2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.6%

Time: 32.0s

Alternatives: 12

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: 62.0% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := 1 + t\_2\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{t\_1 \cdot t\_3}{t\_0} + \left(-0.5 \cdot \left(-1 - t\_2\right) - 0.16666666666666666 \cdot t\_2\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot t\_3}{\cos x}\right)\right) + \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t\_1\right)\right)}{t\_0}\right) \end{array} \end{array} \]

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))
        (t_3 (+ 1.0 t_2)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (*
         eps
         (-
          (+
           (/ (* t_1 t_3) t_0)
           (- (* -0.5 (- -1.0 t_2)) (* 0.16666666666666666 t_2)))
          0.16666666666666666))
        (/ (* (sin x) t_3) (cos x)))))
     (/ (expm1 (log1p t_1)) t_0)))))

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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((sin(x) * t_3) / cos(x))))) + (expm1(log1p(t_1)) / t_0));
}

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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((Math.sin(x) * t_3) / Math.cos(x))))) + (Math.expm1(Math.log1p(t_1)) / t_0));
}

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
	t_3 = 1.0 + t_2
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((math.sin(x) * t_3) / math.cos(x))))) + (math.expm1(math.log1p(t_1)) / t_0))

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	t_3 = Float64(1.0 + t_2)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(Float64(Float64(t_1 * t_3) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(0.16666666666666666 * t_2))) - 0.16666666666666666)) + Float64(Float64(sin(x) * t_3) / cos(x))))) + Float64(expm1(log1p(t_1)) / t_0)))
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]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin x}^{2}\right)\right)}}{{\cos x}^{2}}\right) \]

   2. expm1-undefine 99.4%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\sin x}^{2}\right)} - 1}}{{\cos x}^{2}}\right) \]

6. Applied egg-rr 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\sin x}^{2}\right)} - 1}}{{\cos x}^{2}}\right) \]
7. Step-by-step derivation

   1. expm1-define 99.5%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin x}^{2}\right)\right)}}{{\cos x}^{2}}\right) \]

8. Simplified 99.5%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin x}^{2}\right)\right)}}{{\cos x}^{2}}\right) \]

9. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin x}^{2}\right)\right)}{{\cos x}^{2}}\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := 1 + t\_2\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{t\_1 \cdot t\_3}{t\_0} + \left(-0.5 \cdot \left(-1 - t\_2\right) - 0.16666666666666666 \cdot t\_2\right)\right) - 0.16666666666666666\right) + \frac{\sin x \cdot t\_3}{\cos x}\right)\right) + t\_2\right) \end{array} \end{array} \]

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))
        (t_3 (+ 1.0 t_2)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (*
         eps
         (-
          (+
           (/ (* t_1 t_3) t_0)
           (- (* -0.5 (- -1.0 t_2)) (* 0.16666666666666666 t_2)))
          0.16666666666666666))
        (/ (* (sin x) t_3) (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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((sin(x) * t_3) / 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
    real(8) :: t_3
    t_0 = cos(x) ** 2.0d0
    t_1 = sin(x) ** 2.0d0
    t_2 = t_1 / t_0
    t_3 = 1.0d0 + t_2
    code = eps * ((1.0d0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + (((-0.5d0) * ((-1.0d0) - t_2)) - (0.16666666666666666d0 * t_2))) - 0.16666666666666666d0)) + ((sin(x) * t_3) / 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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((Math.sin(x) * t_3) / 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
	t_3 = 1.0 + t_2
	return eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((math.sin(x) * t_3) / 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)
	t_3 = Float64(1.0 + t_2)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(Float64(Float64(t_1 * t_3) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(0.16666666666666666 * t_2))) - 0.16666666666666666)) + Float64(Float64(sin(x) * t_3) / 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;
	t_3 = 1.0 + t_2;
	tmp = eps * ((1.0 + (eps * ((eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_2))) - 0.16666666666666666)) + ((sin(x) * t_3) / 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]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(eps * N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ t_3 := 1 + t\_2\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\frac{\sin x \cdot t\_3}{\cos x} + \varepsilon \cdot \left(\left(\frac{t\_1 \cdot t\_3}{t\_0} + \left(-0.5 \cdot \left(-1 - t\_2\right) - 0.16666666666666666 \cdot t\_1\right)\right) - 0.16666666666666666\right)\right)\right) + t\_2\right) \end{array} \end{array} \]

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))
        (t_3 (+ 1.0 t_2)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (/ (* (sin x) t_3) (cos x))
        (*
         eps
         (-
          (+
           (/ (* t_1 t_3) t_0)
           (- (* -0.5 (- -1.0 t_2)) (* 0.16666666666666666 t_1)))
          0.16666666666666666)))))
     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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + 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
    real(8) :: t_3
    t_0 = cos(x) ** 2.0d0
    t_1 = sin(x) ** 2.0d0
    t_2 = t_1 / t_0
    t_3 = 1.0d0 + t_2
    code = eps * ((1.0d0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + (((-0.5d0) * ((-1.0d0) - t_2)) - (0.16666666666666666d0 * t_1))) - 0.16666666666666666d0))))) + 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;
	double t_3 = 1.0 + t_2;
	return eps * ((1.0 + (eps * (((Math.sin(x) * t_3) / Math.cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + 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
	t_3 = 1.0 + t_2
	return eps * ((1.0 + (eps * (((math.sin(x) * t_3) / math.cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + 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)
	t_3 = Float64(1.0 + t_2)
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(Float64(sin(x) * t_3) / cos(x)) + Float64(eps * Float64(Float64(Float64(Float64(t_1 * t_3) / t_0) + Float64(Float64(-0.5 * Float64(-1.0 - t_2)) - Float64(0.16666666666666666 * t_1))) - 0.16666666666666666))))) + 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;
	t_3 = 1.0 + t_2;
	tmp = eps * ((1.0 + (eps * (((sin(x) * t_3) / cos(x)) + (eps * ((((t_1 * t_3) / t_0) + ((-0.5 * (-1.0 - t_2)) - (0.16666666666666666 * t_1))) - 0.16666666666666666))))) + 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]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$3), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.16666666666666666 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * 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] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

