2tan (problem 3.3.2)

Percentage Accurate: 42.4% → 99.6%

Time: 18.5s

Alternatives: 10

Speedup: 1.8×

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: 42.4% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

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

   1. tan-sum 64.1%

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

   2. div-inv 64.0%

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

3. Applied egg-rr 64.0%

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

   1. associate-*r/ 64.1%

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

   2. *-rgt-identity 64.1%

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

5. Simplified 64.1%

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

6. Taylor expanded in x around inf 63.9%

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

   1. associate--l+ 79.1%

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

   2. *-commutative 79.1%

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

   3. associate-/r* 79.1%

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

   4. *-commutative 79.1%

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

   5. associate-/l* 79.1%

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

8. Simplified 79.1%

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

   1. clear-num 76.4%

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

   2. frac-sub 76.7%

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

10. Applied egg-rr 77.1%

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

    1. associate--r- 94.3%

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

    2. rgt-mult-inverse 99.5%

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

    3. metadata-eval 99.5%

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

    4. metadata-eval 99.5%

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

    5. +-commutative 99.5%

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

    6. metadata-eval 99.5%

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

    7. div-sub 99.5%

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

    8. *-commutative 99.5%

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

    9. associate-/l* 99.5%

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

    10. *-rgt-identity 99.5%

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

    11. associate-*r/ 99.5%

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

    12. rgt-mult-inverse 99.5%

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

12. Simplified 99.5%

    \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\frac{\sin x}{\frac{\cos x}{\sin \varepsilon}}}{\cos \varepsilon}\right)} + \color{blue}{\frac{\tan x \cdot \tan \varepsilon + 0}{\frac{1}{\tan x} - \frac{\tan \varepsilon}{1}}} \]
13. Step-by-step derivation

    1. *-un-lft-identity 99.5%

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

    2. associate-/r* 99.5%

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

    3. tan-quot 99.6%

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

    4. associate-/r/ 99.6%

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

    5. associate-*r/ 99.6%

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

    6. tan-quot 99.7%

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

    7. tan-quot 99.6%

       \[\leadsto 1 \cdot \frac{\tan \varepsilon}{1 - \tan x \cdot \color{blue}{\tan \varepsilon}} + \frac{\tan x \cdot \tan \varepsilon + 0}{\frac{1}{\tan x} - \frac{\tan \varepsilon}{1}} \]

14. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{1 \cdot \frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \frac{\tan x \cdot \tan \varepsilon + 0}{\frac{1}{\tan x} - \frac{\tan \varepsilon}{1}} \]
15. Step-by-step derivation

    1. *-lft-identity 99.6%

       \[\leadsto \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \frac{\tan x \cdot \tan \varepsilon + 0}{\frac{1}{\tan x} - \frac{\tan \varepsilon}{1}} \]

16. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{\tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} + \frac{\tan x \cdot \tan \varepsilon + 0}{\frac{1}{\tan x} - \frac{\tan \varepsilon}{1}} \]

17. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin \varepsilon}{\frac{1}{\tan x} \cdot \cos \varepsilon}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -3.5e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 2.4e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (-
        (/ t_0 (- 1.0 (/ (sin eps) (* (/ 1.0 (tan x)) (cos eps)))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 2.4e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (sin(eps) / ((1.0 / tan(x)) * cos(eps))))) - 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) :: tmp
    t_0 = tan(eps) + tan(x)
    if (eps <= (-3.5d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else if (eps <= 2.4d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 / (1.0d0 - (sin(eps) / ((1.0d0 / tan(x)) * cos(eps))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else if (eps <= 2.4e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (Math.sin(eps) / ((1.0 / Math.tan(x)) * Math.cos(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -3.5e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	elif eps <= 2.4e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 / (1.0 - (math.sin(eps) / ((1.0 / math.tan(x)) * math.cos(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -3.5e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 2.4e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(eps) / Float64(Float64(1.0 / tan(x)) * cos(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -3.5e-9)
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	elseif (eps <= 2.4e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 / (1.0 - (sin(eps) / ((1.0 / tan(x)) * cos(eps))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.5e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.4e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[eps], $MachinePrecision] / N[(N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.4999999999999999e-9

   1. Initial program 58.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   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/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

      1. log1p-expm1-u 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x\right)\right)} \cdot \tan \varepsilon} - \tan x \]
   8. Step-by-step derivation

      1. log1p-expm1 99.5%

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

      2. remove-double-div 99.5%

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

      3. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -3.4999999999999999e-9 < eps < 2.4e-9

   1. Initial program 32.0%

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

   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 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      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. unpow2 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 2.4e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

      1. log1p-expm1-u 98.3%

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

   7. Applied egg-rr 98.3%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x\right)\right)} \cdot \tan \varepsilon} - \tan x \]
   8. Step-by-step derivation

