2tan (problem 3.3.2)

Percentage Accurate: 42.5% → 99.4%

Time: 19.9s

Alternatives: 9

Speedup: 2.0×

Specification

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: 42.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.35 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.35e-9) (not (<= eps 3.2e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.35e-9) || !(eps <= 3.2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.35d-9)) .or. (.not. (eps <= 3.2d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.35e-9) || !(eps <= 3.2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.35e-9) or not (eps <= 3.2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.35e-9) || !(eps <= 3.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.35e-9) || ~((eps <= 3.2e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.35e-9], N[Not[LessEqual[eps, 3.2e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.34999999999999981e-9 or 3.20000000000000012e-9 < eps

   1. Initial program 53.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. tan-quot 99.4%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

      4. clear-num 99.4%

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

      5. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

   if -3.34999999999999981e-9 < eps < 3.20000000000000012e-9

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. times-frac 99.5%

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

      8. tan-quot 99.6%

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

      9. tan-quot 99.6%

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

      10. pow2 99.6%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2} + \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.35 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ \frac{t_0 + \frac{\cos x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\sin x}}{\frac{1 - t_0}{\tan x}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. tan-sum 71.2%

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

   2. div-inv 71.1%

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

3. Applied egg-rr 71.1%

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

   1. associate-*r/ 71.2%

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

   2. *-rgt-identity 71.2%

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

5. Simplified 71.2%

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

   1. tan-quot 71.1%

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

   2. clear-num 71.1%

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

   3. un-div-inv 71.2%

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

   4. clear-num 71.1%

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

   5. tan-quot 71.2%

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

7. Applied egg-rr 71.2%

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan x}{\frac{1}{\tan \varepsilon}}}} - \tan x \]
8. Step-by-step derivation

   1. tan-quot 70.9%

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

   2. frac-sub 70.9%

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

   3. div-inv 70.8%

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

   4. remove-double-div 70.8%

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

   5. div-inv 70.9%

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

   6. remove-double-div 70.9%

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

   7. frac-sub 70.9%

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

   8. clear-num 70.8%

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

9. Applied egg-rr 70.9%

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

    1. associate--r- 73.4%

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

    2. +-commutative 73.4%

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

    3. associate-*r/ 74.0%

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

    4. associate-/l* 74.0%

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

    5. rem-square-sqrt 37.4%

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

    6. associate-*r/ 37.4%

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

    7. /-rgt-identity 37.4%

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

    8. rem-square-sqrt 74.0%

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

    9. associate-*r/ 74.0%

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

    10. associate-/l* 74.0%

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

    11. rem-square-sqrt 38.6%

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

    12. associate-*r/ 38.6%

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

11. Simplified 74.0%

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

12. Taylor expanded in x around inf 99.2%

    \[\leadsto \frac{\tan x \cdot \tan \varepsilon + \color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon \cdot \sin x}}}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x}} \]
13. Step-by-step derivation

    1. times-frac 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

    \[\leadsto \frac{\tan x \cdot \tan \varepsilon + \frac{\cos x}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\sin x}}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x}} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.7e-9) (not (<= eps 2.05e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 2.05e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.7d-9)) .or. (.not. (eps <= 2.05d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.7e-9) || !(eps <= 2.05e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.7e-9) or not (eps <= 2.05e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.7e-9) || !(eps <= 2.05e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.7e-9) || ~((eps <= 2.05e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.7e-9], N[Not[LessEqual[eps, 2.05e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.7e-9 or 2.0500000000000002e-9 < eps

   1. Initial program 53.3%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

   if -3.7e-9 < eps < 2.0500000000000002e-9

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. times-frac 99.5%

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

      8. tan-quot 99.6%

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

      9. tan-quot 99.6%

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

      10. pow2 99.6%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2} + \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.05 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.6e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 3.6e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 3.6e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / t_1) - tan(x);
	}
	return tmp;
}

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) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-4.6d-9)) then
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    else if (eps <= 3.6d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 / t_1) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	} else if (eps <= 3.6e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 / t_1) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -4.6e-9:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	elif eps <= 3.6e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 / t_1) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 3.6e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -4.6e-9)
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	elseif (eps <= 3.6e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 / t_1) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.6e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.6e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 50.8%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. *-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -4.5999999999999998e-9 < eps < 3.6e-9

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. times-frac 99.5%

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

      8. tan-quot 99.6%

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

      9. tan-quot 99.6%

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

      10. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 3.6e-9 < eps

   1. Initial program 56.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 5: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.7e-7)
   (tan eps)
   (if (<= eps 3e-9)
     (+ eps (* eps (pow (tan x) 2.0)))
     (-
      (/
       (+ (tan x) (tan eps))
       (- 1.0 (/ (tan x) (+ (* eps -0.3333333333333333) (/ 1.0 eps)))))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.7e-7) {
		tmp = tan(eps);
	} else if (eps <= 3e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / ((eps * -0.3333333333333333) + (1.0 / eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.7d-7)) then
        tmp = tan(eps)
    else if (eps <= 3d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) / ((eps * (-0.3333333333333333d0)) + (1.0d0 / eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.7e-7) {
		tmp = Math.tan(eps);
	} else if (eps <= 3e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) / ((eps * -0.3333333333333333) + (1.0 / eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.7e-7:
		tmp = math.tan(eps)
	elif eps <= 3e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) / ((eps * -0.3333333333333333) + (1.0 / eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.7e-7)
		tmp = tan(eps);
	elseif (eps <= 3e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) / Float64(Float64(eps * -0.3333333333333333) + Float64(1.0 / eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.7e-7)
		tmp = tan(eps);
	elseif (eps <= 3e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) / ((eps * -0.3333333333333333) + (1.0 / eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.7e-7], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(N[(eps * -0.3333333333333333), $MachinePrecision] + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.69999999999999987e-7

