2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 99.3%

Time: 16.8s

Alternatives: 11

Speedup: 1.0×

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: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\frac{t_0}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon \cdot \cos x}{\sin x}}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -0.00019)
     (- (/ t_0 (+ 1.0 (+ 1.0 (- -1.0 (* (tan x) (tan eps)))))) (tan x))
     (if (<= eps 2.9e-15)
       (+
        (/
         (/ (sin eps) (cos eps))
         (- 1.0 (/ (sin eps) (/ (* (cos eps) (cos x)) (sin x)))))
        (+
         (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0)))
         (/ (* eps eps) (/ (pow (cos x) 3.0) (pow (sin x) 3.0)))))
       (- (/ t_0 (- (fma (tan x) (tan eps) -1.0))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -0.00019) {
		tmp = (t_0 / (1.0 + (1.0 + (-1.0 - (tan(x) * tan(eps)))))) - tan(x);
	} else if (eps <= 2.9e-15) {
		tmp = ((sin(eps) / cos(eps)) / (1.0 - (sin(eps) / ((cos(eps) * cos(x)) / sin(x))))) + ((eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0))) + ((eps * eps) / (pow(cos(x), 3.0) / pow(sin(x), 3.0))));
	} else {
		tmp = (t_0 / -fma(tan(x), tan(eps), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -0.00019)
		tmp = Float64(Float64(t_0 / Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(tan(x) * tan(eps)))))) - tan(x));
	elseif (eps <= 2.9e-15)
		tmp = Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - Float64(sin(eps) / Float64(Float64(cos(eps) * cos(x)) / sin(x))))) + Float64(Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))) + Float64(Float64(eps * eps) / Float64((cos(x) ^ 3.0) / (sin(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(x), tan(eps), -1.0))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00019], N[(N[(t$95$0 / N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-15], N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[Cos[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] / N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / (-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 < -1.9000000000000001e-4

   1. Initial program 52.2%

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

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. expm1-log1p-u 90.5%

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

      2. expm1-udef 90.5%

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

      3. log1p-udef 90.5%

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

      4. add-exp-log 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -1.9000000000000001e-4 < eps < 2.90000000000000019e-15

   1. Initial program 26.5%

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

      1. tan-sum 26.8%

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

      2. div-inv 26.8%

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

      3. fma-neg 26.9%

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

   3. Applied egg-rr 26.9%

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

      1. fma-neg 26.8%

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

      2. associate-*r/ 26.8%

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

      3. *-rgt-identity 26.8%

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

   5. Simplified 26.8%

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

   6. Taylor expanded in x around inf 26.8%

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

      1. associate--l+ 57.6%

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

      2. associate-/r* 57.6%

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

      3. *-commutative 57.6%

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

      4. associate-/l* 57.6%

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

   8. Simplified 57.6%

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

   9. Taylor expanded in eps around 0 99.8%

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

       1. associate-/l* 99.8%

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

       2. associate-/l* 99.8%

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

       3. unpow2 99.8%

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

   11. Simplified 99.8%

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

   if 2.90000000000000019e-15 < eps

   1. Initial program 52.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. fma-neg 99.1%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

      1. expm1-log1p-u 87.0%

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

      2. expm1-udef 87.0%

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

      3. log1p-udef 87.0%

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

      4. add-exp-log 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. associate--l+ 99.1%

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

      2. fma-neg 99.2%

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

      3. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

       1. sub-neg 99.2%

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

       2. associate--r+ 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{0} - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} + \left(-\tan x\right) \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} + \left(-\tan x\right)} \]
   12. Step-by-step derivation

       1. sub-neg 99.2%

          \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x} \]

       2. sub0-neg 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00019:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\frac{\cos \varepsilon \cdot \cos x}{\sin x}}} + \left(\frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}} + \frac{\varepsilon \cdot \varepsilon}{\frac{{\cos x}^{3}}{{\sin x}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.25e-9)
     (- (/ t_0 (+ 1.0 (+ 1.0 (- -1.0 (* (tan x) (tan eps)))))) (tan x))
     (if (<= eps 2.9e-15)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ t_0 (- (fma (tan x) (tan eps) -1.0))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.25e-9) {
		tmp = (t_0 / (1.0 + (1.0 + (-1.0 - (tan(x) * tan(eps)))))) - tan(x);
	} else if (eps <= 2.9e-15) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 / -fma(tan(x), tan(eps), -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.25e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 + Float64(1.0 + Float64(-1.0 - Float64(tan(x) * tan(eps)))))) - tan(x));
	elseif (eps <= 2.9e-15)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 / Float64(-fma(tan(x), tan(eps), -1.0))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.25e-9], N[(N[(t$95$0 / N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-15], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / (-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 < -3.2500000000000002e-9

