2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 98.7%

Time: 20.7s

Alternatives: 13

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: 42.3% 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: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot t_0 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_1, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x \cdot t_0, \frac{\frac{-1}{\cos x}}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x)))
        (t_1 (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
   (if (<= eps -2.6e-7)
     (- (* (/ 1.0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) t_0) (tan x))
     (if (<= eps 3.2e-29)
       (fma eps t_1 (/ (* eps eps) (/ (/ (cos x) (sin x)) t_1)))
       (fma
        (* (cos x) t_0)
        (/ (/ -1.0 (cos x)) (fma (tan x) (tan eps) -1.0))
        (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double t_1 = 1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0));
	double tmp;
	if (eps <= -2.6e-7) {
		tmp = ((1.0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, t_1, ((eps * eps) / ((cos(x) / sin(x)) / t_1)));
	} else {
		tmp = fma((cos(x) * t_0), ((-1.0 / cos(x)) / fma(tan(x), tan(eps), -1.0)), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	t_1 = Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))
	tmp = 0.0
	if (eps <= -2.6e-7)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x));
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, t_1, Float64(Float64(eps * eps) / Float64(Float64(cos(x) / sin(x)) / t_1)));
	else
		tmp = fma(Float64(cos(x) * t_0), Float64(Float64(-1.0 / cos(x)) / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	end
	return 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.6e-7], N[(N[(N[(1.0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps * t$95$1 + N[(N[(eps * eps), $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.59999999999999999e-7

   1. Initial program 56.8%

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

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -2.59999999999999999e-7 < eps < 3.2e-29

   1. Initial program 27.7%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. fma-def 99.7%

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

      2. sub-neg 99.7%

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

      3. mul-1-neg 99.7%

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

      4. remove-double-neg 99.7%

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

      5. associate-/l* 99.7%

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

      6. unpow2 99.7%

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

      7. associate-/r* 99.7%

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

      8. sub-neg 99.7%

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

      9. mul-1-neg 99.7%

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

   4. Simplified 99.7%

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

   if 3.2e-29 < eps

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. div-inv 99.5%

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

      3. tan-quot 99.4%

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

      4. frac-sub 99.5%

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

      5. div-sub 99.5%

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

      6. div-inv 99.4%

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

      7. fma-neg 99.5%

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

   5. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/r* 99.5%

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

      3. fma-def 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{\varepsilon \cdot \varepsilon}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x \cdot \left(\tan \varepsilon + \tan x\right), \frac{\frac{-1}{\cos x}}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. tan-sum 64.6%

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

   2. tan-quot 64.4%

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

   3. clear-num 64.4%

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

   4. frac-sub 64.3%

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

   5. clear-num 64.3%

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

   6. tan-quot 64.4%

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

   7. clear-num 64.4%

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

   8. tan-quot 64.4%

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

3. Applied egg-rr 64.4%

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

   1. associate-*r/ 64.5%

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

   2. *-rgt-identity 64.5%

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

   3. associate-/r/ 64.4%

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

5. Simplified 68.4%

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

6. Taylor expanded in x around inf 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot t_0 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x \cdot t_0, \frac{\frac{-1}{\cos x}}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -3.8e-9)
     (- (* (/ 1.0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) t_0) (tan x))
     (if (<= eps 3.2e-29)
       (fma eps (pow (tan x) 2.0) eps)
       (fma
        (* (cos x) t_0)
        (/ (/ -1.0 (cos x)) (fma (tan x) (tan eps) -1.0))
        (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -3.8e-9) {
		tmp = ((1.0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = fma((cos(x) * t_0), ((-1.0 / cos(x)) / fma(tan(x), tan(eps), -1.0)), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -3.8e-9)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x));
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = fma(Float64(cos(x) * t_0), Float64(Float64(-1.0 / cos(x)) / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.8e-9], N[(N[(N[(1.0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.80000000000000011e-9

