2tan (problem 3.3.2)

Percentage Accurate: 42.3% → 99.4%

Time: 19.4s

Alternatives: 13

Speedup: 1.9×

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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, t_0, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
   (if (<= eps -3e-7)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (sin x) (/ (cos x) (tan eps)))))
      (/ (sin x) (cos x)))
     (if (<= eps 9.5e-8)
       (fma eps t_0 (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) t_0)))
       (-
        (/
         (- (- (tan x)) (tan eps))
         (+ (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))) -1.0))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0));
	double tmp;
	if (eps <= -3e-7) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - (sin(x) / cos(x));
	} else if (eps <= 9.5e-8) {
		tmp = fma(eps, t_0, (pow(eps, 2.0) / ((cos(x) / sin(x)) / t_0)));
	} else {
		tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + -1.0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))
	tmp = 0.0
	if (eps <= -3e-7)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - Float64(sin(x) / cos(x)));
	elseif (eps <= 9.5e-8)
		tmp = fma(eps, t_0, Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / t_0)));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / Float64(Float64(Float64(sin(x) * sin(eps)) / Float64(cos(x) * cos(eps))) + -1.0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = 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, -3e-7], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 9.5e-8], N[(eps * t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.9999999999999999e-7

   1. Initial program 51.8%

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

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

      3. tan-quot 99.3%

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

      4. div-inv 99.2%

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

      5. prod-diff 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.5%

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

      3. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

   if -2.9999999999999999e-7 < eps < 9.50000000000000036e-8

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.6%

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

      1. fma-def 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-/l* 99.6%

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

      6. associate-/r* 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-lft-identity 99.6%

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

   4. Simplified 99.6%

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

   if 9.50000000000000036e-8 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - t_0\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(t_0 + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (if (<= eps -4.1e-7)
     (-
      (/ (+ (tan x) (tan eps)) (- 1.0 (/ (sin x) (/ (cos x) (tan eps)))))
      t_0)
     (if (<= eps 2.3e-7)
       (+
        (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (* (pow eps 2.0) (+ t_0 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (-
        (/
         (- (- (tan x)) (tan eps))
         (+ (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))) -1.0))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double tmp;
	if (eps <= -4.1e-7) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - t_0;
	} else if (eps <= 2.3e-7) {
		tmp = (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * (t_0 + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + -1.0)) - 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) :: tmp
    t_0 = sin(x) / cos(x)
    if (eps <= (-4.1d-7)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (sin(x) / (cos(x) / tan(eps))))) - t_0
    else if (eps <= 2.3d-7) then
        tmp = (eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps ** 2.0d0) * (t_0 + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    else
        tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + (-1.0d0))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) / Math.cos(x);
	double tmp;
	if (eps <= -4.1e-7) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.sin(x) / (Math.cos(x) / Math.tan(eps))))) - t_0;
	} else if (eps <= 2.3e-7) {
		tmp = (eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + (Math.pow(eps, 2.0) * (t_0 + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	} else {
		tmp = ((-Math.tan(x) - Math.tan(eps)) / (((Math.sin(x) * Math.sin(eps)) / (Math.cos(x) * Math.cos(eps))) + -1.0)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	tmp = 0
	if eps <= -4.1e-7:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.sin(x) / (math.cos(x) / math.tan(eps))))) - t_0
	elif eps <= 2.3e-7:
		tmp = (eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + (math.pow(eps, 2.0) * (t_0 + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	else:
		tmp = ((-math.tan(x) - math.tan(eps)) / (((math.sin(x) * math.sin(eps)) / (math.cos(x) * math.cos(eps))) + -1.0)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	tmp = 0.0
	if (eps <= -4.1e-7)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - t_0);
	elseif (eps <= 2.3e-7)
		tmp = Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(t_0 + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / Float64(Float64(Float64(sin(x) * sin(eps)) / Float64(cos(x) * cos(eps))) + -1.0)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(x) / cos(x);
	tmp = 0.0;
	if (eps <= -4.1e-7)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - t_0;
	elseif (eps <= 2.3e-7)
		tmp = (eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps ^ 2.0) * (t_0 + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	else
		tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + -1.0)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.1e-7], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[eps, 2.3e-7], N[(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] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$0 + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.0999999999999999e-7

