2tan (problem 3.3.2)

Percentage Accurate: 41.8% → 99.5%

Time: 27.3s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.8% 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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.4e-7) (not (<= eps 3.1e-7)))
   (fma
    (+ (tan x) (tan eps))
    (/ -1.0 (fma (tan x) (tan eps) -1.0))
    (- (tan x)))
   (+
    (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
    (*
     (* eps eps)
     (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.4e-7) || !(eps <= 3.1e-7)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.4e-7) || !(eps <= 3.1e-7))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.4e-7], N[Not[LessEqual[eps, 3.1e-7]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.4e-7 or 3.1e-7 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. add-sqr-sqrt 42.1%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 38.0%

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

      12. add-sqr-sqrt 60.1%

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

      13. distribute-lft-neg-in 60.1%

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

      14. add-sqr-sqrt 22.0%

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

      15. sqrt-unprod 79.5%

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

      16. sqr-neg 79.5%

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

      17. sqrt-unprod 57.4%

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

      18. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if -8.4e-7 < eps < 3.1e-7

   1. Initial program 29.2%

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

      1. tan-sum 29.6%

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

      2. clear-num 29.3%

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

   3. Applied egg-rr 29.3%

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

   4. Taylor expanded in eps around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. metadata-eval 99.7%

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

      5. *-lft-identity 99.7%

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

      6. distribute-lft-in 99.7%

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

      7. *-rgt-identity 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.4 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.1 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.3 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (or (<= eps -8e-9) (not (<= eps 6.3e-9)))
     (fma
      (+ (tan x) (tan eps))
      (/ -1.0 (fma (tan x) (tan eps) -1.0))
      (- (tan x)))
     (+
      (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
      (/ eps (/ (pow (cos x) 2.0) (pow (sin x) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if ((eps <= -8e-9) || !(eps <= 6.3e-9)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + (eps / (pow(cos(x), 2.0) / pow(sin(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if ((eps <= -8e-9) || !(eps <= 6.3e-9))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(eps / Float64((cos(x) ^ 2.0) / (sin(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -8e-9], N[Not[LessEqual[eps, 6.3e-9]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps / N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.0000000000000005e-9 or 6.3000000000000002e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. add-sqr-sqrt 42.1%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 38.0%

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

      12. add-sqr-sqrt 60.1%

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

      13. distribute-lft-neg-in 60.1%

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

      14. add-sqr-sqrt 22.0%

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

      15. sqrt-unprod 79.5%

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

      16. sqr-neg 79.5%

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

      17. sqrt-unprod 57.4%

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

      18. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if -8.0000000000000005e-9 < eps < 6.3000000000000002e-9

   1. Initial program 29.2%

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

      1. tan-sum 29.6%

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

      2. div-inv 29.6%

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

      3. fma-neg 29.6%

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

   3. Applied egg-rr 29.6%

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

   4. Taylor expanded in x around inf 29.6%

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

      1. associate--l+ 61.0%

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

      2. sub-neg 61.0%

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

      3. *-commutative 61.0%

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

      4. sub-neg 61.0%

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

   6. Simplified 61.0%

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

   7. Taylor expanded in eps around 0 99.5%

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

      1. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 6.3 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon}{\frac{{\cos x}^{2}}{{\sin x}^{2}}}\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -6.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_0 \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (or (<= eps -6.8e-9) (not (<= eps 3.8e-9)))
     (fma
      (+ (tan x) (tan eps))
      (/ -1.0 (fma (tan x) (tan eps) -1.0))
      (- (tan x)))
     (+
      (/ t_0 (- 1.0 (* t_0 (/ (sin x) (cos x)))))
      (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if ((eps <= -6.8e-9) || !(eps <= 3.8e-9)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = (t_0 / (1.0 - (t_0 * (sin(x) / cos(x))))) + ((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if ((eps <= -6.8e-9) || !(eps <= 3.8e-9))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(t_0 * Float64(sin(x) / cos(x))))) + Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -6.8e-9], N[Not[LessEqual[eps, 3.8e-9]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(t$95$0 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.7999999999999997e-9 or 3.80000000000000011e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. add-sqr-sqrt 42.1%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 38.0%

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

      12. add-sqr-sqrt 60.1%

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

      13. distribute-lft-neg-in 60.1%

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

      14. add-sqr-sqrt 22.0%

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

      15. sqrt-unprod 79.5%

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

      16. sqr-neg 79.5%

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

      17. sqrt-unprod 57.4%

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

      18. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if -6.7999999999999997e-9 < eps < 3.80000000000000011e-9

