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

Percentage Accurate: 27.6% → 32.1%

Time: 16.7s

Alternatives: 9

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: 27.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\\ \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot t\_1 \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{16}{\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t}}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x)))
   (if (<= (* (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0)) t_1) 4e+286)
     (* (cos (/ 1.0 (/ 16.0 (* (* (fma a 2.0 1.0) b) t)))) t_1)
     (* 1.0 (* 1.0 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x;
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 4e+286) {
		tmp = cos((1.0 / (16.0 / ((fma(a, 2.0, 1.0) * b) * t)))) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 4e+286)
		tmp = Float64(cos(Float64(1.0 / Float64(16.0 / Float64(Float64(fma(a, 2.0, 1.0) * b) * t)))) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, 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), $MachinePrecision]}, 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], 4e+286], N[(N[Cos[N[(1.0 / N[(16.0 / N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 4.00000000000000013e286

   1. Initial program 50.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 51.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 51.1

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 51.1

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

   4. Applied rewrites 51.1%

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

   if 4.00000000000000013e286 < (*.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 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 5.1%

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

   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 12.2%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\cos \left(\frac{1}{\frac{16}{\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t}}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\\ \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot t\_1 \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(b \cdot t, 0.0625, \left(0.0625 \cdot t\right) \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x)))
   (if (<= (* (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0)) t_1) 4e+286)
     (* (cos (fma (* b t) 0.0625 (* (* 0.0625 t) (* b (* a 2.0))))) t_1)
     (* 1.0 (* 1.0 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = cos(((t * (z * (1.0 + (2.0 * y)))) / 16.0)) * x;
	double tmp;
	if ((cos((((b * ((a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 4e+286) {
		tmp = cos(fma((b * t), 0.0625, ((0.0625 * t) * (b * (a * 2.0))))) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(cos(Float64(Float64(t * Float64(z * Float64(1.0 + Float64(2.0 * y)))) / 16.0)) * x)
	tmp = 0.0
	if (Float64(cos(Float64(Float64(Float64(b * Float64(Float64(a * 2.0) + 1.0)) * t) / 16.0)) * t_1) <= 4e+286)
		tmp = Float64(cos(fma(Float64(b * t), 0.0625, Float64(Float64(0.0625 * t) * Float64(b * Float64(a * 2.0))))) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, 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), $MachinePrecision]}, 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], 4e+286], N[(N[Cos[N[(N[(b * t), $MachinePrecision] * 0.0625 + N[(N[(0.0625 * t), $MachinePrecision] * N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 4.00000000000000013e286

   1. Initial program 50.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt-in N/A

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

      13. div-inv N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. clear-num N/A

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

      22. associate-/r/ N/A

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

      23. lower-*.f64 N/A

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

   4. Applied rewrites 50.8%

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

   if 4.00000000000000013e286 < (*.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 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 5.1%

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

   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 12.2%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(b \cdot t, 0.0625, \left(0.0625 \cdot t\right) \cdot \left(b \cdot \left(a \cdot 2\right)\right)\right)\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ \mathbf{if}\;t\_1 \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(-2, y, -1\right) \cdot t\right) \cdot z\right)\right) \cdot x\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))))
   (if (<= (* t_1 (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x)) 5e+276)
     (* (* (cos (* -0.0625 (* (* (fma -2.0 y -1.0) t) z))) x) t_1)
     (* 1.0 (* 1.0 x)))))

C

double code(double x, 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)) <= 5e+276) {
		tmp = (cos((-0.0625 * ((fma(-2.0, y, -1.0) * t) * z))) * x) * t_1;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, 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)) <= 5e+276)
		tmp = Float64(Float64(cos(Float64(-0.0625 * Float64(Float64(fma(-2.0, y, -1.0) * t) * z))) * x) * t_1);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, 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]}, 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), $MachinePrecision]), $MachinePrecision], 5e+276], N[(N[(N[Cos[N[(-0.0625 * N[(N[(N[(-2.0 * y + -1.0), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * t$95$1), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 5.00000000000000001e276