   1. Initial program 56.8%

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

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -3.80000000000000011e-9 < eps < 3.2e-29

   1. Initial program 27.7%

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

      1. tan-sum 28.1%

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

      2. tan-quot 27.7%

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

      3. clear-num 27.7%

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

      4. frac-sub 27.7%

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

      5. clear-num 27.6%

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

      6. tan-quot 27.9%

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

      7. clear-num 27.9%

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

      8. tan-quot 27.9%

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

   3. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. *-rgt-identity 27.9%

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

      3. associate-/r/ 27.9%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 76.7%

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

   8. Simplified 76.7%

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

      1. distribute-rgt-in 76.6%

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

      2. clear-num 76.6%

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

      3. associate-*l/ 76.7%

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

      4. *-lft-identity 76.7%

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

      5. tan-quot 76.6%

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

      6. tan-quot 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-out 99.1%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   if 3.2e-29 < eps

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. div-inv 99.5%

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

      3. tan-quot 99.4%

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

      4. frac-sub 99.5%

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

      5. div-sub 99.5%

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

      6. div-inv 99.4%

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

      7. fma-neg 99.5%

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

   5. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/r* 99.5%

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

      3. fma-def 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos x \cdot \left(\tan \varepsilon + \tan x\right), \frac{\frac{-1}{\cos x}}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot t_0 - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -1.6e-9)
     (- (* (/ 1.0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) t_0) (tan x))
     (if (<= eps 3.2e-29)
       (fma eps (pow (tan x) 2.0) eps)
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -1.6e-9) {
		tmp = ((1.0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -1.6e-9)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))) * t_0) - tan(x));
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.6e-9], N[(N[(N[(1.0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.60000000000000006e-9

   1. Initial program 56.8%

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

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

      1. *-commutative 99.6%

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

      2. tan-quot 99.6%

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

      3. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -1.60000000000000006e-9 < eps < 3.2e-29

   1. Initial program 27.7%

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

      1. tan-sum 28.1%

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

      2. tan-quot 27.7%

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

      3. clear-num 27.7%

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

      4. frac-sub 27.7%

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

      5. clear-num 27.6%

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

      6. tan-quot 27.9%

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

      7. clear-num 27.9%

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

      8. tan-quot 27.9%

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

   3. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. *-rgt-identity 27.9%

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

      3. associate-/r/ 27.9%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 76.7%

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

   8. Simplified 76.7%

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

      1. distribute-rgt-in 76.6%

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

      2. clear-num 76.6%

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

      3. associate-*l/ 76.7%

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

      4. *-lft-identity 76.7%

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

      5. tan-quot 76.6%

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

      6. tan-quot 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-out 99.1%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   if 3.2e-29 < eps

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. frac-2neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-in 99.5%

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

      8. +-commutative 99.5%

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

      9. remove-double-neg 99.5%

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

      10. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. fma-def 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}} \cdot \left(\tan \varepsilon + \tan x\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan \varepsilon + \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))))
   (if (<= eps -2.7e-9)
     (- (* t_0 (/ 1.0 (- 1.0 (* (tan eps) (tan x))))) (tan x))
     (if (<= eps 3.2e-29)
       (fma eps (pow (tan x) 2.0) eps)
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(eps) + tan(x);
	double tmp;
	if (eps <= -2.7e-9) {
		tmp = (t_0 * (1.0 / (1.0 - (tan(eps) * tan(x))))) - tan(x);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(eps) + tan(x))
	tmp = 0.0
	if (eps <= -2.7e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(eps) * tan(x))))) - tan(x));
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[eps], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.7e-9], 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], If[LessEqual[eps, 3.2e-29], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.7000000000000002e-9

