2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.6%

Time: 17.6s

Alternatives: 8

Speedup: 1.0×

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: 62.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(t\_0 - \left(-1 - \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 + \left(t\_0 + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - t\_0 \cdot -0.3333333333333333\right)\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (-
     t_0
     (-
      -1.0
      (*
       eps
       (+
        (*
         eps
         (+
          0.3333333333333333
          (+
           t_0
           (-
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
            (* t_0 -0.3333333333333333)))))
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (t_0 - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_0 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (t_0 * -0.3333333333333333))))) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))));
}

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 * (t_0 - ((-1.0d0) - (eps * ((eps * (0.3333333333333333d0 + (t_0 + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (t_0 * (-0.3333333333333333d0)))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))))
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 * (t_0 - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_0 + ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (t_0 * -0.3333333333333333))))) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (t_0 - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_0 + ((math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (t_0 * -0.3333333333333333))))) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)))))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(t_0 - Float64(-1.0 - Float64(eps * Float64(Float64(eps * Float64(0.3333333333333333 + Float64(t_0 + Float64(Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - Float64(t_0 * -0.3333333333333333))))) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (t_0 - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_0 + (((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - (t_0 * -0.3333333333333333))))) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))));
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[(t$95$0 - N[(-1.0 - N[(eps * N[(N[(eps * N[(0.3333333333333333 + N[(t$95$0 + N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.3%

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

   1. tan-sum 64.5%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 64.5%

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

   3. fma-neg 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

4. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Final simplification 99.7%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(-1 - \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333\right)\right)\right) + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\tan x}^{2}\right)\right) - \left(-1 + \frac{\varepsilon \cdot \left(\sin x \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (-
   (expm1 (log1p (pow (tan x) 2.0)))
   (+
    -1.0
    (/
     (* eps (* (sin x) (- -1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
     (cos x))))))

C

double code(double x, double eps) {
	return eps * (expm1(log1p(pow(tan(x), 2.0))) - (-1.0 + ((eps * (sin(x) * (-1.0 - (pow(sin(x), 2.0) / pow(cos(x), 2.0))))) / cos(x))));
}

Java

public static double code(double x, double eps) {
	return eps * (Math.expm1(Math.log1p(Math.pow(Math.tan(x), 2.0))) - (-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))));
}

Python

def code(x, eps):
	return eps * (math.expm1(math.log1p(math.pow(math.tan(x), 2.0))) - (-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))))

Julia

function code(x, eps)
	return Float64(eps * Float64(expm1(log1p((tan(x) ^ 2.0))) - Float64(-1.0 + Float64(Float64(eps * Float64(sin(x) * Float64(-1.0 - Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))) / cos(x)))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(Exp[N[Log[1 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] - N[(-1.0 + N[(N[(eps * 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]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. sqr-neg 99.4%

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

   4. mul-1-neg 99.4%

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

   5. mul-1-neg 99.4%

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

   6. sqrt-unprod 43.8%

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

   7. add-sqr-sqrt 97.1%

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

   8. expm1-log1p-u 97.1%

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

   9. expm1-undefine 97.1%

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

5. Applied egg-rr 99.4%

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

   1. rem-exp-log 99.5%

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

   2. log1p-undefine 99.5%

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

   3. expm1-undefine 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\tan x + {\tan x}^{3}\right) + 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (+ (* eps (+ (tan x) (pow (tan x) 3.0))) 1.0))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + ((eps * (tan(x) + pow(tan(x), 3.0))) + 1.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + ((eps * (tan(x) + (tan(x) ** 3.0d0))) + 1.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + ((eps * (Math.tan(x) + Math.pow(Math.tan(x), 3.0))) + 1.0));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + ((eps * (math.tan(x) + math.pow(math.tan(x), 3.0))) + 1.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(Float64(eps * Float64(tan(x) + (tan(x) ^ 3.0))) + 1.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + ((eps * (tan(x) + (tan(x) ^ 3.0))) + 1.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. associate-/l* 99.4%

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

   2. *-commutative 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. times-frac 99.4%

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

5. Applied egg-rr 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. *-commutative 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. unpow2 99.4%

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

   6. cube-mult 99.4%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\tan x + \color{blue}{{\tan x}^{3}}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

7. Simplified 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \left(\tan x + {\tan x}^{3}\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

8. Final simplification 99.4%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(\varepsilon \cdot \left(\tan x + {\tan x}^{3}\right) + 1\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 5: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in eps around 0 98.7%

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

   1. sub-neg 98.7%

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

   2. mul-1-neg 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in x around 0 97.3%

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

4. Final simplification 97.3%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. Add Preprocessing

Alternative 7: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. *-commutative 64.3%

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

   3. tan-quot 64.3%

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

   4. div-inv 64.2%

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

   5. prod-diff 64.3%

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

4. Applied egg-rr 64.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan \left(x + \varepsilon\right), 1, -\frac{1}{\cos x} \cdot \sin x\right) + \mathsf{fma}\left(-\frac{1}{\cos x}, \sin x, \frac{1}{\cos x} \cdot \sin x\right)} \]
5. Step-by-step derivation

   1. fma-undefine 64.3%

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

   2. *-rgt-identity 64.3%

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

   3. +-commutative 64.3%

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

   4. fma-undefine 64.2%

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

   5. distribute-lft-neg-in 64.2%

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

   6. neg-mul-1 64.2%

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

   7. distribute-lft1-in 64.2%

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

   8. metadata-eval 64.2%

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

   9. associate-*l/ 64.2%

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

   10. *-lft-identity 64.2%

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

   11. mul0-lft 64.2%

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

   12. associate-+l+ 64.2%

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

6. Simplified 64.3%

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

7. Taylor expanded in x around 0 62.4%

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

8. Final simplification 62.4%

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

Alternative 8: 7.9% accurate, 102.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := (-x)

Derivation

1. Initial program 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. *-commutative 64.3%

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

   3. tan-quot 64.3%

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

   4. div-inv 64.2%

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

   5. prod-diff 64.3%

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

4. Applied egg-rr 64.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\tan \left(x + \varepsilon\right), 1, -\frac{1}{\cos x} \cdot \sin x\right) + \mathsf{fma}\left(-\frac{1}{\cos x}, \sin x, \frac{1}{\cos x} \cdot \sin x\right)} \]
5. Step-by-step derivation

   1. fma-undefine 64.3%

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

   2. *-rgt-identity 64.3%

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

   3. +-commutative 64.3%

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

   4. fma-undefine 64.2%

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

   5. distribute-lft-neg-in 64.2%

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

   6. neg-mul-1 64.2%

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

   7. distribute-lft1-in 64.2%

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

   8. metadata-eval 64.2%

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

   9. associate-*l/ 64.2%

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

   10. *-lft-identity 64.2%

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

   11. mul0-lft 64.2%

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

   12. associate-+l+ 64.2%

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

6. Simplified 64.3%

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

7. Taylor expanded in x around 0 62.4%

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

8. Taylor expanded in x around inf 8.4%

   \[\leadsto \color{blue}{-1 \cdot x} \]
9. Step-by-step derivation

   1. neg-mul-1 8.4%

      \[\leadsto \color{blue}{-x} \]

10. Simplified 8.4%

    \[\leadsto \color{blue}{-x} \]

11. Final simplification 8.4%

    \[\leadsto -x \]
12. 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 2024066 
(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)))