   1. Initial program 51.0%

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

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 51.1%

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

   if 5.00000000000000001e276 < (*.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 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 5.1%

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

   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 12.1%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 5 \cdot 10^{+276}:\\ \;\;\;\;\left(\cos \left(-0.0625 \cdot \left(\left(\mathsf{fma}\left(-2, y, -1\right) \cdot t\right) \cdot z\right)\right) \cdot x\right) \cdot \cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 32.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\left(\cos \left(\left(\mathsf{fma}\left(z, 2 \cdot y, z\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t\right) \cdot -0.0625\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (*
       (cos (/ (* (* b (+ (* a 2.0) 1.0)) t) 16.0))
       (* (cos (/ (* t (* z (+ 1.0 (* 2.0 y)))) 16.0)) x))
      4e+286)
   (*
    (*
     (cos (* (* (fma z (* 2.0 y) z) t) -0.0625))
     (cos (* (* (* (fma a 2.0 1.0) b) t) -0.0625)))
    x)
   (* 1.0 (* 1.0 x))))

C

double code(double x, 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)) <= 4e+286) {
		tmp = (cos(((fma(z, (2.0 * y), z) * t) * -0.0625)) * cos((((fma(a, 2.0, 1.0) * b) * t) * -0.0625))) * x;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, 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)) <= 4e+286)
		tmp = Float64(Float64(cos(Float64(Float64(fma(z, Float64(2.0 * y), z) * t) * -0.0625)) * cos(Float64(Float64(Float64(fma(a, 2.0, 1.0) * b) * t) * -0.0625))) * x);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := 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), $MachinePrecision]), $MachinePrecision], 4e+286], N[(N[(N[Cos[N[(N[(N[(z * N[(2.0 * y), $MachinePrecision] + z), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(N[(a * 2.0 + 1.0), $MachinePrecision] * b), $MachinePrecision] * t), $MachinePrecision] * -0.0625), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * N[(1.0 * x), $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)))) < 4.00000000000000013e286

   1. Initial program 50.7%

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

      1. 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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 39.8%

      \[\leadsto \left(x \cdot \cos \color{blue}{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left({\left(\mathsf{fma}\left(2, y, -1\right)\right)}^{-1} \cdot \left(\left(0.0625 \cdot t\right) \cdot z\right)\right)\right)}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]

   6. Applied rewrites 50.7%

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

   if 4.00000000000000013e286 < (*.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 0.0%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 5.1%

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

   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 12.2%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{\left(b \cdot \left(a \cdot 2 + 1\right)\right) \cdot t}{16}\right) \cdot \left(\cos \left(\frac{t \cdot \left(z \cdot \left(1 + 2 \cdot y\right)\right)}{16}\right) \cdot x\right) \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\left(\cos \left(\left(\mathsf{fma}\left(z, 2 \cdot y, z\right) \cdot t\right) \cdot -0.0625\right) \cdot \cos \left(\left(\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t\right) \cdot -0.0625\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\frac{\frac{-1}{\frac{1}{\mathsf{fma}\left(y, 2, 1\right) \cdot t} \cdot \frac{-1}{z}}}{16}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 1.4e-103)
   (*
    (* (cos (/ (/ -1.0 (* (/ 1.0 (* (fma y 2.0 1.0) t)) (/ -1.0 z))) 16.0)) x)
    1.0)
   (* 1.0 (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.4e-103) {
		tmp = (cos(((-1.0 / ((1.0 / (fma(y, 2.0, 1.0) * t)) * (-1.0 / z))) / 16.0)) * x) * 1.0;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 1.4e-103)
		tmp = Float64(Float64(cos(Float64(Float64(-1.0 / Float64(Float64(1.0 / Float64(fma(y, 2.0, 1.0) * t)) * Float64(-1.0 / z))) / 16.0)) * x) * 1.0);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 1.4e-103], N[(N[(N[Cos[N[(N[(-1.0 / N[(N[(1.0 / N[(N[(y * 2.0 + 1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 16.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000011e-103

   1. Initial program 35.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 36.7%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-*l/ N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