   1. Initial program 52.9%

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

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. expm1-log1p-u 90.6%

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

      2. expm1-udef 90.6%

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

      3. log1p-udef 90.7%

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

      4. add-exp-log 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -3.2500000000000002e-9 < eps < 2.90000000000000019e-15

   1. Initial program 25.9%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. *-rgt-identity 99.8%

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

   4. Simplified 99.8%

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

   if 2.90000000000000019e-15 < eps

   1. Initial program 52.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. fma-neg 99.1%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

      1. expm1-log1p-u 87.0%

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

      2. expm1-udef 87.0%

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

      3. log1p-udef 87.0%

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

      4. add-exp-log 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. associate--l+ 99.1%

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

      2. fma-neg 99.2%

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

      3. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

       1. sub-neg 99.2%

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

       2. associate--r+ 99.2%

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

       3. metadata-eval 99.2%

          \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{0} - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} + \left(-\tan x\right) \]

   11. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} + \left(-\tan x\right)} \]
   12. Step-by-step derivation

       1. sub-neg 99.2%

          \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{0 - \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x} \]

       2. sub0-neg 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{-\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_1}{1 + \left(1 + \left(-1 - t_0\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -3.7e-9)
     (- (/ t_1 (+ 1.0 (+ 1.0 (- -1.0 t_0)))) (tan x))
     (if (<= eps 2.9e-15)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (/ t_1 (- 1.0 t_0)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_1 / (1.0 + (1.0 + (-1.0 - t_0)))) - tan(x);
	} else if (eps <= 2.9e-15) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_1 / (1.0 - t_0)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) * tan(eps)
    t_1 = tan(x) + tan(eps)
    if (eps <= (-3.7d-9)) then
        tmp = (t_1 / (1.0d0 + (1.0d0 + ((-1.0d0) - t_0)))) - tan(x)
    else if (eps <= 2.9d-15) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_1 / (1.0d0 - t_0)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	double t_1 = Math.tan(x) + Math.tan(eps);
	double tmp;
	if (eps <= -3.7e-9) {
		tmp = (t_1 / (1.0 + (1.0 + (-1.0 - t_0)))) - Math.tan(x);
	} else if (eps <= 2.9e-15) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_1 / (1.0 - t_0)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	t_1 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3.7e-9:
		tmp = (t_1 / (1.0 + (1.0 + (-1.0 - t_0)))) - math.tan(x)
	elif eps <= 2.9e-15:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_1 / (1.0 - t_0)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.7e-9)
		tmp = Float64(Float64(t_1 / Float64(1.0 + Float64(1.0 + Float64(-1.0 - t_0)))) - tan(x));
	elseif (eps <= 2.9e-15)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_1 / Float64(1.0 - t_0)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) * tan(eps);
	t_1 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -3.7e-9)
		tmp = (t_1 / (1.0 + (1.0 + (-1.0 - t_0)))) - tan(x);
	elseif (eps <= 2.9e-15)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_1 / (1.0 - t_0)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.7e-9], N[(N[(t$95$1 / N[(1.0 + N[(1.0 + N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-15], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.7e-9

   1. Initial program 52.9%

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

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. expm1-log1p-u 90.6%

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

      2. expm1-udef 90.6%

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

      3. log1p-udef 90.7%

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

      4. add-exp-log 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -3.7e-9 < eps < 2.90000000000000019e-15

   1. Initial program 25.9%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. *-rgt-identity 99.8%

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

   4. Simplified 99.8%

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

   if 2.90000000000000019e-15 < eps

   1. Initial program 52.5%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. fma-neg 99.1%

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

      2. associate-*r/ 99.1%

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

      3. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 + \left(1 + \left(-1 - \tan x \cdot \tan \varepsilon\right)\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-15}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.3e-9) (not (<= eps 2.9e-15)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.3e-9) || !(eps <= 2.9e-15)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.3d-9)) .or. (.not. (eps <= 2.9d-15))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.3e-9) || !(eps <= 2.9e-15)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.3e-9) or not (eps <= 2.9e-15):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.3e-9) || !(eps <= 2.9e-15))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.3e-9) || ~((eps <= 2.9e-15)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.3e-9], N[Not[LessEqual[eps, 2.9e-15]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.2999999999999999e-9 or 2.90000000000000019e-15 < eps

   1. Initial program 52.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. fma-neg 99.2%

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

      2. associate-*r/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

   if -2.2999999999999999e-9 < eps < 2.90000000000000019e-15

   1. Initial program 25.9%

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

   2. Taylor expanded in eps around 0 99.7%

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

      1. cancel-sign-sub-inv 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-lft-identity 99.7%

