2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.4%

Time: 33.8s

Alternatives: 13

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: 62.1% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{t\_1 \cdot \left(1 + t\_1\right)}{t\_0} + \left(-0.5 \cdot \left(-1 - t\_1\right) - 0.16666666666666666 \cdot t\_1\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot \left(1 + t\_2\right)}{\cos x}\right)\right) + t\_2\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (pow (sin x) 2.0)) (t_2 (/ t_1 t_0)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (*
         eps
         (-
          (+
           (/ (* t_1 (+ 1.0 t_1)) t_0)
           (- (* -0.5 (- -1.0 t_1)) (* 0.16666666666666666 t_1)))
          0.16666666666666666))
        (/ (* (sin x) (+ 1.0 t_2)) (cos x)))))
     t_2))))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = t_1 / t_0;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * (1.0 + t_1)) / t_0) + ((-0.5 * (-1.0 - t_1)) - (0.16666666666666666 * t_1))) - 0.16666666666666666)) + ((sin(x) * (1.0 + t_2)) / cos(x))))) + t_2);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = cos(x) ** 2.0d0
    t_1 = sin(x) ** 2.0d0
    t_2 = t_1 / t_0
    code = eps * ((1.0d0 + (eps * ((eps * ((((t_1 * (1.0d0 + t_1)) / t_0) + (((-0.5d0) * ((-1.0d0) - t_1)) - (0.16666666666666666d0 * t_1))) - 0.16666666666666666d0)) + ((sin(x) * (1.0d0 + t_2)) / cos(x))))) + t_2)
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.cos(x), 2.0);
	double t_1 = Math.pow(Math.sin(x), 2.0);
	double t_2 = t_1 / t_0;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * (1.0 + t_1)) / t_0) + ((-0.5 * (-1.0 - t_1)) - (0.16666666666666666 * t_1))) - 0.16666666666666666)) + ((Math.sin(x) * (1.0 + t_2)) / Math.cos(x))))) + t_2);
}

Python

def code(x, eps):
	t_0 = math.pow(math.cos(x), 2.0)
	t_1 = math.pow(math.sin(x), 2.0)
	t_2 = t_1 / t_0
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * (1.0 + t_1)) / t_0) + ((-0.5 * (-1.0 - t_1)) - (0.16666666666666666 * t_1))) - 0.16666666666666666)) + ((math.sin(x) * (1.0 + t_2)) / math.cos(x))))) + t_2)

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(Float64(Float64(t_1 * Float64(1.0 + t_1)) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_1)) - Float64(0.16666666666666666 * t_1))) - 0.16666666666666666)) + Float64(Float64(sin(x) * Float64(1.0 + t_2)) / cos(x))))) + t_2))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) ^ 2.0;
	t_1 = sin(x) ^ 2.0;
	t_2 = t_1 / t_0;
	tmp = eps * ((1.0 + (eps * ((eps * ((((t_1 * (1.0 + t_1)) / t_0) + ((-0.5 * (-1.0 - t_1)) - (0.16666666666666666 * t_1))) - 0.16666666666666666)) + ((sin(x) * (1.0 + t_2)) / cos(x))))) + t_2);
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(N[(t$95$1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(1 + {\sin x}^{2}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(-1 - {\sin x}^{2}\right) - 0.16666666666666666 \cdot {\sin x}^{2}\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Add Preprocessing

Alternative 2: 99.3% accurate, 0.2× speedup

Math

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

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 -0.3333333333333333))))
     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 * -0.3333333333333333)))) + 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 * (-0.3333333333333333d0))))) + 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 * -0.3333333333333333)))) + 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 * -0.3333333333333333)))) + 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 * -0.3333333333333333)))) + 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 * -0.3333333333333333)))) + 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 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.2% accurate, 0.2× speedup

Math

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

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))) 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))) - 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))) - 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))) - 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))) - t_0))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 - Float64(Float64(eps * Float64(Float64(sin(x) * Float64(-1.0 - t_0)) / cos(x))) - 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))) - 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[(1.0 - N[(N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Final simplification 99.8%

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

Alternative 4: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+ 1.0 (+ (* 0.3333333333333333 (pow eps 2.0)) (* eps x)))
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * ((1.0 + ((0.3333333333333333 * pow(eps, 2.0)) + (eps * 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 + ((0.3333333333333333d0 * (eps ** 2.0d0)) + (eps * x))) + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * ((1.0 + ((0.3333333333333333 * Math.pow(eps, 2.0)) + (eps * x))) + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
}

Python

def code(x, eps):
	return eps * ((1.0 + ((0.3333333333333333 * math.pow(eps, 2.0)) + (eps * x))) + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(1.0 + Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(eps * x))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((1.0 + ((0.3333333333333333 * (eps ^ 2.0)) + (eps * x))) + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * 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]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in eps around 0 99.9%

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

4. Taylor expanded in x around 0 99.5%

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

5. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
6. Add Preprocessing

Alternative 5: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 - (((cos((x * 2.0)) / 2.0) - 0.5) / 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 - (((cos((x * 2.0d0)) / 2.0d0) - 0.5d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Step-by-step derivation