   7. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   9. Applied rewrites 37.4%

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

   11. Applied rewrites 37.8%

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

   if 1.40000000000000011e-103 < t

   1. Initial program 15.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 18.2%

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

   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 20.5%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\frac{\frac{-1}{\frac{1}{\mathsf{fma}\left(y, 2, 1\right) \cdot t} \cdot \frac{-1}{z}}}{16}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\frac{1}{\frac{\frac{16}{\mathsf{fma}\left(y, 2, 1\right) \cdot t}}{z}}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 1.4e-103)
   (* (* (cos (/ 1.0 (/ (/ 16.0 (* (fma y 2.0 1.0) t)) z))) x) 1.0)
   (* 1.0 (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.4e-103) {
		tmp = (cos((1.0 / ((16.0 / (fma(y, 2.0, 1.0) * t)) / z))) * x) * 1.0;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 1.4e-103)
		tmp = Float64(Float64(cos(Float64(1.0 / Float64(Float64(16.0 / Float64(fma(y, 2.0, 1.0) * t)) / z))) * x) * 1.0);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 1.4e-103], N[(N[(N[Cos[N[(1.0 / N[(N[(16.0 / N[(N[(y * 2.0 + 1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000011e-103

   1. Initial program 35.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 36.7%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-*l/ N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

   7. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}}{16}\right)\right) \cdot 1 \]

      5. lower-/.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}}{16}\right)\right) \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\mathsf{neg}\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \color{blue}{\left(t \cdot z\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      8. associate-*r* N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot z}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\color{blue}{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(\mathsf{neg}\left(z\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\color{blue}{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(\mathsf{neg}\left(z\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\color{blue}{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      12. lower-neg.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \color{blue}{\left(-z\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(2, y, -1\right)\right)}}{16}\right)\right) \cdot 1 \]

      13. lift-fma.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot y + -1\right)}\right)}}{16}\right)\right) \cdot 1 \]

      14. +-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{\mathsf{neg}\left(\color{blue}{\left(-1 + 2 \cdot y\right)}\right)}}{16}\right)\right) \cdot 1 \]

      15. distribute-neg-in N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(2 \cdot y\right)\right)}}}{16}\right)\right) \cdot 1 \]

      16. metadata-eval N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{\color{blue}{1} + \left(\mathsf{neg}\left(2 \cdot y\right)\right)}}{16}\right)\right) \cdot 1 \]

      17. *-rgt-identity N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{1 + \left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot y\right) \cdot 1}\right)\right)}}{16}\right)\right) \cdot 1 \]

      18. *-commutative N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{\left(\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot t\right) \cdot \left(-z\right)}{1 + \left(\mathsf{neg}\left(\color{blue}{\left(y \cdot 2\right)} \cdot 1\right)\right)}}{16}\right)\right) \cdot 1 \]

      19. sub-neg N/A

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

      20. *-commutative N/A

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

      21. *-rgt-identity N/A

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 37.4%

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

   if 1.40000000000000011e-103 < t

   1. Initial program 15.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 18.2%

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

   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 20.5%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\frac{1}{\frac{\frac{16}{\mathsf{fma}\left(y, 2, 1\right) \cdot t}}{z}}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 30.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\left(\cos \left(\frac{-0.0625}{\frac{-1}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\right) \cdot z}}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 6.4e-105)
   (* (* (cos (/ -0.0625 (/ -1.0 (* (* (fma 2.0 y 1.0) t) z)))) x) 1.0)
   (* 1.0 (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.4e-105) {
		tmp = (cos((-0.0625 / (-1.0 / ((fma(2.0, y, 1.0) * t) * z)))) * x) * 1.0;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 6.4e-105)
		tmp = Float64(Float64(cos(Float64(-0.0625 / Float64(-1.0 / Float64(Float64(fma(2.0, y, 1.0) * t) * z)))) * x) * 1.0);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 6.4e-105], N[(N[(N[Cos[N[(-0.0625 / N[(-1.0 / N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.39999999999999962e-105