   1. Initial program 51.8%

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

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

      3. tan-quot 99.3%

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

      4. div-inv 99.2%

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

      5. prod-diff 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.5%

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

      3. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

   if -4.0999999999999999e-7 < eps < 2.29999999999999995e-7

   1. Initial program 32.1%

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

      1. tan-sum 32.8%

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

      2. div-inv 32.8%

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

      3. tan-quot 32.6%

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

      4. div-inv 32.8%

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

      5. prod-diff 32.2%

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

   3. Applied egg-rr 32.2%

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

      1. +-commutative 32.2%

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

      2. fma-udef 32.2%

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

      3. associate-+r+ 32.8%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

      4. cancel-sign-sub-inv 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-lft-identity 99.6%

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

   8. Simplified 99.6%

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

   if 2.29999999999999995e-7 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.6e-9)
   (-
    (/ (+ (tan x) (tan eps)) (- 1.0 (/ (sin x) (/ (cos x) (tan eps)))))
    (/ (sin x) (cos x)))
   (if (<= eps 4e-9)
     (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
     (-
      (/
       (- (- (tan x)) (tan eps))
       (+ (/ (* (sin x) (sin eps)) (* (cos x) (cos eps))) -1.0))
      (tan x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - (sin(x) / cos(x));
	} else if (eps <= 4e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + -1.0)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-4.6d-9)) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (sin(x) / (cos(x) / tan(eps))))) - (sin(x) / cos(x))
    else if (eps <= 4d-9) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + (-1.0d0))) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -4.6e-9) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.sin(x) / (Math.cos(x) / Math.tan(eps))))) - (Math.sin(x) / Math.cos(x));
	} else if (eps <= 4e-9) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = ((-Math.tan(x) - Math.tan(eps)) / (((Math.sin(x) * Math.sin(eps)) / (Math.cos(x) * Math.cos(eps))) + -1.0)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -4.6e-9:
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.sin(x) / (math.cos(x) / math.tan(eps))))) - (math.sin(x) / math.cos(x))
	elif eps <= 4e-9:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = ((-math.tan(x) - math.tan(eps)) / (((math.sin(x) * math.sin(eps)) / (math.cos(x) * math.cos(eps))) + -1.0)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.6e-9)
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - Float64(sin(x) / cos(x)));
	elseif (eps <= 4e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / Float64(Float64(Float64(sin(x) * sin(eps)) / Float64(cos(x) * cos(eps))) + -1.0)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -4.6e-9)
		tmp = ((tan(x) + tan(eps)) / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - (sin(x) / cos(x));
	elseif (eps <= 4e-9)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = ((-tan(x) - tan(eps)) / (((sin(x) * sin(eps)) / (cos(x) * cos(eps))) + -1.0)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.6e-9], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4e-9], 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], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 51.8%

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

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

      3. tan-quot 99.3%

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

      4. div-inv 99.2%

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

      5. prod-diff 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.5%

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

      3. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

   if -4.5999999999999998e-9 < eps < 4.00000000000000025e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

   if 4.00000000000000025e-9 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + -1} - \tan x\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -5.5e-9)
     (- (/ t_0 (- 1.0 (/ (sin x) (/ (cos x) (tan eps))))) (/ (sin x) (cos x)))
     (if (<= eps 4.2e-9)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 (/ -1.0 (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 <= -5.5e-9) {
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) / tan(eps))))) - (sin(x) / cos(x));
	} else if (eps <= 4.2e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (-1.0 / 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 <= -5.5e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(x) / Float64(cos(x) / tan(eps))))) - Float64(sin(x) / cos(x)));
	elseif (eps <= 4.2e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(-1.0 / 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, -5.5e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.2e-9], 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], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.4999999999999996e-9

   1. Initial program 51.8%

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

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

      3. tan-quot 99.3%

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

      4. div-inv 99.2%

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

      5. prod-diff 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.5%

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

      3. associate-*r/ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

   if -5.4999999999999996e-9 < eps < 4.20000000000000039e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

   if 4.20000000000000039e-9 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\sin x}{\frac{\cos x}{\tan \varepsilon}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.8e-9)
     (- (/ t_0 (- 1.0 (/ (tan eps) (/ 1.0 (tan x))))) (/ (sin x) (cos x)))
     (if (<= eps 1.65e-9)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 (/ -1.0 (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 <= -4.8e-9) {
		tmp = (t_0 / (1.0 - (tan(eps) / (1.0 / tan(x))))) - (sin(x) / cos(x));
	} else if (eps <= 1.65e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (-1.0 / 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 <= -4.8e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(eps) / Float64(1.0 / tan(x))))) - Float64(sin(x) / cos(x)));
	elseif (eps <= 1.65e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(-1.0 / 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, -4.8e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[eps], $MachinePrecision] / N[(1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.65e-9], 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], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8e-9