   1. unpow2 99.4%

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

   2. sin-mult 99.4%

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

7. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. +-inverses 99.4%

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

   3. cos-0 99.4%

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

   4. metadata-eval 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* eps (* (- 0.5 (* 0.5 (cos (* x 2.0)))) (pow (cos x) -2.0)))))

C

double code(double x, double eps) {
	return eps + (eps * ((0.5 - (0.5 * cos((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 + (eps * ((0.5d0 - (0.5d0 * cos((x * 2.0d0)))) * (cos(x) ** (-2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Step-by-step derivation

   1. unpow2 99.4%

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

   2. sin-mult 99.4%

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

7. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. +-inverses 99.4%

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

   3. cos-0 99.4%

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

   4. metadata-eval 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

9. Simplified 99.4%

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

    1. distribute-rgt-in 99.4%

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

    2. *-un-lft-identity 99.4%

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

    3. div-inv 99.4%

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

    4. div-inv 99.4%

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

    5. metadata-eval 99.4%

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

    6. pow-flip 99.4%

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

    7. metadata-eval 99.4%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \varepsilon + \varepsilon \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(x \cdot 2\right)\right) \cdot {\cos x}^{-2}\right) \]
13. Add Preprocessing

Alternative 7: 98.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (* eps (+ 1.0 (/ (- 0.5 (/ t_0 2.0)) (/ (+ 1.0 t_0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = cos((x * 2.0));
	return eps * (1.0 + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 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 = cos((x * 2.0d0))
    code = eps * (1.0d0 + ((0.5d0 - (t_0 / 2.0d0)) / ((1.0d0 + t_0) / 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((x * 2.0));
	return eps * (1.0 + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)));
}

Python

def code(x, eps):
	t_0 = math.cos((x * 2.0))
	return eps * (1.0 + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)))

Julia

function code(x, eps)
	t_0 = cos(Float64(x * 2.0))
	return Float64(eps * Float64(1.0 + Float64(Float64(0.5 - Float64(t_0 / 2.0)) / Float64(Float64(1.0 + t_0) / 2.0))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos((x * 2.0));
	tmp = eps * (1.0 + ((0.5 - (t_0 / 2.0)) / ((1.0 + t_0) / 2.0)));
end

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Step-by-step derivation

   1. unpow2 99.4%

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

   2. sin-mult 99.4%

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

7. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. +-inverses 99.4%

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

   3. cos-0 99.4%

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

   4. metadata-eval 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

9. Simplified 99.4%

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

    1. unpow2 99.4%

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

    2. cos-mult 99.4%

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

11. Applied egg-rr 99.4%

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

    1. +-commutative 99.4%

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

    2. +-inverses 99.4%

       \[\leadsto \varepsilon \cdot \left(1 + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

    3. cos-0 99.4%

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

    4. count-2 99.4%

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

    5. *-commutative 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 8: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Step-by-step derivation

   1. unpow2 99.4%

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

   2. sin-mult 99.4%

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

7. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. +-inverses 99.4%

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

   3. cos-0 99.4%

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

   4. metadata-eval 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around 0 98.9%

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

    1. mul-1-neg 98.9%

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

    2. unsub-neg 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 9: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* (pow x 2.0) (+ 1.0 (* (pow x 2.0) 0.6666666666666666))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(x, 2.0) * (1.0 + (pow(x, 2.0) * 0.6666666666666666))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * 0.6666666666666666d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * 0.6666666666666666))));
}

Python

def code(x, eps):
	return eps * (1.0 + (math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * 0.6666666666666666))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * 0.6666666666666666)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((x ^ 2.0) * (1.0 + ((x ^ 2.0) * 0.6666666666666666))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Taylor expanded in x around 0 98.8%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(1 + 0.6666666666666666 \cdot {x}^{2}\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto \varepsilon \cdot \left(1 + {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.6666666666666666}\right)\right) \]

8. Simplified 98.8%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)}\right) \]

9. Final simplification 98.8%

   \[\leadsto \varepsilon \cdot \left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.6666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 10: 98.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Step-by-step derivation

   1. unpow2 99.4%

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

   2. sin-mult 99.4%

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

7. Applied egg-rr 99.4%

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

   1. div-sub 99.4%

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

   2. +-inverses 99.4%

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

   3. cos-0 99.4%

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

   4. metadata-eval 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around 0 98.7%

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

11. Final simplification 98.7%

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

Alternative 11: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow x 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(x, 2.0));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps + (eps * math.pow(x, 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (x ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 61.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Taylor expanded in x around 0 98.7%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 98.7%

      \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \varepsilon} \]

8. Simplified 98.7%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \varepsilon} \]

9. Final simplification 98.7%

   \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
10. Add Preprocessing

Alternative 12: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 98.4%

      \[\leadsto \color{blue}{1 \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. quot-tan 98.4%

      \[\leadsto 1 \cdot \color{blue}{\tan \varepsilon} \]

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 98.4%

      \[\leadsto \color{blue}{\tan \varepsilon} \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{\tan \varepsilon} \]

8. Final simplification 98.4%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 13: 97.6% 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 61.7%

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

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

4. Taylor expanded in eps around 0 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \varepsilon \]
6. 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 2024058 
(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)))