2tan (problem 3.3.2)

Percentage Accurate: 42.6% → 99.4%

Time: 31.7s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.6% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ t_2 := \frac{\sin x}{\cos x}\\ t_3 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ t_4 := \tan x + \tan \varepsilon\\ t_5 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_4, \frac{1}{1 - \log \left({\left(e^{\tan \varepsilon}\right)}^{\tan x}\right)}, t_0\right) + t_1\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;t_1 + \left(\varepsilon \cdot \left(1 + t_5\right) + \left({\varepsilon}^{2} \cdot \left(t_2 + t_3\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t_5 + \frac{\sin x \cdot \left(t_3 - t_2 \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{t_4}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x)))
        (t_1 (fma -1.0 (tan x) (tan x)))
        (t_2 (/ (sin x) (cos x)))
        (t_3 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (t_4 (+ (tan x) (tan eps)))
        (t_5 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (if (<= eps -4.8e-5)
     (+ (fma t_4 (/ 1.0 (- 1.0 (log (pow (exp (tan eps)) (tan x))))) t_0) t_1)
     (if (<= eps 5.2e-11)
       (+
        t_1
        (+
         (* eps (+ 1.0 t_5))
         (+
          (* (pow eps 2.0) (+ t_2 t_3))
          (*
           (pow eps 3.0)
           (+
            0.3333333333333333
            (+
             t_5
             (/ (* (sin x) (- t_3 (* t_2 -0.3333333333333333))) (cos x))))))))
       (- t_0 (/ t_4 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = fma(-1.0, tan(x), tan(x));
	double t_2 = sin(x) / cos(x);
	double t_3 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double t_4 = tan(x) + tan(eps);
	double t_5 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double tmp;
	if (eps <= -4.8e-5) {
		tmp = fma(t_4, (1.0 / (1.0 - log(pow(exp(tan(eps)), tan(x))))), t_0) + t_1;
	} else if (eps <= 5.2e-11) {
		tmp = t_1 + ((eps * (1.0 + t_5)) + ((pow(eps, 2.0) * (t_2 + t_3)) + (pow(eps, 3.0) * (0.3333333333333333 + (t_5 + ((sin(x) * (t_3 - (t_2 * -0.3333333333333333))) / cos(x)))))));
	} else {
		tmp = t_0 - (t_4 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = fma(-1.0, tan(x), tan(x))
	t_2 = Float64(sin(x) / cos(x))
	t_3 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	t_4 = Float64(tan(x) + tan(eps))
	t_5 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	tmp = 0.0
	if (eps <= -4.8e-5)
		tmp = Float64(fma(t_4, Float64(1.0 / Float64(1.0 - log((exp(tan(eps)) ^ tan(x))))), t_0) + t_1);
	elseif (eps <= 5.2e-11)
		tmp = Float64(t_1 + Float64(Float64(eps * Float64(1.0 + t_5)) + Float64(Float64((eps ^ 2.0) * Float64(t_2 + t_3)) + Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_5 + Float64(Float64(sin(x) * Float64(t_3 - Float64(t_2 * -0.3333333333333333))) / cos(x))))))));
	else
		tmp = Float64(t_0 - Float64(t_4 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.8e-5], N[(N[(t$95$4 * N[(1.0 / N[(1.0 - N[Log[N[Power[N[Exp[N[Tan[eps], $MachinePrecision]], $MachinePrecision], N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 5.2e-11], N[(t$95$1 + N[(N[(eps * N[(1.0 + t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$5 + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$3 - N[(t$95$2 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(t$95$4 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.8000000000000001e-5

   1. Initial program 51.4%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing
   3. 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. *-commutative 99.3%

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

      5. prod-diff 99.3%

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

      6. *-un-lft-identity 99.3%

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

      7. metadata-eval 99.3%

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

      8. *-un-lft-identity 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. add-log-exp 99.4%

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

      2. *-commutative 99.4%

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

      3. exp-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

   if -4.8000000000000001e-5 < eps < 5.2000000000000001e-11

   1. Initial program 28.9%

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

      1. tan-sum 29.5%

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

      2. div-inv 29.5%

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

      3. *-un-lft-identity 29.5%

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

      4. *-commutative 29.5%

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

      5. prod-diff 29.5%

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

      6. *-un-lft-identity 29.5%

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

      7. metadata-eval 29.5%

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

      8. *-un-lft-identity 29.5%

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

   4. Applied egg-rr 29.5%

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

   5. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(-1 \cdot \frac{\sin x \cdot \left(-1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}} + -0.3333333333333333 \cdot \frac{\sin x}{\cos x}\right)}{\cos x} + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)} + \mathsf{fma}\left(-1, \tan x, \tan x\right) \]

   if 5.2000000000000001e-11 < eps

   1. Initial program 52.2%

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

      1. log1p-expm1-u 50.6%

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

   4. Applied egg-rr 50.6%

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

      1. log1p-expm1-u 52.2%

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

      2. tan-sum 99.4%

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

      3. frac-2neg 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. distribute-lft-neg-in 99.4%

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

      8. add-sqr-sqrt 53.7%

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

      9. sqrt-unprod 78.2%

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

      10. sqr-neg 78.2%

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

      11. sqrt-unprod 24.4%

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

      12. add-sqr-sqrt 54.8%

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

      13. distribute-lft-neg-in 54.8%

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

      14. add-sqr-sqrt 30.4%

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

      15. sqrt-unprod 76.0%

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

      16. sqr-neg 76.0%

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

      17. sqrt-unprod 45.5%

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

      18. add-sqr-sqrt 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. neg-sub0 99.4%