   1. Initial program 56.8%

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

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

   if -2.7000000000000002e-9 < eps < 3.2e-29

   1. Initial program 27.7%

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

      1. tan-sum 28.1%

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

      2. tan-quot 27.7%

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

      3. clear-num 27.7%

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

      4. frac-sub 27.7%

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

      5. clear-num 27.6%

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

      6. tan-quot 27.9%

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

      7. clear-num 27.9%

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

      8. tan-quot 27.9%

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

   3. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. *-rgt-identity 27.9%

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

      3. associate-/r/ 27.9%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 76.7%

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

   8. Simplified 76.7%

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

      1. distribute-rgt-in 76.6%

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

      2. clear-num 76.6%

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

      3. associate-*l/ 76.7%

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

      4. *-lft-identity 76.7%

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

      5. tan-quot 76.6%

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

      6. tan-quot 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-out 99.1%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   if 3.2e-29 < eps

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. frac-2neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. distribute-neg-in 99.5%

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

      8. +-commutative 99.5%

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

      9. remove-double-neg 99.5%

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

      10. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. fma-def 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan \varepsilon + \tan x}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 6: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_0}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -5.2e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 4.7e-26)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ 1.0 (/ t_1 t_0)) (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 <= -5.2e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 4.7e-26) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (1.0 / (t_1 / t_0)) - 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 <= -5.2e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 4.7e-26)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(1.0 / Float64(t_1 / t_0)) - tan(x));
	end
	return 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, -5.2e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.7e-26], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(1.0 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.2000000000000002e-9

   1. Initial program 56.8%

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

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

   if -5.2000000000000002e-9 < eps < 4.69999999999999989e-26

   1. Initial program 28.9%

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

      1. tan-sum 29.2%

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

      2. tan-quot 28.9%

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

      3. clear-num 28.8%

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

      4. frac-sub 28.8%

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

      5. clear-num 28.8%

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

      6. tan-quot 29.1%

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

      7. clear-num 29.1%

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

      8. tan-quot 29.1%

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

   3. Applied egg-rr 29.1%

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

      1. associate-*r/ 29.1%

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

      2. *-rgt-identity 29.1%

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

      3. associate-/r/ 29.1%

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

   5. Simplified 37.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 77.0%

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

   8. Simplified 77.0%

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

      1. distribute-rgt-in 77.0%

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

      2. clear-num 76.9%

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

      3. associate-*l/ 77.0%

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

      4. *-lft-identity 77.0%

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

      5. tan-quot 76.9%

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

      6. tan-quot 77.0%

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

   10. Applied egg-rr 77.0%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.2%

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

       5. *-commutative 99.2%

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

       6. distribute-rgt-out 99.2%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   if 4.69999999999999989e-26 < eps

   1. Initial program 59.9%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan \varepsilon \cdot \tan x}{\tan \varepsilon + \tan x}} - \tan x\\ \end{array} \]

Alternative 7: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.4e-9) (not (<= eps 3.2e-29)))
   (- (/ (+ (tan eps) (tan x)) (- 1.0 (* (tan eps) (tan x)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.4e-9) || !(eps <= 3.2e-29)) {
		tmp = ((tan(eps) + tan(x)) / (1.0 - (tan(eps) * tan(x)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.4e-9) || !(eps <= 3.2e-29))
		tmp = Float64(Float64(Float64(tan(eps) + tan(x)) / Float64(1.0 - Float64(tan(eps) * tan(x)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.4e-9], N[Not[LessEqual[eps, 3.2e-29]], $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[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.3999999999999998e-9 or 3.2e-29 < eps

   1. Initial program 58.8%

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

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.4%

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

      2. *-commutative 99.4%

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

      3. unsub-neg 99.4%

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

      4. associate-*l/ 99.5%

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

      5. *-lft-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 -3.3999999999999998e-9 < eps < 3.2e-29

   1. Initial program 27.7%

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

      1. tan-sum 28.1%

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

      2. tan-quot 27.7%

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

      3. clear-num 27.7%

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

      4. frac-sub 27.7%

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

      5. clear-num 27.6%

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

      6. tan-quot 27.9%

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

      7. clear-num 27.9%

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

      8. tan-quot 27.9%

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

   3. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. *-rgt-identity 27.9%

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

      3. associate-/r/ 27.9%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 76.7%

