2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.1%

Time: 36.7s

Alternatives: 17

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(x + \varepsilon\right) - \tan x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

C

double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = tan((x + eps)) - tan(x)
end function

Java

public static double code(double x, double eps) {
	return Math.tan((x + eps)) - Math.tan(x);
}

Python

def code(x, eps):
	return math.tan((x + eps)) - math.tan(x)

Julia

function code(x, eps)
	return Float64(tan(Float64(x + eps)) - tan(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = tan((x + eps)) - tan(x);
end

Wolfram

code[x_, eps_] := N[(N[Tan[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(\left(1 - \varepsilon \cdot \left(\frac{\sin x \cdot \left(-1 - t\_0\right)}{\cos x} - \varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot 0.6666666666666666 + 1.8888888888888888 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) + \left(0.3333333333333333 + {x}^{2} \cdot \left(1.3333333333333333 + {x}^{2} \cdot \left(1.8888888888888888 - {x}^{2} \cdot -1.837037037037037\right)\right)\right)\right)\right)\right) + t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     (-
      1.0
      (*
       eps
       (-
        (/ (* (sin x) (- -1.0 t_0)) (cos x))
        (*
         eps
         (+
          (*
           x
           (+
            (* eps 0.6666666666666666)
            (* 1.8888888888888888 (* eps (pow x 2.0)))))
          (+
           0.3333333333333333
           (*
            (pow x 2.0)
            (+
             1.3333333333333333
             (*
              (pow x 2.0)
              (-
               1.8888888888888888
               (* (pow x 2.0) -1.837037037037037)))))))))))
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * ((1.0 - (eps * (((sin(x) * (-1.0 - t_0)) / cos(x)) - (eps * ((x * ((eps * 0.6666666666666666) + (1.8888888888888888 * (eps * pow(x, 2.0))))) + (0.3333333333333333 + (pow(x, 2.0) * (1.3333333333333333 + (pow(x, 2.0) * (1.8888888888888888 - (pow(x, 2.0) * -1.837037037037037))))))))))) + t_0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * ((1.0d0 - (eps * (((sin(x) * ((-1.0d0) - t_0)) / cos(x)) - (eps * ((x * ((eps * 0.6666666666666666d0) + (1.8888888888888888d0 * (eps * (x ** 2.0d0))))) + (0.3333333333333333d0 + ((x ** 2.0d0) * (1.3333333333333333d0 + ((x ** 2.0d0) * (1.8888888888888888d0 - ((x ** 2.0d0) * (-1.837037037037037d0)))))))))))) + t_0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * ((1.0 - (eps * (((Math.sin(x) * (-1.0 - t_0)) / Math.cos(x)) - (eps * ((x * ((eps * 0.6666666666666666) + (1.8888888888888888 * (eps * Math.pow(x, 2.0))))) + (0.3333333333333333 + (Math.pow(x, 2.0) * (1.3333333333333333 + (Math.pow(x, 2.0) * (1.8888888888888888 - (Math.pow(x, 2.0) * -1.837037037037037))))))))))) + t_0);
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * ((1.0 - (eps * (((math.sin(x) * (-1.0 - t_0)) / math.cos(x)) - (eps * ((x * ((eps * 0.6666666666666666) + (1.8888888888888888 * (eps * math.pow(x, 2.0))))) + (0.3333333333333333 + (math.pow(x, 2.0) * (1.3333333333333333 + (math.pow(x, 2.0) * (1.8888888888888888 - (math.pow(x, 2.0) * -1.837037037037037))))))))))) + t_0)

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(Float64(1.0 - Float64(eps * Float64(Float64(Float64(sin(x) * Float64(-1.0 - t_0)) / cos(x)) - Float64(eps * Float64(Float64(x * Float64(Float64(eps * 0.6666666666666666) + Float64(1.8888888888888888 * Float64(eps * (x ^ 2.0))))) + Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(1.3333333333333333 + Float64((x ^ 2.0) * Float64(1.8888888888888888 - Float64((x ^ 2.0) * -1.837037037037037))))))))))) + t_0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * ((1.0 - (eps * (((sin(x) * (-1.0 - t_0)) / cos(x)) - (eps * ((x * ((eps * 0.6666666666666666) + (1.8888888888888888 * (eps * (x ^ 2.0))))) + (0.3333333333333333 + ((x ^ 2.0) * (1.3333333333333333 + ((x ^ 2.0) * (1.8888888888888888 - ((x ^ 2.0) * -1.837037037037037))))))))))) + t_0);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(1.0 - N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(x * N[(N[(eps * 0.6666666666666666), $MachinePrecision] + N[(1.8888888888888888 * N[(eps * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.8888888888888888 - N[(N[Power[x, 2.0], $MachinePrecision] * -1.837037037037037), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\color{blue}{x \cdot \left(-1.2777777777777777 \cdot {x}^{2} - 0.3333333333333333\right)}}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{x \cdot \left(-1.2777777777777777 \cdot {x}^{2} - 0.3333333333333333\right)}{\cos x}\right)\right)\right) - \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(-1.837037037037037 \cdot {x}^{2} - 1.8888888888888888\right) - 1.3333333333333333\right) - 0.3333333333333333\right)}\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

6. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{x \cdot \left(0.6666666666666666 \cdot \varepsilon + 1.8888888888888888 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(-1.837037037037037 \cdot {x}^{2} - 1.8888888888888888\right) - 1.3333333333333333\right) - 0.3333333333333333\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

7. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 - \varepsilon \cdot \left(\frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} - \varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot 0.6666666666666666 + 1.8888888888888888 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) + \left(0.3333333333333333 + {x}^{2} \cdot \left(1.3333333333333333 + {x}^{2} \cdot \left(1.8888888888888888 - {x}^{2} \cdot -1.837037037037037\right)\right)\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(\left(1 - \varepsilon \cdot \left(\frac{\sin x \cdot \left(-1 - t\_0\right)}{\cos x} - \varepsilon \cdot \left(0.6666666666666666 \cdot \left(\varepsilon \cdot x\right) + \left(0.3333333333333333 + {x}^{2} \cdot \left(1.3333333333333333 + {x}^{2} \cdot \left(1.8888888888888888 - {x}^{2} \cdot -1.837037037037037\right)\right)\right)\right)\right)\right) + t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     (-
      1.0
      (*
       eps
       (-
        (/ (* (sin x) (- -1.0 t_0)) (cos x))
        (*
         eps
         (+
          (* 0.6666666666666666 (* eps x))
          (+
           0.3333333333333333
           (*
            (pow x 2.0)
            (+
             1.3333333333333333
             (*
              (pow x 2.0)
              (-
               1.8888888888888888
               (* (pow x 2.0) -1.837037037037037)))))))))))
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * ((1.0 - (eps * (((sin(x) * (-1.0 - t_0)) / cos(x)) - (eps * ((0.6666666666666666 * (eps * x)) + (0.3333333333333333 + (pow(x, 2.0) * (1.3333333333333333 + (pow(x, 2.0) * (1.8888888888888888 - (pow(x, 2.0) * -1.837037037037037))))))))))) + t_0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * ((1.0d0 - (eps * (((sin(x) * ((-1.0d0) - t_0)) / cos(x)) - (eps * ((0.6666666666666666d0 * (eps * x)) + (0.3333333333333333d0 + ((x ** 2.0d0) * (1.3333333333333333d0 + ((x ** 2.0d0) * (1.8888888888888888d0 - ((x ** 2.0d0) * (-1.837037037037037d0)))))))))))) + t_0)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * ((1.0 - (eps * (((Math.sin(x) * (-1.0 - t_0)) / Math.cos(x)) - (eps * ((0.6666666666666666 * (eps * x)) + (0.3333333333333333 + (Math.pow(x, 2.0) * (1.3333333333333333 + (Math.pow(x, 2.0) * (1.8888888888888888 - (Math.pow(x, 2.0) * -1.837037037037037))))))))))) + t_0);
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * ((1.0 - (eps * (((math.sin(x) * (-1.0 - t_0)) / math.cos(x)) - (eps * ((0.6666666666666666 * (eps * x)) + (0.3333333333333333 + (math.pow(x, 2.0) * (1.3333333333333333 + (math.pow(x, 2.0) * (1.8888888888888888 - (math.pow(x, 2.0) * -1.837037037037037))))))))))) + t_0)

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(Float64(1.0 - Float64(eps * Float64(Float64(Float64(sin(x) * Float64(-1.0 - t_0)) / cos(x)) - Float64(eps * Float64(Float64(0.6666666666666666 * Float64(eps * x)) + Float64(0.3333333333333333 + Float64((x ^ 2.0) * Float64(1.3333333333333333 + Float64((x ^ 2.0) * Float64(1.8888888888888888 - Float64((x ^ 2.0) * -1.837037037037037))))))))))) + t_0))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * ((1.0 - (eps * (((sin(x) * (-1.0 - t_0)) / cos(x)) - (eps * ((0.6666666666666666 * (eps * x)) + (0.3333333333333333 + ((x ^ 2.0) * (1.3333333333333333 + ((x ^ 2.0) * (1.8888888888888888 - ((x ^ 2.0) * -1.837037037037037))))))))))) + t_0);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(1.0 - N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(0.6666666666666666 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.3333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.8888888888888888 - N[(N[Power[x, 2.0], $MachinePrecision] * -1.837037037037037), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

4. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\color{blue}{x \cdot \left(-1.2777777777777777 \cdot {x}^{2} - 0.3333333333333333\right)}}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{x \cdot \left(-1.2777777777777777 \cdot {x}^{2} - 0.3333333333333333\right)}{\cos x}\right)\right)\right) - \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(-1.837037037037037 \cdot {x}^{2} - 1.8888888888888888\right) - 1.3333333333333333\right) - 0.3333333333333333\right)}\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

6. Taylor expanded in x around 0 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\color{blue}{0.6666666666666666 \cdot \left(\varepsilon \cdot x\right)} - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(-1.837037037037037 \cdot {x}^{2} - 1.8888888888888888\right) - 1.3333333333333333\right) - 0.3333333333333333\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