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

      4. distribute-lft-in 99.8%

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

      5. *-rgt-identity 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.55 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00175\right):\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon - x \cdot \sin \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.55e-5) (not (<= eps 0.00175)))
   (- (/ (sin (+ eps x)) (- (cos eps) (* x (sin eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.55e-5) || !(eps <= 0.00175)) {
		tmp = (sin((eps + x)) / (cos(eps) - (x * sin(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.55d-5)) .or. (.not. (eps <= 0.00175d0))) then
        tmp = (sin((eps + x)) / (cos(eps) - (x * sin(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.55e-5) || !(eps <= 0.00175)) {
		tmp = (Math.sin((eps + x)) / (Math.cos(eps) - (x * Math.sin(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.55e-5) or not (eps <= 0.00175):
		tmp = (math.sin((eps + x)) / (math.cos(eps) - (x * math.sin(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.55e-5) || !(eps <= 0.00175))
		tmp = Float64(Float64(sin(Float64(eps + x)) / Float64(cos(eps) - Float64(x * sin(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.55e-5) || ~((eps <= 0.00175)))
		tmp = (sin((eps + x)) / (cos(eps) - (x * sin(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.55e-5], N[Not[LessEqual[eps, 0.00175]], $MachinePrecision]], N[(N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[eps], $MachinePrecision] - N[(x * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.54999999999999998e-5 or 0.00175000000000000004 < eps

   1. Initial program 52.7%

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

      1. tan-quot 52.6%

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

   3. Applied egg-rr 52.6%

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

   4. Taylor expanded in x around 0 56.2%

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

      1. mul-1-neg 56.2%

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

      2. unsub-neg 56.2%

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

      3. *-commutative 56.2%

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

   6. Simplified 56.2%

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

   if -2.54999999999999998e-5 < eps < 0.00175000000000000004

   1. Initial program 26.3%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.55 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00175\right):\\ \;\;\;\;\frac{\sin \left(\varepsilon + x\right)}{\cos \varepsilon - x \cdot \sin \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 6: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_0 \cdot {1}^{0.3333333333333333} + \left(-\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0021:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (<= eps -4e-6)
     (+ (* t_0 (pow 1.0 0.3333333333333333)) (- (tan x)))
     (if (<= eps 0.0021)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -4e-6) {
		tmp = (t_0 * pow(1.0, 0.3333333333333333)) + -tan(x);
	} else if (eps <= 0.0021) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    if (eps <= (-4d-6)) then
        tmp = (t_0 * (1.0d0 ** 0.3333333333333333d0)) + -tan(x)
    else if (eps <= 0.0021d0) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) / Math.cos(eps);
	double tmp;
	if (eps <= -4e-6) {
		tmp = (t_0 * Math.pow(1.0, 0.3333333333333333)) + -Math.tan(x);
	} else if (eps <= 0.0021) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	tmp = 0
	if eps <= -4e-6:
		tmp = (t_0 * math.pow(1.0, 0.3333333333333333)) + -math.tan(x)
	elif eps <= 0.0021:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -4e-6)
		tmp = Float64(Float64(t_0 * (1.0 ^ 0.3333333333333333)) + Float64(-tan(x)));
	elseif (eps <= 0.0021)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	tmp = 0.0;
	if (eps <= -4e-6)
		tmp = (t_0 * (1.0 ^ 0.3333333333333333)) + -tan(x);
	elseif (eps <= 0.0021)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4e-6], N[(N[(t$95$0 * N[Power[1.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 0.0021], N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.99999999999999982e-6

   1. Initial program 52.9%

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

      1. add-cube-cbrt 52.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)} \cdot \sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot \sqrt[3]{\tan \left(x + \varepsilon\right)}} - \tan x \]

      2. pow3 52.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^{3}} - \tan x \]

      3. add-exp-log 20.8%

         \[\leadsto {\color{blue}{\left(e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}\right)}}^{3} - \tan x \]

      4. pow-exp 20.9%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot 3}} - \tan x \]

   3. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot 3}} - \tan x \]

   4. Taylor expanded in x around 0 55.6%

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot {1}^{0.3333333333333333}} - \tan x \]