   1. Initial program 51.8%

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

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

      3. tan-quot 99.3%

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

      4. div-inv 99.2%

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

      5. prod-diff 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.5%

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

      3. clear-num 99.5%

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

      4. un-div-inv 99.5%

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

      5. clear-num 99.5%

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

      6. tan-quot 99.4%

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

   7. Applied egg-rr 99.4%

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

   if -4.8e-9 < eps < 1.65000000000000009e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

   if 1.65000000000000009e-9 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon}{\frac{1}{\tan x}}} - \frac{\sin x}{\cos x}\\ \mathbf{elif}\;\varepsilon \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{t_0} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (tan x) (tan eps) -1.0)))
   (if (<= eps -4e-9)
     (- (/ (- (- (tan x)) (tan eps)) t_0) (tan x))
     (if (<= eps 2.8e-9)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* (+ (tan x) (tan eps)) (/ -1.0 t_0)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = fma(tan(x), tan(eps), -1.0);
	double tmp;
	if (eps <= -4e-9) {
		tmp = ((-tan(x) - tan(eps)) / t_0) - tan(x);
	} else if (eps <= 2.8e-9) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = ((tan(x) + tan(eps)) * (-1.0 / t_0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(tan(x), tan(eps), -1.0)
	tmp = 0.0
	if (eps <= -4e-9)
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / t_0) - tan(x));
	elseif (eps <= 2.8e-9)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) * Float64(-1.0 / t_0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -4e-9], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.8e-9], 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], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000025e-9

   1. Initial program 51.8%

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

      1. expm1-log1p-u 45.6%

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

   3. Applied egg-rr 45.6%

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

      1. expm1-log1p-u 51.8%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 54.5%

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

      12. sqrt-unprod 76.5%

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

      13. sqr-neg 76.5%

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

      14. sqrt-prod 22.0%

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

      15. add-sqr-sqrt 55.7%

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

      16. tan-quot 55.7%

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

      17. tan-quot 55.7%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.4%

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

      8. metadata-eval 99.4%

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

   7. Simplified 99.4%

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

   if -4.00000000000000025e-9 < eps < 2.79999999999999984e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

   if 2.79999999999999984e-9 < eps

   1. Initial program 55.4%

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

      1. expm1-log1p-u 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. expm1-log1p-u 55.4%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 46.7%

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

      12. sqrt-unprod 77.8%

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

      13. sqr-neg 77.8%

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

      14. sqrt-prod 31.1%

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

      15. add-sqr-sqrt 58.3%

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

      16. tan-quot 58.3%

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

      17. tan-quot 58.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.2e-9) (not (<= eps 4e-9)))
   (- (/ (- (- (tan x)) (tan eps)) (fma (tan x) (tan eps) -1.0)) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.2e-9) || !(eps <= 4e-9)) {
		tmp = ((-tan(x) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x);
	} else {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.2e-9) || !(eps <= 4e-9))
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / fma(tan(x), tan(eps), -1.0)) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.2e-9], N[Not[LessEqual[eps, 4e-9]], $MachinePrecision]], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.20000000000000012e-9 or 4.00000000000000025e-9 < eps

   1. Initial program 53.6%

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

      1. expm1-log1p-u 47.7%

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

   3. Applied egg-rr 47.7%

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

      1. expm1-log1p-u 53.6%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 50.7%

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

      12. sqrt-unprod 77.2%

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

      13. sqr-neg 77.2%

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

      14. sqrt-prod 26.4%

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

      15. add-sqr-sqrt 57.0%

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

      16. tan-quot 57.0%

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

      17. tan-quot 57.0%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   if -3.20000000000000012e-9 < eps < 4.00000000000000025e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 8: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.9e-9) (not (<= eps 1.5e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.9e-9) || !(eps <= 1.5e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= (-2.9d-9)) .or. (.not. (eps <= 1.5d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps * (1.0d0 + ((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 <= -2.9e-9) || !(eps <= 1.5e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= -2.9e-9) or not (eps <= 1.5e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps * (1.0 + (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 <= -2.9e-9) || !(eps <= 1.5e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + 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 <= -2.9e-9) || ~((eps <= 1.5e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.9e-9], N[Not[LessEqual[eps, 1.5e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(1.0 + 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 < -2.89999999999999991e-9 or 1.49999999999999999e-9 < eps