7. Final simplification 100.0%

   \[\leadsto \varepsilon \cdot \left(\left(1 - \varepsilon \cdot \left(\frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} - \varepsilon \cdot \left(0.6666666666666666 \cdot \left(\varepsilon \cdot x\right) + \left(0.3333333333333333 + {x}^{2} \cdot \left(1.3333333333333333 + {x}^{2} \cdot \left(1.8888888888888888 - {x}^{2} \cdot -1.837037037037037\right)\right)\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Add Preprocessing

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ \varepsilon \cdot \left(1 - \left(\frac{t\_0}{\frac{-1 - \cos \left(x \cdot 2\right)}{2}} + \varepsilon \cdot \frac{\sin x \cdot \left(-1 - \frac{t\_0}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)))
   (*
    eps
    (-
     1.0
     (+
      (/ t_0 (/ (- -1.0 (cos (* x 2.0))) 2.0))
      (* eps (/ (* (sin x) (- -1.0 (/ t_0 (pow (cos x) 2.0)))) (cos x))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	return eps * (1.0 - ((t_0 / ((-1.0 - cos((x * 2.0))) / 2.0)) + (eps * ((sin(x) * (-1.0 - (t_0 / pow(cos(x), 2.0)))) / cos(x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(x) ** 2.0d0
    code = eps * (1.0d0 - ((t_0 / (((-1.0d0) - cos((x * 2.0d0))) / 2.0d0)) + (eps * ((sin(x) * ((-1.0d0) - (t_0 / (cos(x) ** 2.0d0)))) / cos(x)))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	return eps * (1.0 - ((t_0 / ((-1.0 - Math.cos((x * 2.0))) / 2.0)) + (eps * ((Math.sin(x) * (-1.0 - (t_0 / Math.pow(Math.cos(x), 2.0)))) / Math.cos(x)))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	return eps * (1.0 - ((t_0 / ((-1.0 - math.cos((x * 2.0))) / 2.0)) + (eps * ((math.sin(x) * (-1.0 - (t_0 / math.pow(math.cos(x), 2.0)))) / math.cos(x)))))

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	return Float64(eps * Float64(1.0 - Float64(Float64(t_0 / Float64(Float64(-1.0 - cos(Float64(x * 2.0))) / 2.0)) + Float64(eps * Float64(Float64(sin(x) * Float64(-1.0 - Float64(t_0 / (cos(x) ^ 2.0)))) / cos(x))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	tmp = eps * (1.0 - ((t_0 / ((-1.0 - cos((x * 2.0))) / 2.0)) + (eps * ((sin(x) * (-1.0 - (t_0 / (cos(x) ^ 2.0)))) / cos(x)))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 - N[(N[(t$95$0 / N[(N[(-1.0 - N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * N[(-1.0 - N[(t$95$0 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\color{blue}{\cos x \cdot \cos x}}\right)\right)\right) \]

   2. cos-mult 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right)\right)\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\frac{\color{blue}{\cos \left(x - x\right) + \cos \left(x + x\right)}}{2}}\right)\right)\right) \]

   2. +-inverses 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right)\right)\right) \]

   3. cos-0 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\frac{\color{blue}{1} + \cos \left(x + x\right)}{2}}\right)\right)\right) \]

   4. count-2 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(2 \cdot x\right)}}{2}}\right)\right)\right) \]

   5. *-commutative 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\frac{1 + \cos \color{blue}{\left(x \cdot 2\right)}}{2}}\right)\right)\right) \]

9. Simplified 99.8%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{\color{blue}{\frac{1 + \cos \left(x \cdot 2\right)}{2}}}\right)\right)\right) \]

10. Final simplification 99.8%

    \[\leadsto \varepsilon \cdot \left(1 - \left(\frac{{\sin x}^{2}}{\frac{-1 - \cos \left(x \cdot 2\right)}{2}} + \varepsilon \cdot \frac{\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 - \left(\varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot \left(0.5083333333333333 + {x}^{2} \cdot 0.2748015873015873\right)\right)\right)}{\cos x} - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   1.0
   (-
    (*
     eps
     (/
      (*
       x
       (-
        -1.0
        (*
         (pow x 2.0)
         (+
          0.8333333333333334
          (*
           (pow x 2.0)
           (+ 0.5083333333333333 (* (pow x 2.0) 0.2748015873015873)))))))
      (cos x)))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	return eps * (1.0 - ((eps * ((x * (-1.0 - (pow(x, 2.0) * (0.8333333333333334 + (pow(x, 2.0) * (0.5083333333333333 + (pow(x, 2.0) * 0.2748015873015873))))))) / cos(x))) - (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 - ((eps * ((x * ((-1.0d0) - ((x ** 2.0d0) * (0.8333333333333334d0 + ((x ** 2.0d0) * (0.5083333333333333d0 + ((x ** 2.0d0) * 0.2748015873015873d0))))))) / cos(x))) - ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 - ((eps * ((x * (-1.0 - (Math.pow(x, 2.0) * (0.8333333333333334 + (Math.pow(x, 2.0) * (0.5083333333333333 + (Math.pow(x, 2.0) * 0.2748015873015873))))))) / Math.cos(x))) - (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))));
}