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

   8. Simplified 76.7%

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

      1. distribute-rgt-in 76.6%

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

      2. clear-num 76.6%

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

      3. associate-*l/ 76.7%

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

      4. *-lft-identity 76.7%

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

      5. tan-quot 76.6%

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

      6. tan-quot 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-out 99.1%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 8: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \varepsilon + \tan x\\ t_1 := 1 - \tan \varepsilon \cdot \tan x\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan eps) (tan x))) (t_1 (- 1.0 (* (tan eps) (tan x)))))
   (if (<= eps -3.1e-9)
     (- (* t_0 (/ 1.0 t_1)) (tan x))
     (if (<= eps 3.2e-29)
       (fma eps (pow (tan x) 2.0) eps)
       (- (/ t_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 <= -3.1e-9) {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (t_0 / t_1) - 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 <= -3.1e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	end
	return 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, -3.1e-9], N[(N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $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 < -3.10000000000000005e-9

   1. Initial program 56.8%

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

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

   if -3.10000000000000005e-9 < eps < 3.2e-29

   1. Initial program 27.7%

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

      1. tan-sum 28.1%

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

      2. tan-quot 27.7%

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

      3. clear-num 27.7%

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

      4. frac-sub 27.7%

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

      5. clear-num 27.6%

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

      6. tan-quot 27.9%

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

      7. clear-num 27.9%

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

      8. tan-quot 27.9%

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

   3. Applied egg-rr 27.9%

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

      1. associate-*r/ 27.9%

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

      2. *-rgt-identity 27.9%

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

      3. associate-/r/ 27.9%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in eps around 0 99.3%

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

      1. associate-*r* 76.7%

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

   8. Simplified 76.7%

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

      1. distribute-rgt-in 76.6%

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

      2. clear-num 76.6%

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

      3. associate-*l/ 76.7%

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

      4. *-lft-identity 76.7%

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

      5. tan-quot 76.6%

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

      6. tan-quot 76.7%

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

   10. Applied egg-rr 76.7%

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

       1. associate-*l/ 99.3%

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

       2. *-rgt-identity 99.3%

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

       3. associate-*r/ 99.2%

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

       4. associate-*r* 99.1%

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

       5. *-commutative 99.1%

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

       6. distribute-rgt-out 99.1%

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

       7. associate-*r/ 99.3%

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

       8. *-rgt-identity 99.3%

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

   12. Simplified 99.3%

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

       1. associate-/l* 99.1%

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

       2. associate-/r/ 99.2%

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

       3. tan-quot 99.3%

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

       4. distribute-rgt-in 99.4%

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

       5. div-inv 99.2%

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

       6. associate-*l* 99.4%

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

       7. *-commutative 99.4%

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

       8. rgt-mult-inverse 99.6%

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

       9. *-rgt-identity 99.6%

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

       10. associate-*l* 99.5%

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

       11. pow2 99.5%

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

   14. Applied egg-rr 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   if 3.2e-29 < eps

   1. Initial program 61.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.3%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-udef 99.3%

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

      2. *-commutative 99.3%

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

      3. unsub-neg 99.3%

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

      4. associate-*l/ 99.5%

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

      5. *-lft-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan \varepsilon + \tan x\right) \cdot \frac{1}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x\\ \end{array} \]

Alternative 9: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00068)
   (tan eps)
   (if (<= eps 3.2e-29) (fma eps (pow (tan x) 2.0) eps) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00068) {
		tmp = tan(eps);
	} else if (eps <= 3.2e-29) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00068)
		tmp = tan(eps);
	elseif (eps <= 3.2e-29)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00068], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.8e-4 or 3.2e-29 < eps