   1. Initial program 53.6%

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

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

      3. *-un-lft-identity 99.3%

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

      4. prod-diff 99.3%

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

      5. *-commutative 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. *-commutative 99.3%

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

      8. *-un-lft-identity 99.3%

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

   3. Applied egg-rr 99.3%

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

   5. Simplified 99.3%

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

   if -2.89999999999999991e-9 < eps < 1.49999999999999999e-9

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 9: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{-1} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -7.5e-5) (not (<= eps 3.2e-5)))
   (- (/ (- (- (tan x)) (tan eps)) -1.0) (tan x))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -7.5e-5) || !(eps <= 3.2e-5)) {
		tmp = ((-tan(x) - tan(eps)) / -1.0) - tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= (-7.5d-5)) .or. (.not. (eps <= 3.2d-5))) then
        tmp = ((-tan(x) - tan(eps)) / (-1.0d0)) - tan(x)
    else
        tmp = eps * (1.0d0 + ((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 <= -7.5e-5) || !(eps <= 3.2e-5)) {
		tmp = ((-Math.tan(x) - Math.tan(eps)) / -1.0) - Math.tan(x);
	} else {
		tmp = eps * (1.0 + (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 <= -7.5e-5) or not (eps <= 3.2e-5):
		tmp = ((-math.tan(x) - math.tan(eps)) / -1.0) - math.tan(x)
	else:
		tmp = eps * (1.0 + (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 <= -7.5e-5) || !(eps <= 3.2e-5))
		tmp = Float64(Float64(Float64(Float64(-tan(x)) - tan(eps)) / -1.0) - tan(x));
	else
		tmp = Float64(eps * Float64(1.0 + 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 <= -7.5e-5) || ~((eps <= 3.2e-5)))
		tmp = ((-tan(x) - tan(eps)) / -1.0) - tan(x);
	else
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -7.5e-5], N[Not[LessEqual[eps, 3.2e-5]], $MachinePrecision]], N[(N[(N[((-N[Tan[x], $MachinePrecision]) - N[Tan[eps], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.49999999999999934e-5 or 3.19999999999999986e-5 < eps

   1. Initial program 53.6%

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

      1. expm1-log1p-u 47.7%

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

   3. Applied egg-rr 47.7%

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

      1. expm1-log1p-u 53.6%

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

      2. tan-sum 99.3%

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

      3. div-inv 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. associate-*r/ 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. add-sqr-sqrt 50.7%

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

      12. sqrt-unprod 77.2%

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

      13. sqr-neg 77.2%

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

      14. sqrt-prod 26.4%

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

      15. add-sqr-sqrt 57.0%

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

      16. tan-quot 57.0%

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

      17. tan-quot 57.0%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. distribute-neg-frac 99.3%

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

      4. +-commutative 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. fma-neg 99.3%

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

      8. metadata-eval 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 57.5%

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

   if -7.49999999999999934e-5 < eps < 3.19999999999999986e-5

   1. Initial program 32.1%

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

   2. Taylor expanded in eps around 0 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.5 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\left(-\tan x\right) - \tan \varepsilon}{-1} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]

Alternative 10: 58.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in x around 0 59.5%

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

3. Final simplification 59.5%

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

Alternative 11: 55.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;x + \tan \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.3e-6) (not (<= eps 2.6e-6))) (+ x (tan (+ eps x))) eps))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-6) || !(eps <= 2.6e-6)) {
		tmp = x + tan((eps + x));
	} else {
		tmp = eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.3d-6)) .or. (.not. (eps <= 2.6d-6))) then
        tmp = x + tan((eps + x))
    else
        tmp = eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.3e-6) || !(eps <= 2.6e-6)) {
		tmp = x + Math.tan((eps + x));
	} else {
		tmp = eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.3e-6) or not (eps <= 2.6e-6):
		tmp = x + math.tan((eps + x))
	else:
		tmp = eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.3e-6) || !(eps <= 2.6e-6))
		tmp = Float64(x + tan(Float64(eps + x)));
	else
		tmp = eps;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.3e-6) || ~((eps <= 2.6e-6)))
		tmp = x + tan((eps + x));
	else
		tmp = eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.3e-6], N[Not[LessEqual[eps, 2.6e-6]], $MachinePrecision]], N[(x + N[Tan[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], eps]

