. Initial program 49.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\frac{1}{8} \cdot \left(a \cdot \left(b \cdot t\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{1}{8} \cdot \color{blue}{\left(\left(b \cdot t\right) \cdot a\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\frac{1}{8} \cdot \left(b \cdot t\right)\right) \cdot a\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \color{blue}{\left(\left(\frac{1}{8} \cdot \left(b \cdot t\right)\right) \cdot a\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\color{blue}{\left(\frac{1}{8} \cdot \left(b \cdot t\right)\right)} \cdot a\right) \]

      5. *-commutative N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\left(\frac{1}{8} \cdot \color{blue}{\left(t \cdot b\right)}\right) \cdot a\right) \]

      6. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.5%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.1%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 10^{+283}:\\ \;\;\;\;\cos \left(\left(\left(b \cdot t\right) \cdot 0.125\right) \cdot a\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t\right)\right)\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))
        (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m))
       5e+295)
    (*
     (*
      (cos (* (* (* (fma 2.0 y 1.0) z) t) -0.0625))
      (cos (* -0.0625 (* (* (fma a 2.0 1.0) b) t))))
     x_m)
    (* 1.0 (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295) {
		tmp = (cos((((fma(2.0, y, 1.0) * z) * t) * -0.0625)) * cos((-0.0625 * ((fma(a, 2.0, 1.0) * b) * t)))) * x_m;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)) <= 5e+295)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(2.0, y, 1.0) * z) * t) * -0.0625)) * cos(Float64(-0.0625 * Float64(Float64(fma(a, 2.0, 1.0) * b) * t)))) * x_m);
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 5e+295], N[(N[(N[Cos[N[(N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.0625 * N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.99999999999999991e295

   1. Initial program 49.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 49.0

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

   4. Applied rewrites 49.0%

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

   6. Applied rewrites 49.3%

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

   if 4.99999999999999991e295 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.1%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot z\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t\right)\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot x\_m\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))))
   (*
    x_s
    (if (<=
         (* t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m))
         5e+295)
      (* (* (cos (* 0.0625 (* t z))) x_m) t_1)
      (* 1.0 (* 1.0 x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0));
	double tmp;
	if ((t_1 * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295) {
		tmp = (cos((0.0625 * (t * z))) * x_m) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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) :: t_1
    real(8) :: tmp
    t_1 = cos((((b * ((a * 2.0d0) + 1.0d0)) * t) / 16.0d0))
    if ((t_1 * (cos(((t * (z * (1.0d0 + (2.0d0 * y)))) / 16.0d0)) * x_m)) <= 5d+295) then
        tmp = (cos((0.0625d0 * (t * z))) * x_m) * t_1
    else
        tmp = 1.0d0 * (1.0d0 * x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double t_1 = Math.cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0));
	double tmp;
	if ((t_1 * (Math.cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295) {
		tmp = (Math.cos((0.0625 * (t * z))) * x_m) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	t_1 = math.cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0))
	tmp = 0
	if (t_1 * (math.cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295:
		tmp = (math.cos((0.0625 * (t * z))) * x_m) * t_1
	else:
		tmp = 1.0 * (1.0 * x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	t_1 = cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0))
	tmp = 0.0
	if (Float64(t_1 * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)) <= 5e+295)
		tmp = Float64(Float64(cos(Float64(0.0625 * Float64(t * z))) * x_m) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t, a, b)
	t_1 = cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0));
	tmp = 0.0;
	if ((t_1 * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295)
		tmp = (cos((0.0625 * (t * z))) * x_m) * t_1;
	else
		tmp = 1.0 * (1.0 * x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(t$95$1 * N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 5e+295], N[(N[(N[Cos[N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.99999999999999991e295

   1. Initial program 49.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 49.1

         \[\leadsto \left(x \cdot \cos \left(\color{blue}{\left(z \cdot t\right)} \cdot 0.0625\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   5. Applied rewrites 49.1%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\left(z \cdot t\right) \cdot 0.0625\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   if 4.99999999999999991e295 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.1%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\left(\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot x\right) \cdot \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\cos \left(\left(\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right) \cdot t\right) \cdot 0.0625\right) \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))
        (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m))
       5e+295)
    (*
     (cos (* (* (* (fma y 2.0 1.0) z) t) 0.0625))
     (* (cos (* 0.0625 (* b t))) x_m))
    (* 1.0 (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 5e+295) {
		tmp = cos((((fma(y, 2.0, 1.0) * z) * t) * 0.0625)) * (cos((0.0625 * (b * t))) * x_m);
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)) <= 5e+295)
		tmp = Float64(cos(Float64(Float64(Float64(fma(y, 2.0, 1.0) * z) * t) * 0.0625)) * Float64(cos(Float64(0.0625 * Float64(b * t))) * x_m));
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 5e+295], N[(N[Cos[N[(N[(N[(N[(y * 2.0 + 1.0), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(0.0625 * N[(b * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 4.99999999999999991e295

   1. Initial program 49.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   4. Applied rewrites 21.9%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 48.9