      1. log1p-expm1 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. clear-num 99.5%

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

      4. tan-quot 99.4%

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

      5. frac-times 99.6%

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

      6. *-un-lft-identity 99.6%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin \varepsilon}{\frac{1}{\tan x} \cdot \cos \varepsilon}} - \tan x\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -3.5e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 2.8e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (* t_0 (/ 1.0 t_1)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 - (tan(eps) * tan(x));
	double tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 2.8e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / 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(eps) + tan(x)
    t_1 = 1.0d0 - (tan(eps) * tan(x))
    if (eps <= (-3.5d-9)) then
        tmp = (t_0 / t_1) - tan(x)
    else if (eps <= 2.8d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) + Math.tan(x);
	double t_1 = 1.0 - (Math.tan(eps) * Math.tan(x));
	double tmp;
	if (eps <= -3.5e-9) {
		tmp = (t_0 / t_1) - Math.tan(x);
	} else if (eps <= 2.8e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) + math.tan(x)
	t_1 = 1.0 - (math.tan(eps) * math.tan(x))
	tmp = 0
	if eps <= -3.5e-9:
		tmp = (t_0 / t_1) - math.tan(x)
	elif eps <= 2.8e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 - Float64(tan(eps) * tan(x)))
	tmp = 0.0
	if (eps <= -3.5e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 2.8e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) + tan(x);
	t_1 = 1.0 - (tan(eps) * tan(x));
	tmp = 0.0;
	if (eps <= -3.5e-9)
		tmp = (t_0 / t_1) - tan(x);
	elseif (eps <= 2.8e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.5e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.4999999999999999e-9

   1. Initial program 58.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   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/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   if -3.4999999999999999e-9 < eps < 2.79999999999999984e-9

   1. Initial program 32.0%

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

   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 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      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. unpow2 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 2.79999999999999984e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.6%

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

      3. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -1.75e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 3.2e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (* t_0 (/ 1.0 (- 1.0 (* (tan eps) (tan x))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -1.75e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 3.2e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (tan(eps) * tan(x))))) - 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) :: tmp
    t_0 = tan(eps) + tan(x)
    if (eps <= (-1.75d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else if (eps <= 3.2d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - (tan(eps) * tan(x))))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(eps) + Math.tan(x);
	double tmp;
	if (eps <= -1.75e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else if (eps <= 3.2e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (Math.tan(eps) * Math.tan(x))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(eps) + math.tan(x)
	tmp = 0
	if eps <= -1.75e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	elif eps <= 3.2e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 * (1.0 / (1.0 - (math.tan(eps) * math.tan(x))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -1.75e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 3.2e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(eps) + tan(x);
	tmp = 0.0;
	if (eps <= -1.75e-9)
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	elseif (eps <= 3.2e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 * (1.0 / (1.0 - (tan(eps) * tan(x))))) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.75e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.75e-9

   1. Initial program 58.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   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/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

      1. log1p-expm1-u 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan x\right)\right)} \cdot \tan \varepsilon} - \tan x \]
   8. Step-by-step derivation

      1. log1p-expm1 99.5%

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

      2. remove-double-div 99.5%

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

      3. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -1.75e-9 < eps < 3.20000000000000012e-9

   1. Initial program 32.0%

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

   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 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      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. unpow2 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 3.20000000000000012e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.6%

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

      3. *-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.45e-9) (not (<= eps 3.4e-9)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.45e-9) || !(eps <= 3.4e-9)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - 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 <= (-3.45d-9)) .or. (.not. (eps <= 3.4d-9))) then
        tmp = ((tan(eps) + tan(x)) / (1.0d0 - (tan(eps) * tan(x)))) - 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 <= -3.45e-9) || !(eps <= 3.4e-9)) {
		tmp = ((Math.tan(eps) + Math.tan(x)) / (1.0 - (Math.tan(eps) * Math.tan(x)))) - 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 <= -3.45e-9) or not (eps <= 3.4e-9):
		tmp = ((math.tan(eps) + math.tan(x)) / (1.0 - (math.tan(eps) * math.tan(x)))) - 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 <= -3.45e-9) || !(eps <= 3.4e-9))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - 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 <= -3.45e-9) || ~((eps <= 3.4e-9)))
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.45e-9], N[Not[LessEqual[eps, 3.4e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $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 < -3.44999999999999987e-9 or 3.3999999999999998e-9 < eps

   1. Initial program 56.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   if -3.44999999999999987e-9 < eps < 3.3999999999999998e-9