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. frac-2neg 62.0%

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

      2. distribute-frac-neg 62.0%

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

      3. neg-sub0 62.0%

         \[\leadsto \color{blue}{0 - \frac{\sin \varepsilon}{-\cos \varepsilon}} \]

      4. remove-double-neg 62.0%

         \[\leadsto 0 - \color{blue}{\left(-\left(-\frac{\sin \varepsilon}{-\cos \varepsilon}\right)\right)} \]

      5. distribute-frac-neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{-\sin \varepsilon}{-\cos \varepsilon}}\right) \]

      6. frac-2neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

      7. tan-quot 62.3%

         \[\leadsto 0 - \left(-\color{blue}{\tan \varepsilon}\right) \]

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{0 - \left(-\tan \varepsilon\right)} \]
   5. Step-by-step derivation

      1. neg-sub0 62.3%

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

      2. remove-double-neg 62.3%

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

   6. Simplified 62.3%

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

   if -6.8e-4 < eps < 3.2e-29

   1. Initial program 27.5%

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

      1. tan-sum 28.6%

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

      2. tan-quot 28.2%

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

      3. clear-num 28.2%

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

      4. frac-sub 28.2%

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

      5. clear-num 28.1%

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

      6. tan-quot 28.4%

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

      7. clear-num 28.4%

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

      8. tan-quot 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. associate-*r/ 28.4%

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

      2. *-rgt-identity 28.4%

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

      3. associate-/r/ 28.4%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in eps around 0 98.7%

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

      1. associate-*r* 76.3%

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

   8. Simplified 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. clear-num 76.2%

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

      3. associate-*l/ 76.3%

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

      4. *-lft-identity 76.3%

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

      5. tan-quot 76.2%

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

      6. tan-quot 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. associate-*l/ 98.7%

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

       2. *-rgt-identity 98.7%

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

       3. associate-*r/ 98.6%

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

       4. associate-*r* 98.6%

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

       5. *-commutative 98.6%

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

       6. distribute-rgt-out 98.6%

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

       7. associate-*r/ 98.7%

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

       8. *-rgt-identity 98.7%

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

   12. Simplified 98.7%

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

       1. associate-/l* 98.6%

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

       2. associate-/r/ 98.7%

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

       3. tan-quot 98.8%

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

       4. distribute-rgt-in 98.9%

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

       5. div-inv 98.7%

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

       6. associate-*l* 98.9%

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

       7. *-commutative 98.9%

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

       8. rgt-mult-inverse 99.0%

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

       9. *-rgt-identity 99.0%

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

       10. associate-*l* 99.0%

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

       11. pow2 99.0%

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

   14. Applied egg-rr 99.0%

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

       1. +-commutative 99.0%

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

       2. fma-def 99.1%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)} \]

   16. Simplified 99.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 10: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00068)
   (tan eps)
   (if (<= eps 3.2e-29) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00068) {
		tmp = tan(eps);
	} else if (eps <= 3.2e-29) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = tan(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 <= (-0.00068d0)) then
        tmp = tan(eps)
    else if (eps <= 3.2d-29) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00068) {
		tmp = Math.tan(eps);
	} else if (eps <= 3.2e-29) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00068:
		tmp = math.tan(eps)
	elif eps <= 3.2e-29:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00068)
		tmp = tan(eps);
	elseif (eps <= 3.2e-29)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00068)
		tmp = tan(eps);
	elseif (eps <= 3.2e-29)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00068], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.8e-4 or 3.2e-29 < eps

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. frac-2neg 62.0%

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

      2. distribute-frac-neg 62.0%

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

      3. neg-sub0 62.0%

         \[\leadsto \color{blue}{0 - \frac{\sin \varepsilon}{-\cos \varepsilon}} \]

      4. remove-double-neg 62.0%

         \[\leadsto 0 - \color{blue}{\left(-\left(-\frac{\sin \varepsilon}{-\cos \varepsilon}\right)\right)} \]