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

   7. Applied rewrites 48.9%

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

   if 4.99999999999999991e295 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 0.2%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 4.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 10.1%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\cos \left(\left(\left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right) \cdot t\right) \cdot 0.0625\right) \cdot \left(\cos \left(0.0625 \cdot \left(b \cdot t\right)\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 31.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\_m\right) \leq 10^{+283}:\\ \;\;\;\;\left(\cos \left(\left(\left(0.0625 \cdot z\right) \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot x\_m\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<=
       (*
        (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))
        (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x_m))
       1e+283)
    (* (* (cos (* (* (* 0.0625 z) t) (fma 2.0 y 1.0))) x_m) 1.0)
    (* 1.0 (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * (cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x_m)) <= 1e+283) {
		tmp = (cos((((0.0625 * z) * t) * fma(2.0, y, 1.0))) * x_m) * 1.0;
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x_m)) <= 1e+283)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(0.0625 * z) * t) * fma(2.0, y, 1.0))) * x_m) * 1.0);
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[N[(N[Cos[N[(N[(N[(b * N[(N[(a * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(t * N[(z * N[(1.0 + N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], 1e+283], N[(N[(N[Cos[N[(N[(N[(0.0625 * z), $MachinePrecision] * t), $MachinePrecision] * N[(2.0 * y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64)))) < 9.99999999999999955e282

   1. Initial program 49.1%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 48.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-*l* N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. *-rgt-identity N/A

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

      15. lift-/.f64 N/A

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

      16. div-inv N/A

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

      17. times-frac N/A

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

      18. clear-num N/A

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

      19. /-rgt-identity N/A

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

      20. lower-*.f64 N/A

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

      21. div-inv N/A

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

      22. lower-*.f64 N/A

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

      23. metadata-eval 48.4

          \[\leadsto \left(x \cdot \cos \left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot \color{blue}{0.0625}\right) \cdot t\right)\right)\right) \cdot 1 \]

   7. Applied rewrites 48.4%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot \left(\left(z \cdot 0.0625\right) \cdot t\right)\right)}\right) \cdot 1 \]

   if 9.99999999999999955e282 < (*.f64 (*.f64 x (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 y #s(literal 2 binary64)) #s(literal 1 binary64)) z) t) #s(literal 16 binary64)))) (cos.f64 (/.f64 (*.f64 (*.f64 (+.f64 (*.f64 a #s(literal 2 binary64)) #s(literal 1 binary64)) b) t) #s(literal 16 binary64))))

   1. Initial program 1.5%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 11.1%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 10^{+283}:\\ \;\;\;\;\left(\cos \left(\left(\left(0.0625 \cdot z\right) \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{+80}:\\ \;\;\;\;\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b)
 :precision binary64
 (*
  x_s
  (if (<= t 4.7e+80)
    (*
     (cos (* 0.0625 (* t z)))
     (* (cos (* (* (* (fma a 2.0 1.0) t) b) 0.0625)) x_m))
    (* 1.0 (* 1.0 x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 4.7e+80) {
		tmp = cos((0.0625 * (t * z))) * (cos((((fma(a, 2.0, 1.0) * t) * b) * 0.0625)) * x_m);
	} else {
		tmp = 1.0 * (1.0 * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	tmp = 0.0
	if (t <= 4.7e+80)
		tmp = Float64(cos(Float64(0.0625 * Float64(t * z))) * Float64(cos(Float64(Float64(Float64(fma(a, 2.0, 1.0) * t) * b) * 0.0625)) * x_m));
	else
		tmp = Float64(1.0 * Float64(1.0 * x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * If[LessEqual[t, 4.7e+80], N[(N[Cos[N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * t), $MachinePrecision] * b), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.70000000000000009e80

   1. Initial program 37.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   4. Applied rewrites 16.5%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   7. Applied rewrites 38.7%

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

   if 4.70000000000000009e80 < t

   1. Initial program 4.0%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 9.8%

      \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.7 \cdot 10^{+80}:\\ \;\;\;\;\cos \left(0.0625 \cdot \left(t \cdot z\right)\right) \cdot \left(\cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right) \cdot b\right) \cdot 0.0625\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.0% accurate, 24.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(1 \cdot \left(1 \cdot x\_m\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t a b) :precision binary64 (* x_s (* 1.0 (* 1.0 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (1.0 * (1.0 * x_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t, a, b)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_s * (1.0d0 * (1.0d0 * x_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t, double a, double b) {
	return x_s * (1.0 * (1.0 * x_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t, a, b):
	return x_s * (1.0 * (1.0 * x_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t, a, b)
	return Float64(x_s * Float64(1.0 * Float64(1.0 * x_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t, a, b)
	tmp = x_s * (1.0 * (1.0 * x_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_, a_, b_] := N[(x$95$s * N[(1.0 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 30.4%

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

6. Taylor expanded in b around 0

   \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 32.2%

   \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1} \]

9. Final simplification 32.2%

   \[\leadsto 1 \cdot \left(1 \cdot x\right) \]
10. Add Preprocessing

Developer Target 1: 30.5% accurate, 1.1× 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 2024249 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2)))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))