   1. Initial program 29.2%

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

      1. tan-sum 29.6%

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

      2. div-inv 29.6%

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

      3. fma-neg 29.6%

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

   3. Applied egg-rr 29.6%

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

   4. Taylor expanded in x around inf 29.6%

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

      1. associate--l+ 61.0%

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

      2. sub-neg 61.0%

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

      3. *-commutative 61.0%

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

      4. sub-neg 61.0%

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

   6. Simplified 61.0%

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

   7. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.8 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5.5e-9) (not (<= eps 2.8e-9)))
   (fma
    (+ (tan x) (tan eps))
    (/ -1.0 (fma (tan x) (tan eps) -1.0))
    (- (tan x)))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5.5e-9) || !(eps <= 2.8e-9)) {
		tmp = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5.5e-9) || !(eps <= 2.8e-9))
		tmp = fma(Float64(tan(x) + tan(eps)), Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), Float64(-tan(x)));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5.5e-9], N[Not[LessEqual[eps, 2.8e-9]], $MachinePrecision]], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.4999999999999996e-9 or 2.79999999999999984e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. div-inv 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. add-sqr-sqrt 42.1%

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

      9. sqrt-unprod 80.1%

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

      10. sqr-neg 80.1%

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

      11. sqrt-unprod 38.0%

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

      12. add-sqr-sqrt 60.1%

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

      13. distribute-lft-neg-in 60.1%

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

      14. add-sqr-sqrt 22.0%

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

      15. sqrt-unprod 79.5%

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

      16. sqr-neg 79.5%

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

      17. sqrt-unprod 57.4%

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

      18. add-sqr-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if -5.4999999999999996e-9 < eps < 2.79999999999999984e-9

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 2.8 \cdot 10^{-9}\right):\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \frac{1}{1 - \tan x \cdot \tan \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1 - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (/ 1.0 (- 1.0 (* (tan x) (tan eps))))))
   (if (<= eps -3.6e-9)
     (fma t_0 t_1 (- (tan x)))
     (if (<= eps 3.55e-9)
       (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 t_1) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 / (1.0 - (tan(x) * tan(eps)));
	double tmp;
	if (eps <= -3.6e-9) {
		tmp = fma(t_0, t_1, -tan(x));
	} else if (eps <= 3.55e-9) {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * t_1) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))
	tmp = 0.0
	if (eps <= -3.6e-9)
		tmp = fma(t_0, t_1, Float64(-tan(x)));
	elseif (eps <= 3.55e-9)
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * t_1) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -3.6e-9], N[(t$95$0 * t$95$1 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 3.55e-9], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.6e-9

   1. Initial program 66.7%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -3.6e-9 < eps < 3.54999999999999994e-9

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

   if 3.54999999999999994e-9 < eps

   1. Initial program 48.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4e-9) (not (<= eps 4.2e-9)))
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 1.0 (* (tan x) (tan eps))))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e-9) || !(eps <= 4.2e-9)) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-4d-9)) .or. (.not. (eps <= 4.2d-9))) then
        tmp = ((tan(x) + tan(eps)) * (1.0d0 / (1.0d0 - (tan(x) * tan(eps))))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4e-9) || !(eps <= 4.2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) * (1.0 / (1.0 - (Math.tan(x) * Math.tan(eps))))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4e-9) or not (eps <= 4.2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) * (1.0 / (1.0 - (math.tan(x) * math.tan(eps))))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4e-9) || !(eps <= 4.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) * Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4e-9) || ~((eps <= 4.2e-9)))
		tmp = ((tan(x) + tan(eps)) * (1.0 / (1.0 - (tan(x) * tan(eps))))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4e-9], N[Not[LessEqual[eps, 4.2e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.00000000000000025e-9 or 4.20000000000000039e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

   if -4.00000000000000025e-9 < eps < 4.20000000000000039e-9

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 7: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.82 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.3e-9) (not (<= eps 1.82e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-9) || !(eps <= 1.82e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.3d-9)) .or. (.not. (eps <= 1.82d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.3e-9) || !(eps <= 1.82e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.3e-9) or not (eps <= 1.82e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.3e-9) || !(eps <= 1.82e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * 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 <= -3.3e-9) || ~((eps <= 1.82e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.3e-9], N[Not[LessEqual[eps, 1.82e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(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 < -3.30000000000000018e-9 or 1.8199999999999999e-9 < eps