Python

def code(x, eps):
	return eps * (1.0 - ((eps * ((x * (-1.0 - (math.pow(x, 2.0) * (0.8333333333333334 + (math.pow(x, 2.0) * (0.5083333333333333 + (math.pow(x, 2.0) * 0.2748015873015873))))))) / math.cos(x))) - (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 - Float64(Float64(eps * Float64(Float64(x * Float64(-1.0 - Float64((x ^ 2.0) * Float64(0.8333333333333334 + Float64((x ^ 2.0) * Float64(0.5083333333333333 + Float64((x ^ 2.0) * 0.2748015873015873))))))) / cos(x))) - Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 - ((eps * ((x * (-1.0 - ((x ^ 2.0) * (0.8333333333333334 + ((x ^ 2.0) * (0.5083333333333333 + ((x ^ 2.0) * 0.2748015873015873))))))) / cos(x))) - ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 - N[(N[(eps * N[(N[(x * N[(-1.0 - N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.8333333333333334 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.5083333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.2748015873015873), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot \left(0.5083333333333333 + 0.2748015873015873 \cdot {x}^{2}\right)\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot \left(0.5083333333333333 + \color{blue}{{x}^{2} \cdot 0.2748015873015873}\right)\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot \left(0.5083333333333333 + {x}^{2} \cdot 0.2748015873015873\right)\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

9. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(1 - \left(\varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot \left(0.5083333333333333 + {x}^{2} \cdot 0.2748015873015873\right)\right)\right)}{\cos x} - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
10. Add Preprocessing

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot 0.8333333333333334 - \varepsilon \cdot -0.5\\ \varepsilon \cdot \left(1 - \left(x \cdot \left({x}^{2} \cdot \left(\varepsilon \cdot -0.5 + \left({x}^{2} \cdot \left(\left(-0.5 \cdot t\_0 + \varepsilon \cdot 0.041666666666666664\right) + \left({x}^{2} \cdot \left(\left(-0.5 \cdot \left(\varepsilon \cdot 0.5083333333333333 + \left(-0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.8333333333333334\right) - \varepsilon \cdot 0.041666666666666664\right)\right) + \left(\varepsilon \cdot -0.001388888888888889 + t\_0 \cdot 0.041666666666666664\right)\right) - \varepsilon \cdot 0.2748015873015873\right) - \varepsilon \cdot 0.5083333333333333\right)\right) - \varepsilon \cdot 0.8333333333333334\right)\right) - \varepsilon\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps 0.8333333333333334) (* eps -0.5))))
   (*
    eps
    (-
     1.0
     (-
      (*
       x
       (-
        (*
         (pow x 2.0)
         (+
          (* eps -0.5)
          (-
           (*
            (pow x 2.0)
            (+
             (+ (* -0.5 t_0) (* eps 0.041666666666666664))
             (-
              (*
               (pow x 2.0)
               (-
                (+
                 (*
                  -0.5
                  (+
                   (* eps 0.5083333333333333)
                   (-
                    (* -0.5 (- (* eps -0.5) (* eps 0.8333333333333334)))
                    (* eps 0.041666666666666664))))
                 (+
                  (* eps -0.001388888888888889)
                  (* t_0 0.041666666666666664)))
                (* eps 0.2748015873015873)))
              (* eps 0.5083333333333333))))
           (* eps 0.8333333333333334))))
        eps))
      (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (eps * 0.8333333333333334) - (eps * -0.5);
	return eps * (1.0 - ((x * ((pow(x, 2.0) * ((eps * -0.5) + ((pow(x, 2.0) * (((-0.5 * t_0) + (eps * 0.041666666666666664)) + ((pow(x, 2.0) * (((-0.5 * ((eps * 0.5083333333333333) + ((-0.5 * ((eps * -0.5) - (eps * 0.8333333333333334))) - (eps * 0.041666666666666664)))) + ((eps * -0.001388888888888889) + (t_0 * 0.041666666666666664))) - (eps * 0.2748015873015873))) - (eps * 0.5083333333333333)))) - (eps * 0.8333333333333334)))) - eps)) - (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (eps * 0.8333333333333334d0) - (eps * (-0.5d0))
    code = eps * (1.0d0 - ((x * (((x ** 2.0d0) * ((eps * (-0.5d0)) + (((x ** 2.0d0) * ((((-0.5d0) * t_0) + (eps * 0.041666666666666664d0)) + (((x ** 2.0d0) * ((((-0.5d0) * ((eps * 0.5083333333333333d0) + (((-0.5d0) * ((eps * (-0.5d0)) - (eps * 0.8333333333333334d0))) - (eps * 0.041666666666666664d0)))) + ((eps * (-0.001388888888888889d0)) + (t_0 * 0.041666666666666664d0))) - (eps * 0.2748015873015873d0))) - (eps * 0.5083333333333333d0)))) - (eps * 0.8333333333333334d0)))) - eps)) - ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = (eps * 0.8333333333333334) - (eps * -0.5);
	return eps * (1.0 - ((x * ((Math.pow(x, 2.0) * ((eps * -0.5) + ((Math.pow(x, 2.0) * (((-0.5 * t_0) + (eps * 0.041666666666666664)) + ((Math.pow(x, 2.0) * (((-0.5 * ((eps * 0.5083333333333333) + ((-0.5 * ((eps * -0.5) - (eps * 0.8333333333333334))) - (eps * 0.041666666666666664)))) + ((eps * -0.001388888888888889) + (t_0 * 0.041666666666666664))) - (eps * 0.2748015873015873))) - (eps * 0.5083333333333333)))) - (eps * 0.8333333333333334)))) - eps)) - (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))));
}