   1. Initial program 35.5%

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

   3. Taylor expanded in 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 36.8%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-*l/ N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

   7. Applied rewrites 28.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{\mathsf{neg}\left(\frac{1}{16}\right)}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)}\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\frac{-1}{16}}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)\right) \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\frac{1}{-16}}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)\right) \cdot 1 \]

      9. metadata-eval N/A

         \[\leadsto \left(x \cdot \cos \left(\frac{\frac{1}{\color{blue}{\mathsf{neg}\left(16\right)}}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)\right) \cdot 1 \]

      10. lower-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{\frac{1}{\mathsf{neg}\left(16\right)}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)}\right) \cdot 1 \]

      11. metadata-eval N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{1}{\color{blue}{-16}}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)\right) \cdot 1 \]

      12. metadata-eval N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{\frac{-1}{16}}}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}\right)}\right)\right) \cdot 1 \]

      13. /-rgt-identity N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{-1}{16}}{\mathsf{neg}\left(\color{blue}{\frac{\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}}{1}}\right)}\right)\right) \cdot 1 \]

      14. clear-num N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{-1}{16}}{\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{1}{\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}}}}\right)}\right)\right) \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{-1}{16}}{\mathsf{neg}\left(\frac{1}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}}}}\right)}\right)\right) \cdot 1 \]

      16. distribute-neg-frac N/A

          \[\leadsto \left(x \cdot \cos \left(\frac{\frac{-1}{16}}{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{1}{\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}}}}}\right)\right) \cdot 1 \]

      17. metadata-eval N/A

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

   9. Applied rewrites 38.0%

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

   if 6.39999999999999962e-105 < t

   1. Initial program 15.5%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 18.2%

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

   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 20.5%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\left(\cos \left(\frac{-0.0625}{\frac{-1}{\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\right) \cdot z}}\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\right) \cdot z\right) \cdot 0.0625\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 1.4e-103)
   (* (* (cos (* (* (* (fma 2.0 y 1.0) t) z) 0.0625)) x) 1.0)
   (* 1.0 (* 1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 1.4e-103) {
		tmp = (cos((((fma(2.0, y, 1.0) * t) * z) * 0.0625)) * x) * 1.0;
	} else {
		tmp = 1.0 * (1.0 * x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 1.4e-103)
		tmp = Float64(Float64(cos(Float64(Float64(Float64(fma(2.0, y, 1.0) * t) * z) * 0.0625)) * x) * 1.0);
	else
		tmp = Float64(1.0 * Float64(1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 1.4e-103], N[(N[(N[Cos[N[(N[(N[(N[(2.0 * y + 1.0), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000011e-103

   1. Initial program 35.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 36.7%

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

      1. lift-*.f64 N/A

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

      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. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. flip-+ N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-*l/ N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

   7. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 28.0

         \[\leadsto \left(x \cdot \cos \color{blue}{\left(\frac{1}{\frac{\mathsf{fma}\left(2, y, -1\right)}{\mathsf{fma}\left(4 \cdot y, y, -1\right) \cdot \left(t \cdot z\right)}} \cdot 0.0625\right)}\right) \cdot 1 \]

   9. Applied rewrites 37.5%

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

   if 1.40000000000000011e-103 < t

   1. Initial program 15.4%

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right) \]
   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 18.2%

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

   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 20.5%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-103}:\\ \;\;\;\;\left(\cos \left(\left(\left(\mathsf{fma}\left(2, y, 1\right) \cdot t\right) \cdot z\right) \cdot 0.0625\right) \cdot x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(1 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.7% accurate, 24.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \left(1 \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 (* 1.0 x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * (1.0 * x);
}

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 = 1.0d0 * (1.0d0 * x)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * (1.0 * x);
}

Python

def code(x, y, z, t, a, b):
	return 1.0 * (1.0 * x)

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * Float64(1.0 * x))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * (1.0 * x);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * N[(1.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.3%

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

3. Taylor expanded in 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 29.6%

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

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 32.4%

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

9. Final simplification 32.4%

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

Developer Target 1: 30.1% 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 2024259 
(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))))