   1. Initial program 57.4%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

      3. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-neg 99.5%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

   if -3.30000000000000018e-9 < eps < 1.8199999999999999e-9

   1. Initial program 29.2%

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

   2. Taylor expanded in eps around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. *-lft-identity 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. *-rgt-identity 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 1.82 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 8: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.0016\right):\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.2e-5) (not (<= eps 0.0016)))
   (- (/ 1.0 (/ (cos eps) (sin eps))) (tan x))
   (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.2e-5) || !(eps <= 0.0016)) {
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	} else {
		tmp = eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8.2d-5)) .or. (.not. (eps <= 0.0016d0))) then
        tmp = (1.0d0 / (cos(eps) / sin(eps))) - tan(x)
    else
        tmp = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.2e-5) || !(eps <= 0.0016)) {
		tmp = (1.0 / (Math.cos(eps) / Math.sin(eps))) - Math.tan(x);
	} else {
		tmp = eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8.2e-5) or not (eps <= 0.0016):
		tmp = (1.0 / (math.cos(eps) / math.sin(eps))) - math.tan(x)
	else:
		tmp = eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.2e-5) || !(eps <= 0.0016))
		tmp = Float64(Float64(1.0 / Float64(cos(eps) / sin(eps))) - tan(x));
	else
		tmp = Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8.2e-5) || ~((eps <= 0.0016)))
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	else
		tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.2e-5], N[Not[LessEqual[eps, 0.0016]], $MachinePrecision]], N[(N[(1.0 / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.20000000000000009e-5 or 0.00160000000000000008 < eps

   1. Initial program 58.3%

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

      1. tan-sum 99.7%

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

      2. clear-num 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in x around 0 60.7%

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

   if -8.20000000000000009e-5 < eps < 0.00160000000000000008

   1. Initial program 28.8%

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

      1. tan-sum 30.5%

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

      2. div-inv 30.5%

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

      3. fma-neg 30.6%

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

   3. Applied egg-rr 30.6%

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

   4. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.2 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.0016\right):\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right)\\ \end{array} \]

Alternative 9: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000265 \lor \neg \left(\varepsilon \leq 0.0016\right):\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000265) (not (<= eps 0.0016)))
   (- (/ 1.0 (/ (cos eps) (sin eps))) (tan x))
   (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000265) || !(eps <= 0.0016)) {
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	} else {
		tmp = eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.000265d0)) .or. (.not. (eps <= 0.0016d0))) then
        tmp = (1.0d0 / (cos(eps) / sin(eps))) - tan(x)
    else
        tmp = eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000265) || !(eps <= 0.0016)) {
		tmp = (1.0 / (Math.cos(eps) / Math.sin(eps))) - Math.tan(x);
	} else {
		tmp = eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000265) or not (eps <= 0.0016):
		tmp = (1.0 / (math.cos(eps) / math.sin(eps))) - math.tan(x)
	else:
		tmp = eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000265) || !(eps <= 0.0016))
		tmp = Float64(Float64(1.0 / Float64(cos(eps) / sin(eps))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000265) || ~((eps <= 0.0016)))
		tmp = (1.0 / (cos(eps) / sin(eps))) - tan(x);
	else
		tmp = eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000265], N[Not[LessEqual[eps, 0.0016]], $MachinePrecision]], N[(N[(1.0 / N[(N[Cos[eps], $MachinePrecision] / N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.6499999999999999e-4 or 0.00160000000000000008 < eps

   1. Initial program 58.3%

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

      1. tan-sum 99.7%

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

      2. clear-num 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in x around 0 60.7%

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

   if -2.6499999999999999e-4 < eps < 0.00160000000000000008

   1. Initial program 28.8%

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

   2. Taylor expanded in eps around 0 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-lft-identity 98.4%

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

      4. distribute-lft-in 98.5%

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

      5. *-rgt-identity 98.5%

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

   4. Simplified 98.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000265 \lor \neg \left(\varepsilon \leq 0.0016\right):\\ \;\;\;\;\frac{1}{\frac{\cos \varepsilon}{\sin \varepsilon}} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 10: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

2. Taylor expanded in x around 0 60.0%

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

   1. tan-quot 60.2%

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

   2. expm1-log1p-u 51.4%

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

   3. expm1-udef 22.9%

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

4. Applied egg-rr 22.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \varepsilon\right)} - 1} \]
5. Step-by-step derivation

   1. expm1-def 51.4%

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

   2. expm1-log1p 60.2%

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

6. Simplified 60.2%

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

7. Final simplification 60.2%

   \[\leadsto \tan \varepsilon \]

Alternative 11: 30.8% 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 42.7%

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

2. Taylor expanded in x around 0 60.0%

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

3. Taylor expanded in eps around 0 33.2%

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

4. Final simplification 33.2%

   \[\leadsto \varepsilon \]

Developer target: 76.5% 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 2023278 
(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)))