Python

def code(x, eps):
	t_0 = (eps * 0.8333333333333334) - (eps * -0.5)
	return eps * (1.0 - ((x * ((math.pow(x, 2.0) * ((eps * -0.5) + ((math.pow(x, 2.0) * (((-0.5 * t_0) + (eps * 0.041666666666666664)) + ((math.pow(x, 2.0) * (((-0.5 * ((eps * 0.5083333333333333) + ((-0.5 * ((eps * -0.5) - (eps * 0.8333333333333334))) - (eps * 0.041666666666666664)))) + ((eps * -0.001388888888888889) + (t_0 * 0.041666666666666664))) - (eps * 0.2748015873015873))) - (eps * 0.5083333333333333)))) - (eps * 0.8333333333333334)))) - eps)) - (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))))

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * 0.8333333333333334) - Float64(eps * -0.5))
	return Float64(eps * Float64(1.0 - Float64(Float64(x * Float64(Float64((x ^ 2.0) * Float64(Float64(eps * -0.5) + Float64(Float64((x ^ 2.0) * Float64(Float64(Float64(-0.5 * t_0) + Float64(eps * 0.041666666666666664)) + Float64(Float64((x ^ 2.0) * Float64(Float64(Float64(-0.5 * Float64(Float64(eps * 0.5083333333333333) + Float64(Float64(-0.5 * Float64(Float64(eps * -0.5) - Float64(eps * 0.8333333333333334))) - Float64(eps * 0.041666666666666664)))) + Float64(Float64(eps * -0.001388888888888889) + Float64(t_0 * 0.041666666666666664))) - Float64(eps * 0.2748015873015873))) - Float64(eps * 0.5083333333333333)))) - Float64(eps * 0.8333333333333334)))) - eps)) - Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (eps * 0.8333333333333334) - (eps * -0.5);
	tmp = eps * (1.0 - ((x * (((x ^ 2.0) * ((eps * -0.5) + (((x ^ 2.0) * (((-0.5 * t_0) + (eps * 0.041666666666666664)) + (((x ^ 2.0) * (((-0.5 * ((eps * 0.5083333333333333) + ((-0.5 * ((eps * -0.5) - (eps * 0.8333333333333334))) - (eps * 0.041666666666666664)))) + ((eps * -0.001388888888888889) + (t_0 * 0.041666666666666664))) - (eps * 0.2748015873015873))) - (eps * 0.5083333333333333)))) - (eps * 0.8333333333333334)))) - eps)) - ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * 0.8333333333333334), $MachinePrecision] - N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(1.0 - N[(N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(eps * -0.5), $MachinePrecision] + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(-0.5 * t$95$0), $MachinePrecision] + N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(eps * 0.5083333333333333), $MachinePrecision] + N[(N[(-0.5 * N[(N[(eps * -0.5), $MachinePrecision] - N[(eps * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * -0.001388888888888889), $MachinePrecision] + N[(t$95$0 * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.2748015873015873), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.5083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{x \cdot \left(\varepsilon + {x}^{2} \cdot \left(\left(0.8333333333333334 \cdot \varepsilon + {x}^{2} \cdot \left(\left(0.5083333333333333 \cdot \varepsilon + {x}^{2} \cdot \left(0.2748015873015873 \cdot \varepsilon - \left(-0.5 \cdot \left(0.5083333333333333 \cdot \varepsilon - \left(-0.5 \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right) + 0.041666666666666664 \cdot \varepsilon\right)\right) + \left(-0.001388888888888889 \cdot \varepsilon + 0.041666666666666664 \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)\right) - \left(-0.5 \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right) + 0.041666666666666664 \cdot \varepsilon\right)\right)\right) - -0.5 \cdot \varepsilon\right)\right)} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

7. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(1 - \left(x \cdot \left({x}^{2} \cdot \left(\varepsilon \cdot -0.5 + \left({x}^{2} \cdot \left(\left(-0.5 \cdot \left(\varepsilon \cdot 0.8333333333333334 - \varepsilon \cdot -0.5\right) + \varepsilon \cdot 0.041666666666666664\right) + \left({x}^{2} \cdot \left(\left(-0.5 \cdot \left(\varepsilon \cdot 0.5083333333333333 + \left(-0.5 \cdot \left(\varepsilon \cdot -0.5 - \varepsilon \cdot 0.8333333333333334\right) - \varepsilon \cdot 0.041666666666666664\right)\right) + \left(\varepsilon \cdot -0.001388888888888889 + \left(\varepsilon \cdot 0.8333333333333334 - \varepsilon \cdot -0.5\right) \cdot 0.041666666666666664\right)\right) - \varepsilon \cdot 0.2748015873015873\right) - \varepsilon \cdot 0.5083333333333333\right)\right) - \varepsilon \cdot 0.8333333333333334\right)\right) - \varepsilon\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
8. Add Preprocessing