   1. Initial program 32.0%

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

   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 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      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. unpow2 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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 6: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.35e-8) (not (<= eps 2.5e-6)))
   (/ (sin eps) (cos eps))
   (* eps (+ 1.0 (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-8) || !(eps <= 2.5e-6)) {
		tmp = sin(eps) / cos(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 <= (-1.35d-8)) .or. (.not. (eps <= 2.5d-6))) then
        tmp = sin(eps) / cos(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 <= -1.35e-8) || !(eps <= 2.5e-6)) {
		tmp = Math.sin(eps) / Math.cos(eps);
	} else {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.35e-8) or not (eps <= 2.5e-6):
		tmp = math.sin(eps) / math.cos(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 <= -1.35e-8) || !(eps <= 2.5e-6))
		tmp = Float64(sin(eps) / cos(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 <= -1.35e-8) || ~((eps <= 2.5e-6)))
		tmp = sin(eps) / cos(eps);
	else
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.35e-8], N[Not[LessEqual[eps, 2.5e-6]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $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 < -1.35000000000000001e-8 or 2.5000000000000002e-6 < eps

   1. Initial program 56.4%

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

   2. Taylor expanded in x around 0 58.4%

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

   if -1.35000000000000001e-8 < eps < 2.5000000000000002e-6

   1. Initial program 32.0%

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

   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. pow1 99.3%

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

      2. unpow2 99.3%

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

      3. unpow2 99.3%

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

      4. times-frac 99.3%

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

      5. tan-quot 99.4%

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

      6. tan-quot 99.4%

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

      7. unpow2 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. unpow1 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \end{array} \]

Alternative 7: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.35e-8) (not (<= eps 3.7e-6)))
   (/ (sin eps) (cos eps))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-8) || !(eps <= 3.7e-6)) {
		tmp = sin(eps) / cos(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 <= (-1.35d-8)) .or. (.not. (eps <= 3.7d-6))) then
        tmp = sin(eps) / cos(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 <= -1.35e-8) || !(eps <= 3.7e-6)) {
		tmp = Math.sin(eps) / Math.cos(eps);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.35e-8) or not (eps <= 3.7e-6):
		tmp = math.sin(eps) / math.cos(eps)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.35e-8) || !(eps <= 3.7e-6))
		tmp = Float64(sin(eps) / cos(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 <= -1.35e-8) || ~((eps <= 3.7e-6)))
		tmp = sin(eps) / cos(eps);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.35e-8], N[Not[LessEqual[eps, 3.7e-6]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $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 < -1.35000000000000001e-8 or 3.7000000000000002e-6 < eps

   1. Initial program 56.4%

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

   2. Taylor expanded in x around 0 58.4%

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

   if -1.35000000000000001e-8 < eps < 3.7000000000000002e-6

   1. Initial program 32.0%

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

   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 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. unpow2 99.5%

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

      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. unpow2 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 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 8: 58.4% 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 43.5%

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

2. Taylor expanded in x around 0 59.7%

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

3. Final simplification 59.7%

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

Alternative 9: 55.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 6.6 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.35e-8) (not (<= eps 6.6e-7))) (- (tan (+ eps x)) x) eps))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-8) || !(eps <= 6.6e-7)) {
		tmp = tan((eps + x)) - x;
	} else {
		tmp = eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.35d-8)) .or. (.not. (eps <= 6.6d-7))) then
        tmp = tan((eps + x)) - x
    else
        tmp = eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.35e-8) || !(eps <= 6.6e-7)) {
		tmp = Math.tan((eps + x)) - x;
	} else {
		tmp = eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.35e-8) or not (eps <= 6.6e-7):
		tmp = math.tan((eps + x)) - x
	else:
		tmp = eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.35e-8) || !(eps <= 6.6e-7))
		tmp = Float64(tan(Float64(eps + x)) - x);
	else
		tmp = eps;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.35e-8) || ~((eps <= 6.6e-7)))
		tmp = tan((eps + x)) - x;
	else
		tmp = eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.35e-8], N[Not[LessEqual[eps, 6.6e-7]], $MachinePrecision]], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision], eps]

Derivation

1. Split input into 2 regimes

2. if eps < -1.35000000000000001e-8 or 6.6000000000000003e-7 < eps

   1. Initial program 56.4%

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

   2. Taylor expanded in x around 0 53.0%

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

   if -1.35000000000000001e-8 < eps < 6.6000000000000003e-7

   1. Initial program 32.0%

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

   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. Taylor expanded in x around 0 60.9%

      \[\leadsto \color{blue}{\varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 6.6 \cdot 10^{-7}\right):\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 10: 31.5% 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 43.5%

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

2. Taylor expanded in eps around 0 55.1%

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

   1. sub-neg 55.1%

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

   2. mul-1-neg 55.1%

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

   3. remove-double-neg 55.1%

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

4. Simplified 55.1%

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

5. Taylor expanded in x around 0 34.9%

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

6. Final simplification 34.9%

   \[\leadsto \varepsilon \]

Developer target: 76.2% 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 2023336 
(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)))