      5. distribute-frac-neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{-\sin \varepsilon}{-\cos \varepsilon}}\right) \]

      6. frac-2neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

      7. tan-quot 62.3%

         \[\leadsto 0 - \left(-\color{blue}{\tan \varepsilon}\right) \]

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{0 - \left(-\tan \varepsilon\right)} \]
   5. Step-by-step derivation

      1. neg-sub0 62.3%

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

      2. remove-double-neg 62.3%

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

   6. Simplified 62.3%

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

   if -6.8e-4 < eps < 3.2e-29

   1. Initial program 27.5%

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

      1. tan-sum 28.6%

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

      2. tan-quot 28.2%

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

      3. clear-num 28.2%

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

      4. frac-sub 28.2%

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

      5. clear-num 28.1%

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

      6. tan-quot 28.4%

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

      7. clear-num 28.4%

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

      8. tan-quot 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. associate-*r/ 28.4%

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

      2. *-rgt-identity 28.4%

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

      3. associate-/r/ 28.4%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in eps around 0 98.7%

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

      1. associate-*r* 76.3%

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

   8. Simplified 76.3%

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

      1. distribute-rgt-in 76.3%

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

      2. clear-num 76.2%

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

      3. associate-*l/ 76.3%

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

      4. *-lft-identity 76.3%

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

      5. tan-quot 76.2%

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

      6. tan-quot 76.3%

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

   10. Applied egg-rr 76.3%

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

       1. associate-*l/ 98.7%

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

       2. *-rgt-identity 98.7%

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

       3. associate-*r/ 98.6%

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

       4. associate-*r* 98.6%

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

       5. *-commutative 98.6%

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

       6. distribute-rgt-out 98.6%

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

       7. associate-*r/ 98.7%

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

       8. *-rgt-identity 98.7%

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

   12. Simplified 98.7%

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

       1. associate-/l* 98.6%

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

       2. associate-/r/ 98.7%

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

       3. tan-quot 98.8%

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

       4. +-commutative 98.8%

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

       5. distribute-rgt-in 98.9%

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

       6. associate-*l* 98.8%

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

       7. pow2 98.8%

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

       8. div-inv 98.6%

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

       9. associate-*l* 98.8%

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

       10. *-commutative 98.8%

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

       11. rgt-mult-inverse 99.0%

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

       12. *-rgt-identity 99.0%

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

   14. Applied egg-rr 99.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 11: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\cos x}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00068)
   (tan eps)
   (if (<= eps 3.2e-29) (/ (/ eps (cos x)) (cos x)) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00068) {
		tmp = tan(eps);
	} else if (eps <= 3.2e-29) {
		tmp = (eps / cos(x)) / cos(x);
	} else {
		tmp = tan(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 <= (-0.00068d0)) then
        tmp = tan(eps)
    else if (eps <= 3.2d-29) then
        tmp = (eps / cos(x)) / cos(x)
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00068) {
		tmp = Math.tan(eps);
	} else if (eps <= 3.2e-29) {
		tmp = (eps / Math.cos(x)) / Math.cos(x);
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00068:
		tmp = math.tan(eps)
	elif eps <= 3.2e-29:
		tmp = (eps / math.cos(x)) / math.cos(x)
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00068)
		tmp = tan(eps);
	elseif (eps <= 3.2e-29)
		tmp = Float64(Float64(eps / cos(x)) / cos(x));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00068)
		tmp = tan(eps);
	elseif (eps <= 3.2e-29)
		tmp = (eps / cos(x)) / cos(x);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00068], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.2e-29], N[(N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.8e-4 or 3.2e-29 < eps

   1. Initial program 59.2%

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

   2. Taylor expanded in x around 0 62.0%

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

      1. frac-2neg 62.0%

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

      2. distribute-frac-neg 62.0%

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

      3. neg-sub0 62.0%

         \[\leadsto \color{blue}{0 - \frac{\sin \varepsilon}{-\cos \varepsilon}} \]