Alternative 6: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot 0.5083333333333333\right)\right)}{1 + {x}^{2} \cdot -0.5}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (-
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
    (*
     eps
     (/
      (*
       x
       (-
        -1.0
        (*
         (pow x 2.0)
         (+ 0.8333333333333334 (* (pow x 2.0) 0.5083333333333333)))))
      (+ 1.0 (* (pow x 2.0) -0.5))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) - (eps * ((x * (-1.0 - (pow(x, 2.0) * (0.8333333333333334 + (pow(x, 2.0) * 0.5083333333333333))))) / (1.0 + (pow(x, 2.0) * -0.5))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) - (eps * ((x * ((-1.0d0) - ((x ** 2.0d0) * (0.8333333333333334d0 + ((x ** 2.0d0) * 0.5083333333333333d0))))) / (1.0d0 + ((x ** 2.0d0) * (-0.5d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (Math.pow(x, 2.0) * (0.8333333333333334 + (Math.pow(x, 2.0) * 0.5083333333333333))))) / (1.0 + (Math.pow(x, 2.0) * -0.5))))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (math.pow(x, 2.0) * (0.8333333333333334 + (math.pow(x, 2.0) * 0.5083333333333333))))) / (1.0 + (math.pow(x, 2.0) * -0.5))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - Float64(eps * Float64(Float64(x * Float64(-1.0 - Float64((x ^ 2.0) * Float64(0.8333333333333334 + Float64((x ^ 2.0) * 0.5083333333333333))))) / Float64(1.0 + Float64((x ^ 2.0) * -0.5)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - (eps * ((x * (-1.0 - ((x ^ 2.0) * (0.8333333333333334 + ((x ^ 2.0) * 0.5083333333333333))))) / (1.0 + ((x ^ 2.0) * -0.5))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(x * N[(-1.0 - N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.8333333333333334 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.5083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + 0.5083333333333333 \cdot {x}^{2}\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{{x}^{2} \cdot 0.5083333333333333}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot 0.5083333333333333\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

9. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot 0.5083333333333333\right)\right)}{\color{blue}{1 + -0.5 \cdot {x}^{2}}} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot 0.5083333333333333\right)\right)}{1 + {x}^{2} \cdot -0.5}\right)\right) \]
11. Add Preprocessing

Alternative 7: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + 0.5083333333333333 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (-
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
    (*
     eps
     (/
      (*
       x
       (-
        -1.0
        (* (pow x 2.0) (+ 0.8333333333333334 (* 0.5083333333333333 (* x x))))))
      (cos x)))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) - (eps * ((x * (-1.0 - (pow(x, 2.0) * (0.8333333333333334 + (0.5083333333333333 * (x * x)))))) / cos(x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) - (eps * ((x * ((-1.0d0) - ((x ** 2.0d0) * (0.8333333333333334d0 + (0.5083333333333333d0 * (x * x)))))) / cos(x)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (Math.pow(x, 2.0) * (0.8333333333333334 + (0.5083333333333333 * (x * x)))))) / Math.cos(x)))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (math.pow(x, 2.0) * (0.8333333333333334 + (0.5083333333333333 * (x * x)))))) / math.cos(x)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - Float64(eps * Float64(Float64(x * Float64(-1.0 - Float64((x ^ 2.0) * Float64(0.8333333333333334 + Float64(0.5083333333333333 * Float64(x * x)))))) / cos(x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - (eps * ((x * (-1.0 - ((x ^ 2.0) * (0.8333333333333334 + (0.5083333333333333 * (x * x)))))) / cos(x)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(x * N[(-1.0 - N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.8333333333333334 + N[(0.5083333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + 0.5083333333333333 \cdot {x}^{2}\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{{x}^{2} \cdot 0.5083333333333333}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + {x}^{2} \cdot 0.5083333333333333\right)\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5083333333333333\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5083333333333333\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

11. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - {x}^{2} \cdot \left(0.8333333333333334 + 0.5083333333333333 \cdot \left(x \cdot x\right)\right)\right)}{\cos x}\right)\right) \]
12. Add Preprocessing