5. Taylor expanded in x around 0 99.3%

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

   1. remove-double-neg 99.3%

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

   2. mul-1-neg 99.3%

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

   3. expm1-log1p-u 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right) \]

   4. log1p-define 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(--1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}\right)\right) \]

   5. sub-neg 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \mathsf{expm1}\left(\log \color{blue}{\left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\right)\right) \]

   6. expm1-undefine 99.3%

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

7. Applied egg-rr 99.3%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \color{blue}{\left(\left({\tan x}^{2} + 1\right) - 1\right)}\right) \]
8. Step-by-step derivation

   1. associate--l+ 99.3%

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

   2. metadata-eval 99.3%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \left({\tan x}^{2} + \color{blue}{0}\right)\right) \]

   3. +-rgt-identity 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, eps):
	return eps * ((1.0 + (eps * (((math.sin(x) * (1.0 + math.pow(math.tan(x), 2.0))) / math.cos(x)) - (eps * -0.3333333333333333)))) + (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(eps * Float64(Float64(Float64(sin(x) * Float64(1.0 + (tan(x) ^ 2.0))) / cos(x)) - Float64(eps * -0.3333333333333333)))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(eps * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(eps * -0.3333333333333333), $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]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

5. Taylor expanded in x around 0 99.3%

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

   1. cancel-sign-sub-inv 99.3%

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

   2. metadata-eval 99.3%

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

   3. *-un-lft-identity 99.3%

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

   4. distribute-rgt-in 99.3%

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

   5. *-un-lft-identity 99.3%

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

   6. unpow2 99.3%

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

   7. unpow2 99.3%

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

   8. frac-times 99.3%

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

   9. tan-quot 99.3%

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

   10. tan-quot 99.3%

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

   11. pow2 99.3%

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

7. Applied egg-rr 99.3%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot -0.3333333333333333\right) + \frac{\color{blue}{\sin x + {\tan x}^{2} \cdot \sin x}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Step-by-step derivation

   1. distribute-rgt1-in 99.3%

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

   2. *-commutative 99.3%

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

   3. +-commutative 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, eps):
	return eps * ((1.0 + ((eps * (math.tan(x) * (1.0 + math.pow(math.tan(x), 2.0)))) + (eps * (eps * -0.3333333333333333)))) + (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(eps * Float64(tan(x) * Float64(1.0 + (tan(x) ^ 2.0)))) + Float64(eps * Float64(eps * -0.3333333333333333)))) + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in eps around 0 99.5%

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

5. Taylor expanded in x around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-rgt-in 99.3%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 7: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + x \cdot \left(1.3333333333333333 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2} \cdot 1.3333333333333333\right)\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    1.0
    (+
     (* 0.3333333333333333 (pow eps 2.0))
     (*
      x
      (+
       eps
       (*
        x
        (+
         (* 1.3333333333333333 (* eps x))
         (* (pow eps 2.0) 1.3333333333333333)))))))
   (/ (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)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (pow(eps, 2.0) * 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 + ((0.3333333333333333d0 * (eps ** 2.0d0)) + (x * (eps + (x * ((1.3333333333333333d0 * (eps * x)) + ((eps ** 2.0d0) * 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 + ((0.3333333333333333 * Math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (Math.pow(eps, 2.0) * 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 + ((0.3333333333333333 * math.pow(eps, 2.0)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + (math.pow(eps, 2.0) * 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(Float64(1.0 + Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(x * Float64(eps + Float64(x * Float64(Float64(1.3333333333333333 * Float64(eps * x)) + Float64((eps ^ 2.0) * 1.3333333333333333))))))) + 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)) + (x * (eps + (x * ((1.3333333333333333 * (eps * x)) + ((eps ^ 2.0) * 1.3333333333333333))))))) + ((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[(x * N[(eps + N[(x * N[(N[(1.3333333333333333 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 98.9%

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

5. Final simplification 98.9%

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

Alternative 8: 99.0% 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 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 98.8%

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

5. Final simplification 98.8%

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

Alternative 9: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[(1.0 + N[(eps * x), $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 60.2%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 98.8%

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

5. Taylor expanded in eps around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 10: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

   1. +-commutative 98.6%

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

   2. distribute-rgt-in 98.6%

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

   3. unpow2 98.6%

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

   4. unpow2 98.6%

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

   5. frac-times 98.6%

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

   6. tan-quot 98.6%

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

   7. tan-quot 98.6%

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

   8. pow2 98.6%

      \[\leadsto \color{blue}{{\tan x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon \]

   9. *-un-lft-identity 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 11: 98.2% 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 60.2%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 97.4%

   \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {x}^{2}} \]
7. Add Preprocessing

Alternative 12: 97.8% 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 60.2%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. sub-neg 98.6%

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

   2. mul-1-neg 98.6%

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

   3. remove-double-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 97.0%

   \[\leadsto \color{blue}{\varepsilon} \]
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 2024086 
(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)))