   1. Initial program 50.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 54.5%

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

      1. tan-quot 54.6%

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

      2. *-un-lft-identity 54.6%

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

      3. *-commutative 54.6%

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

   4. Applied egg-rr 54.6%

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

      1. *-rgt-identity 54.6%

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

   6. Simplified 54.6%

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

   if -1.69999999999999987e-7 < eps < 2.99999999999999998e-9

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. times-frac 99.5%

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

      8. tan-quot 99.6%

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

      9. tan-quot 99.6%

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

      10. pow2 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 2.99999999999999998e-9 < eps

   1. Initial program 56.3%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

      1. tan-quot 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.4%

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

      4. clear-num 99.3%

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

      5. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in eps around 0 58.5%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\color{blue}{-0.3333333333333333 \cdot \varepsilon + \frac{1}{\varepsilon}}}} - \tan x \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}}} - \tan x\\ \end{array} \]

Alternative 6: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.65 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-7) (not (<= eps 1.65e-7)))
   (tan eps)
   (* eps (+ 1.0 (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-7) || !(eps <= 1.65e-7)) {
		tmp = tan(eps);
	} else {
		tmp = eps * (1.0 + pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.7d-7)) .or. (.not. (eps <= 1.65d-7))) then
        tmp = tan(eps)
    else
        tmp = eps * (1.0d0 + (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-7) || !(eps <= 1.65e-7)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.7e-7) or not (eps <= 1.65e-7):
		tmp = math.tan(eps)
	else:
		tmp = eps * (1.0 + math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.7e-7) || !(eps <= 1.65e-7))
		tmp = tan(eps);
	else
		tmp = Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.7e-7) || ~((eps <= 1.65e-7)))
		tmp = tan(eps);
	else
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.7e-7], N[Not[LessEqual[eps, 1.65e-7]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps * N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.69999999999999987e-7 or 1.6500000000000001e-7 < eps

   1. Initial program 53.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 55.9%

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

      1. tan-quot 56.2%

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

      2. *-un-lft-identity 56.2%

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

      3. *-commutative 56.2%

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

   4. Applied egg-rr 56.2%

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

      1. *-rgt-identity 56.2%

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

   6. Simplified 56.2%

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

   if -1.69999999999999987e-7 < eps < 1.6500000000000001e-7

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. unpow2 99.3%

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

      2. unpow2 99.3%

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

      3. times-frac 99.4%

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

      4. tan-quot 99.5%

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

      5. tan-quot 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow2 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 1.65 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \]

Alternative 7: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7e-7) (not (<= eps 2.5e-7)))
   (tan eps)
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-7) || !(eps <= 2.5e-7)) {
		tmp = tan(eps);
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.7d-7)) .or. (.not. (eps <= 2.5d-7))) then
        tmp = tan(eps)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.7e-7) || !(eps <= 2.5e-7)) {
		tmp = Math.tan(eps);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.7e-7) or not (eps <= 2.5e-7):
		tmp = math.tan(eps)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.7e-7) || !(eps <= 2.5e-7))
		tmp = tan(eps);
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.7e-7) || ~((eps <= 2.5e-7)))
		tmp = tan(eps);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.7e-7], N[Not[LessEqual[eps, 2.5e-7]], $MachinePrecision]], N[Tan[eps], $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.69999999999999987e-7 or 2.49999999999999989e-7 < eps

   1. Initial program 53.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in x around 0 55.9%

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

      1. tan-quot 56.2%

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

      2. *-un-lft-identity 56.2%

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

      3. *-commutative 56.2%

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

   4. Applied egg-rr 56.2%

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

      1. *-rgt-identity 56.2%

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

   6. Simplified 56.2%

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

   if -1.69999999999999987e-7 < eps < 2.49999999999999989e-7

   1. Initial program 33.7%

      \[\tan \left(x + \varepsilon\right) - \tan x \]

   2. Taylor expanded in eps around 0 99.3%

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

      1. sub-neg 99.3%

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

      2. mul-1-neg 99.3%

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

      3. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

      1. distribute-rgt-in 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. unpow2 99.4%

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

      7. times-frac 99.5%

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

      8. tan-quot 99.6%

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

      9. tan-quot 99.6%

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

      10. pow2 99.6%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\varepsilon \cdot {\tan x}^{2} + \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 8: 58.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 57.8%

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

   1. tan-quot 57.9%

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

   2. *-un-lft-identity 57.9%

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

   3. *-commutative 57.9%

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

4. Applied egg-rr 57.9%

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

   1. *-rgt-identity 57.9%

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

6. Simplified 57.9%

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

7. Final simplification 57.9%

   \[\leadsto \tan \varepsilon \]

Alternative 9: 31.4% 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 44.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]

2. Taylor expanded in x around 0 57.8%

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

3. Taylor expanded in eps around 0 28.8%

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

4. Final simplification 28.8%

   \[\leadsto \varepsilon \]

Developer target: 76.6% 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 2023301 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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