   if -3.99999999999999982e-6 < eps < 0.00209999999999999987

   1. Initial program 26.3%

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

      1. tan-sum 27.3%

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

      2. div-inv 27.3%

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

      3. fma-neg 27.4%

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

   3. Applied egg-rr 27.4%

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

      1. fma-neg 27.3%

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

      2. associate-*r/ 27.3%

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

      3. *-rgt-identity 27.3%

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

   5. Simplified 27.3%

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

   6. Taylor expanded in eps around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. mul-1-neg 99.2%

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

      3. remove-double-neg 99.2%

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

   8. Simplified 99.2%

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

   if 0.00209999999999999987 < eps

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 54.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} \cdot {1}^{0.3333333333333333} + \left(-\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0021:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 7: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -9 \cdot 10^{-6}:\\ \;\;\;\;t_0 \cdot {1}^{0.3333333333333333} + \left(-\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00165:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (<= eps -9e-6)
     (+ (* t_0 (pow 1.0 0.3333333333333333)) (- (tan x)))
     (if (<= eps 0.00165)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if (eps <= -9e-6) {
		tmp = (t_0 * pow(1.0, 0.3333333333333333)) + -tan(x);
	} else if (eps <= 0.00165) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    if (eps <= (-9d-6)) then
        tmp = (t_0 * (1.0d0 ** 0.3333333333333333d0)) + -tan(x)
    else if (eps <= 0.00165d0) then
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) / Math.cos(eps);
	double tmp;
	if (eps <= -9e-6) {
		tmp = (t_0 * Math.pow(1.0, 0.3333333333333333)) + -Math.tan(x);
	} else if (eps <= 0.00165) {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	tmp = 0
	if eps <= -9e-6:
		tmp = (t_0 * math.pow(1.0, 0.3333333333333333)) + -math.tan(x)
	elif eps <= 0.00165:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if (eps <= -9e-6)
		tmp = Float64(Float64(t_0 * (1.0 ^ 0.3333333333333333)) + Float64(-tan(x)));
	elseif (eps <= 0.00165)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	tmp = 0.0;
	if (eps <= -9e-6)
		tmp = (t_0 * (1.0 ^ 0.3333333333333333)) + -tan(x);
	elseif (eps <= 0.00165)
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -9e-6], N[(N[(t$95$0 * N[Power[1.0, 0.3333333333333333], $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 0.00165], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if eps < -9.00000000000000023e-6

   1. Initial program 52.9%

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

      1. add-cube-cbrt 52.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)} \cdot \sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot \sqrt[3]{\tan \left(x + \varepsilon\right)}} - \tan x \]

      2. pow3 52.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^{3}} - \tan x \]

      3. add-exp-log 20.8%

         \[\leadsto {\color{blue}{\left(e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}\right)}}^{3} - \tan x \]

      4. pow-exp 20.9%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot 3}} - \tan x \]

   3. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right) \cdot 3}} - \tan x \]

   4. Taylor expanded in x around 0 55.6%

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot {1}^{0.3333333333333333}} - \tan x \]

   if -9.00000000000000023e-6 < eps < 0.00165

   1. Initial program 26.3%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. cancel-sign-sub-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-lft-identity 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. *-rgt-identity 99.2%

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

   4. Simplified 99.2%

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

   if 0.00165 < eps

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 54.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon} \cdot {1}^{0.3333333333333333} + \left(-\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00165:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \varepsilon}{\cos \varepsilon}\\ \end{array} \]

Alternative 8: 41.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;\sin x - \tan x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.82) (- (sin x) (tan x)) (- (tan (+ eps x)) x)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = sin(x) - tan(x);
	} else {
		tmp = tan((eps + x)) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.82d0)) then
        tmp = sin(x) - tan(x)
    else
        tmp = tan((eps + x)) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = Math.sin(x) - Math.tan(x);
	} else {
		tmp = Math.tan((eps + x)) - x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.82:
		tmp = math.sin(x) - math.tan(x)
	else:
		tmp = math.tan((eps + x)) - x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.82)
		tmp = Float64(sin(x) - tan(x));
	else
		tmp = Float64(tan(Float64(eps + x)) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.82)
		tmp = sin(x) - tan(x);
	else
		tmp = tan((eps + x)) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.82], N[(N[Sin[x], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.819999999999999951

   1. Initial program 7.7%

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

      1. tan-quot 7.4%

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

      2. div-inv 7.5%

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

   3. Applied egg-rr 7.5%

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

   4. Taylor expanded in x around 0 8.4%

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

   5. Taylor expanded in eps around 0 10.3%

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

   if -0.819999999999999951 < x

   1. Initial program 51.4%

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

   2. Taylor expanded in x around 0 49.7%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;\sin x - \tan x\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\varepsilon + x\right) - x\\ \end{array} \]

Alternative 9: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in x around 0 56.2%

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

3. Final simplification 56.2%

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

Alternative 10: 39.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in x around 0 37.5%

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

3. Final simplification 37.5%

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

Alternative 11: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 39.9%

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

   1. add-cube-cbrt 39.6%

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}} \]

   2. pow3 39.6%

      \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^{3}} \]

3. Applied egg-rr 39.6%

   \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^{3}} \]

4. Taylor expanded in eps around 0 4.4%

   \[\leadsto \color{blue}{\frac{\sin x}{\cos x} - {1}^{0.3333333333333333} \cdot \frac{\sin x}{\cos x}} \]
5. Step-by-step derivation

   1. pow-base-1 4.4%

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

   2. *-lft-identity 4.4%

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

   3. +-inverses 4.4%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.4%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.4%

   \[\leadsto 0 \]

Developer target: 76.7% 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 2023261 
(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)))