Derivation

1. Split input into 2 regimes

2. if eps < -1.30000000000000005e-6 or 2.60000000000000009e-6 < eps

   1. Initial program 53.6%

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

      1. add-sqr-sqrt 25.1%

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

      2. sqrt-unprod 53.3%

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

      3. pow2 53.3%

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

   3. Applied egg-rr 53.3%

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

      1. unpow2 53.3%

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

      2. sqr-neg 53.3%

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

      3. sqrt-unprod 28.3%

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

      4. add-sqr-sqrt 53.6%

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

      5. tan-quot 53.6%

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

      6. add-sqr-sqrt 25.3%

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

      7. distribute-rgt-neg-in 25.3%

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

      8. tan-quot 25.3%

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

      9. tan-quot 25.3%

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

   5. Applied egg-rr 25.3%

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

   6. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

   8. Simplified 50.9%

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

      1. sub-neg 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. remove-double-neg 50.9%

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

       2. +-commutative 50.9%

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

   12. Simplified 50.9%

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

   if -1.30000000000000005e-6 < eps < 2.60000000000000009e-6

   1. Initial program 32.1%

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

      1. tan-sum 32.8%

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

      2. div-inv 32.8%

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

      3. tan-quot 32.6%

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

      4. div-inv 32.8%

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

      5. prod-diff 32.2%

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

   3. Applied egg-rr 32.2%

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

      1. +-commutative 32.2%

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

      2. fma-udef 32.2%

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

      3. associate-+r+ 32.8%

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

   5. Simplified 32.6%

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

      1. add-cube-cbrt 32.6%

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

      2. pow3 32.6%

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

   7. Applied egg-rr 32.6%

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

   8. Taylor expanded in eps around 0 62.2%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;x + \tan \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon\\ \end{array} \]

Alternative 12: 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 43.9%

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

   1. expm1-log1p-u 40.3%

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

3. Applied egg-rr 40.3%

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

   1. expm1-log1p-u 43.9%

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

   2. tan-sum 69.4%

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

   3. div-inv 69.4%

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

   4. frac-2neg 69.4%

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

   5. metadata-eval 69.4%

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

   6. associate-*r/ 69.4%

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

   7. sub-neg 69.4%

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

   8. distribute-neg-in 69.4%

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

   9. metadata-eval 69.4%

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

   10. distribute-lft-neg-in 69.4%

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

   11. add-sqr-sqrt 38.6%

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

   12. sqrt-unprod 57.2%

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

   13. sqr-neg 57.2%

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

   14. sqrt-prod 18.6%

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

   15. add-sqr-sqrt 45.8%

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

   16. tan-quot 45.8%

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

   17. tan-quot 45.8%

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

5. Applied egg-rr 69.4%

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

   1. *-commutative 69.4%

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

   2. neg-mul-1 69.4%

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

   3. distribute-neg-frac 69.4%

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

   4. +-commutative 69.4%

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

   5. metadata-eval 69.4%

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

   6. sub-neg 69.4%

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

   7. fma-neg 69.5%

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

   8. metadata-eval 69.5%

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

7. Simplified 69.5%

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

8. Taylor expanded in eps around 0 4.1%

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

   1. distribute-lft1-in 4.1%

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

   2. metadata-eval 4.1%

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

   3. mul0-lft 4.1%

      \[\leadsto -1 \cdot \color{blue}{0} \]

   4. metadata-eval 4.1%

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

10. Simplified 4.1%

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

11. Final simplification 4.1%

    \[\leadsto 0 \]

Alternative 13: 31.6% 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 43.9%

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

   1. tan-sum 69.4%

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

   2. div-inv 69.4%

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

   3. tan-quot 69.3%

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

   4. div-inv 69.3%

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

   5. prod-diff 69.1%

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

3. Applied egg-rr 69.1%

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

   1. +-commutative 69.1%

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

   2. fma-udef 69.1%

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

   3. associate-+r+ 69.4%

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

5. Simplified 69.3%

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

   1. add-cube-cbrt 69.2%

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

   2. pow3 69.2%

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

7. Applied egg-rr 69.2%

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

8. Taylor expanded in eps around 0 30.0%

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

9. Final simplification 30.0%

   \[\leadsto \varepsilon \]

Developer target: 76.8% 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 2023326 
(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)))