Alternative 8: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot 0.8333333333333334\right)}{1 + {x}^{2} \cdot -0.5} + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (+
    (*
     eps
     (/
      (* x (+ 1.0 (* (pow x 2.0) 0.8333333333333334)))
      (+ 1.0 (* (pow x 2.0) -0.5))))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((eps * ((x * (1.0 + (pow(x, 2.0) * 0.8333333333333334))) / (1.0 + (pow(x, 2.0) * -0.5)))) + (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((eps * ((x * (1.0d0 + ((x ** 2.0d0) * 0.8333333333333334d0))) / (1.0d0 + ((x ** 2.0d0) * (-0.5d0))))) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((eps * ((x * (1.0 + (Math.pow(x, 2.0) * 0.8333333333333334))) / (1.0 + (Math.pow(x, 2.0) * -0.5)))) + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((eps * ((x * (1.0 + (math.pow(x, 2.0) * 0.8333333333333334))) / (1.0 + (math.pow(x, 2.0) * -0.5)))) + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(eps * Float64(Float64(x * Float64(1.0 + Float64((x ^ 2.0) * 0.8333333333333334))) / Float64(1.0 + Float64((x ^ 2.0) * -0.5)))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((eps * ((x * (1.0 + ((x ^ 2.0) * 0.8333333333333334))) / (1.0 + ((x ^ 2.0) * -0.5)))) + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(eps * N[(N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + 0.8333333333333334 \cdot {x}^{2}\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.8333333333333334}\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot 0.8333333333333334\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

9. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot 0.8333333333333334\right)}{\color{blue}{1 + -0.5 \cdot {x}^{2}}} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot 0.8333333333333334\right)}{1 + {x}^{2} \cdot -0.5} + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
11. Add Preprocessing

Alternative 9: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - 0.8333333333333334 \cdot \left(x \cdot x\right)\right)}{\cos x}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (-
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
    (* eps (/ (* x (- -1.0 (* 0.8333333333333334 (* x x)))) (cos x)))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) - (eps * ((x * (-1.0 - (0.8333333333333334 * (x * x)))) / cos(x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) - (eps * ((x * ((-1.0d0) - (0.8333333333333334d0 * (x * x)))) / cos(x)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (0.8333333333333334 * (x * x)))) / Math.cos(x)))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) - (eps * ((x * (-1.0 - (0.8333333333333334 * (x * x)))) / math.cos(x)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - Float64(eps * Float64(Float64(x * Float64(-1.0 - Float64(0.8333333333333334 * Float64(x * x)))) / cos(x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) - (eps * ((x * (-1.0 - (0.8333333333333334 * (x * x)))) / cos(x)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(eps * N[(N[(x * N[(-1.0 - N[(0.8333333333333334 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + 0.8333333333333334 \cdot {x}^{2}\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.8333333333333334}\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot 0.8333333333333334\right)}}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5083333333333333\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.8333333333333334\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

11. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \varepsilon \cdot \frac{x \cdot \left(-1 - 0.8333333333333334 \cdot \left(x \cdot x\right)\right)}{\cos x}\right)\right) \]
12. Add Preprocessing

Alternative 10: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + \left(x \cdot x\right) \cdot \left(\varepsilon \cdot 1.3333333333333333\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (+
    (* x (+ eps (* (* x x) (* eps 1.3333333333333333))))
    (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((x * (eps + ((x * x) * (eps * 1.3333333333333333)))) + (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((x * (eps + ((x * x) * (eps * 1.3333333333333333d0)))) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((x * (eps + ((x * x) * (eps * 1.3333333333333333)))) + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0))));
}

Python

def code(x, eps):
	return eps * (1.0 + ((x * (eps + ((x * x) * (eps * 1.3333333333333333)))) + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(x * Float64(eps + Float64(Float64(x * x) * Float64(eps * 1.3333333333333333)))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((x * (eps + ((x * x) * (eps * 1.3333333333333333)))) + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(x * N[(eps + N[(N[(x * x), $MachinePrecision] * N[(eps * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{x \cdot \left(\varepsilon + {x}^{2} \cdot \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
7. Step-by-step derivation

   1. distribute-rgt-out-- 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + {x}^{2} \cdot \color{blue}{\left(\varepsilon \cdot \left(0.8333333333333334 - -0.5\right)\right)}\right) - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

   2. metadata-eval 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + {x}^{2} \cdot \left(\varepsilon \cdot \color{blue}{1.3333333333333333}\right)\right) - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

   3. *-commutative 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + {x}^{2} \cdot \color{blue}{\left(1.3333333333333333 \cdot \varepsilon\right)}\right) - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{x \cdot \left(\varepsilon + {x}^{2} \cdot \left(1.3333333333333333 \cdot \varepsilon\right)\right)} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{x \cdot \left(1 + {x}^{2} \cdot \left(0.8333333333333334 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5083333333333333\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + \color{blue}{\left(x \cdot x\right)} \cdot \left(1.3333333333333333 \cdot \varepsilon\right)\right) - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) \]

11. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(x \cdot \left(\varepsilon + \left(x \cdot x\right) \cdot \left(\varepsilon \cdot 1.3333333333333333\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]
12. Add Preprocessing

Alternative 11: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg 99.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. mul-1-neg 99.4%

      \[\leadsto \varepsilon \cdot \left(1 + \left(-\color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   3. remove-double-neg 99.4%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}}}\right) \]