      4. remove-double-neg 62.0%

         \[\leadsto 0 - \color{blue}{\left(-\left(-\frac{\sin \varepsilon}{-\cos \varepsilon}\right)\right)} \]

      5. distribute-frac-neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{-\sin \varepsilon}{-\cos \varepsilon}}\right) \]

      6. frac-2neg 62.0%

         \[\leadsto 0 - \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

      7. tan-quot 62.3%

         \[\leadsto 0 - \left(-\color{blue}{\tan \varepsilon}\right) \]

   4. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{0 - \left(-\tan \varepsilon\right)} \]
   5. Step-by-step derivation

      1. neg-sub0 62.3%

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

      2. remove-double-neg 62.3%

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

   6. Simplified 62.3%

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

   if -6.8e-4 < eps < 3.2e-29

   1. Initial program 27.5%

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

      1. tan-sum 28.6%

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

      2. tan-quot 28.2%

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

      3. clear-num 28.2%

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

      4. frac-sub 28.2%

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

      5. clear-num 28.1%

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

      6. tan-quot 28.4%

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

      7. clear-num 28.4%

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

      8. tan-quot 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. associate-*r/ 28.4%

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

      2. *-rgt-identity 28.4%

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

      3. associate-/r/ 28.4%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in eps around 0 98.7%

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

      1. associate-*r* 76.3%

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

   8. Simplified 76.3%

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

      1. remove-double-neg 76.3%

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

      2. neg-sub0 76.3%

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

      3. associate-*l* 98.7%

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

      4. distribute-rgt-neg-in 98.7%

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

      5. frac-add 98.6%

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

      6. cos-sin-sum 98.6%

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

      7. associate-*r/ 98.8%

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

      8. *-rgt-identity 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. neg-sub0 98.8%

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

       2. distribute-rgt-neg-in 98.8%

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

       3. remove-double-neg 98.8%

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

       4. associate-/r* 98.8%

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

       5. *-inverses 98.8%

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

       6. associate-*r/ 98.9%

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

       7. *-rgt-identity 98.9%

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

   12. Simplified 98.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00068:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\cos x}}{\cos x}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 57.7% 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 43.6%

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

2. Taylor expanded in x around 0 59.5%

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

   1. frac-2neg 59.5%

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

   2. distribute-frac-neg 59.5%

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

   3. neg-sub0 59.5%

      \[\leadsto \color{blue}{0 - \frac{\sin \varepsilon}{-\cos \varepsilon}} \]

   4. remove-double-neg 59.5%

      \[\leadsto 0 - \color{blue}{\left(-\left(-\frac{\sin \varepsilon}{-\cos \varepsilon}\right)\right)} \]

   5. distribute-frac-neg 59.5%

      \[\leadsto 0 - \left(-\color{blue}{\frac{-\sin \varepsilon}{-\cos \varepsilon}}\right) \]

   6. frac-2neg 59.5%

      \[\leadsto 0 - \left(-\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \]

   7. tan-quot 59.7%

      \[\leadsto 0 - \left(-\color{blue}{\tan \varepsilon}\right) \]

4. Applied egg-rr 59.7%

   \[\leadsto \color{blue}{0 - \left(-\tan \varepsilon\right)} \]
5. Step-by-step derivation

   1. neg-sub0 59.7%

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

   2. remove-double-neg 59.7%

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

6. Simplified 59.7%

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

7. Final simplification 59.7%

   \[\leadsto \tan \varepsilon \]

Alternative 13: 30.7% 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 43.6%

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

2. Taylor expanded in x around 0 59.5%

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

3. Taylor expanded in eps around 0 31.4%

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

4. Final simplification 31.4%

   \[\leadsto \varepsilon \]

Developer target: 76.4% 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 2023297 
(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)))