5. Simplified 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. Add Preprocessing

Alternative 12: 98.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\varepsilon \cdot 1.3333333333333333 + x \cdot 0.6666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   1.0
   (*
    x
    (+
     eps
     (*
      x
      (+
       1.0
       (* x (+ (* eps 1.3333333333333333) (* x 0.6666666666666666))))))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((eps * 1.3333333333333333) + (x * 0.6666666666666666))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (eps + (x * (1.0d0 + (x * ((eps * 1.3333333333333333d0) + (x * 0.6666666666666666d0))))))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((eps * 1.3333333333333333) + (x * 0.6666666666666666))))))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((eps * 1.3333333333333333) + (x * 0.6666666666666666))))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(eps + Float64(x * Float64(1.0 + Float64(x * Float64(Float64(eps * 1.3333333333333333) + Float64(x * 0.6666666666666666)))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (eps + (x * (1.0 + (x * ((eps * 1.3333333333333333) + (x * 0.6666666666666666))))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(eps + N[(x * N[(1.0 + N[(x * N[(N[(eps * 1.3333333333333333), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(0.6666666666666666 \cdot x + 0.8333333333333334 \cdot \varepsilon\right) - -0.5 \cdot \varepsilon\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. associate--l+ 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \color{blue}{\left(0.6666666666666666 \cdot x + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)}\right)\right)\right) \]

   2. *-commutative 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\color{blue}{x \cdot 0.6666666666666666} + \left(0.8333333333333334 \cdot \varepsilon - -0.5 \cdot \varepsilon\right)\right)\right)\right)\right) \]

   3. distribute-rgt-out-- 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \color{blue}{\varepsilon \cdot \left(0.8333333333333334 - -0.5\right)}\right)\right)\right)\right) \]

   4. metadata-eval 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \varepsilon \cdot \color{blue}{1.3333333333333333}\right)\right)\right)\right) \]

   5. *-commutative 99.3%

      \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + \color{blue}{1.3333333333333333 \cdot \varepsilon}\right)\right)\right)\right) \]

8. Simplified 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(x \cdot 0.6666666666666666 + 1.3333333333333333 \cdot \varepsilon\right)\right)\right)}\right) \]

9. Final simplification 99.3%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\varepsilon \cdot 1.3333333333333333 + x \cdot 0.6666666666666666\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 13: 98.3% accurate, 22.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* (* eps x) (+ eps x))))

C

double code(double x, double eps) {
	return eps + ((eps * x) * (eps + x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((eps * x) * (eps + x))
end function

Java

public static double code(double x, double eps) {
	return eps + ((eps * x) * (eps + x));
}

Python

def code(x, eps):
	return eps + ((eps * x) * (eps + x))

Julia

function code(x, eps)
	return Float64(eps + Float64(Float64(eps * x) * Float64(eps + x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((eps * x) * (eps + x));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[(eps * x), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.1%

   \[\leadsto \color{blue}{\varepsilon + x \cdot \left(\varepsilon \cdot x + {\varepsilon}^{2}\right)} \]

7. Taylor expanded in eps around 0 99.1%

   \[\leadsto \varepsilon + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot x + {x}^{2}\right)} \]
8. Step-by-step derivation

   1. distribute-lft-in 99.1%

      \[\leadsto \varepsilon + \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot x\right) + \varepsilon \cdot {x}^{2}\right)} \]

   2. *-commutative 99.1%

      \[\leadsto \varepsilon + \left(\color{blue}{\left(\varepsilon \cdot x\right) \cdot \varepsilon} + \varepsilon \cdot {x}^{2}\right) \]

   3. unpow2 99.1%

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

   4. associate-*l* 99.1%

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

   5. distribute-lft-out 99.1%

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

9. Simplified 99.1%

   \[\leadsto \varepsilon + \color{blue}{\left(\varepsilon \cdot x\right) \cdot \left(\varepsilon + x\right)} \]
10. Add Preprocessing

Alternative 14: 98.3% accurate, 22.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (* x (+ eps x)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * (eps + x)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (eps + x)));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + x)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(eps + x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (eps + x)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 15: 97.9% accurate, 29.3× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (* eps x))))

C

double code(double x, double eps) {
	return eps + (eps * (eps * x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (eps * x))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * (eps * x));
}

Python

def code(x, eps):
	return eps + (eps * (eps * x))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * Float64(eps * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (eps * x));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

8. Simplified 98.7%

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

   1. distribute-rgt-in 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. *-commutative 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

    \[\leadsto \varepsilon + \varepsilon \cdot \left(\varepsilon \cdot x\right) \]
12. Add Preprocessing

Alternative 16: 97.9% accurate, 29.3× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (* eps x))))

C

double code(double x, double eps) {
	return eps * (1.0 + (eps * x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (eps * x))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (eps * x));
}

Python

def code(x, eps):
	return eps * (1.0 + (eps * x))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(eps * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (eps * x));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 17: 97.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{\varepsilon \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. associate-/l* 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\color{blue}{\varepsilon \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   3. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)}{\cos x} - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

   4. mul-1-neg 99.8%

      \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \color{blue}{\left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \frac{\sin x \cdot \left(1 - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x} - \left(-\frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)} \]

6. Taylor expanded in x around 0 98.7%

   \[\leadsto \color{blue}{\varepsilon} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (cos(x) * cos((x + eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (cos(x) * cos((x + eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (Math.cos(x) * Math.cos((x + eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (math.cos(x) * math.cos((x + eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(cos(x) * cos(Float64(x + eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (cos(x) * cos((x + eps)));
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024114 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))