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

      2. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. neg-sub0 99.4%

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

      5. +-commutative 99.4%

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

      6. fma-def 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \log \left({\left(e^{\tan \varepsilon}\right)}^{\tan x}\right)}, -\tan x\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-1, \tan x, \tan x\right) + \left(\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} - \frac{\sin x}{\cos x} \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \log \left({\left(e^{\tan \varepsilon}\right)}^{\tan x}\right)}, t_0\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -5.1e-9)
     (+
      (fma t_1 (/ 1.0 (- 1.0 (log (pow (exp (tan eps)) (tan x))))) t_0)
      (fma -1.0 (tan x) (tan x)))
     (if (<= eps 5.2e-11)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- t_0 (/ t_1 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -5.1e-9) {
		tmp = fma(t_1, (1.0 / (1.0 - log(pow(exp(tan(eps)), tan(x))))), t_0) + fma(-1.0, tan(x), tan(x));
	} else if (eps <= 5.2e-11) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = t_0 - (t_1 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -5.1e-9)
		tmp = Float64(fma(t_1, Float64(1.0 / Float64(1.0 - log((exp(tan(eps)) ^ tan(x))))), t_0) + fma(-1.0, tan(x), tan(x)));
	elseif (eps <= 5.2e-11)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(t_0 - Float64(t_1 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -5.1e-9], N[(N[(t$95$1 * N[(1.0 / N[(1.0 - N[Log[N[Power[N[Exp[N[Tan[eps], $MachinePrecision]], $MachinePrecision], N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + N[(-1.0 * N[Tan[x], $MachinePrecision] + N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-11], 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[(t$95$0 - N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.10000000000000017e-9

   1. Initial program 51.5%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. *-commutative 98.9%

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

      5. prod-diff 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. metadata-eval 98.9%

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

      8. *-un-lft-identity 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. add-log-exp 99.0%

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

      2. *-commutative 99.0%

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

      3. exp-prod 99.1%

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

   6. Applied egg-rr 99.1%

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

   if -5.10000000000000017e-9 < eps < 5.2000000000000001e-11

   1. Initial program 28.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

   if 5.2000000000000001e-11 < eps

   1. Initial program 52.2%

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

      1. log1p-expm1-u 50.6%

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

   4. Applied egg-rr 50.6%

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

      1. log1p-expm1-u 52.2%

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

      2. tan-sum 99.4%

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

      3. frac-2neg 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. distribute-lft-neg-in 99.4%

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

      8. add-sqr-sqrt 53.7%

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

      9. sqrt-unprod 78.2%

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

      10. sqr-neg 78.2%

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

      11. sqrt-unprod 24.4%

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

      12. add-sqr-sqrt 54.8%

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

      13. distribute-lft-neg-in 54.8%

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

      14. add-sqr-sqrt 30.4%

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

      15. sqrt-unprod 76.0%

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

      16. sqr-neg 76.0%

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

      17. sqrt-unprod 45.5%

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

      18. add-sqr-sqrt 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. neg-sub0 99.4%

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

      2. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. neg-sub0 99.4%

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

      5. +-commutative 99.4%

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

      6. fma-def 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \log \left({\left(e^{\tan \varepsilon}\right)}^{\tan x}\right)}, -\tan x\right) + \mathsf{fma}\left(-1, \tan x, \tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.5e-9) (not (<= eps 5.2e-11)))
   (- (- (tan x)) (/ (+ (tan x) (tan eps)) (fma (tan x) (tan eps) -1.0)))
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.5e-9) || !(eps <= 5.2e-11)) {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0));
	} 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 <= -4.5e-9) || !(eps <= 5.2e-11))
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)));
	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, -4.5e-9], N[Not[LessEqual[eps, 5.2e-11]], $MachinePrecision]], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $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 < -4.49999999999999976e-9 or 5.2000000000000001e-11 < eps

   1. Initial program 51.9%

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

      1. log1p-expm1-u 50.3%

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

   4. Applied egg-rr 50.3%

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

      1. log1p-expm1-u 51.9%

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

      2. tan-sum 99.1%

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

      3. frac-2neg 99.1%

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

      4. sub-neg 99.1%

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

      5. distribute-neg-in 99.1%

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

      6. metadata-eval 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. add-sqr-sqrt 54.0%

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

      9. sqrt-unprod 79.4%

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

      10. sqr-neg 79.4%

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

      11. sqrt-unprod 25.4%

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

      12. add-sqr-sqrt 54.8%

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

      13. distribute-lft-neg-in 54.8%

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

      14. add-sqr-sqrt 29.4%

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

      15. sqrt-unprod 74.5%

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

      16. sqr-neg 74.5%

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

      17. sqrt-unprod 45.0%

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

      18. add-sqr-sqrt 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. neg-sub0 99.1%

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

      2. +-commutative 99.1%

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

      3. associate--r+ 99.1%

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

      4. neg-sub0 99.1%

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

      5. +-commutative 99.1%

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

      6. fma-def 99.3%

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

   8. Simplified 99.3%

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

   if -4.49999999999999976e-9 < eps < 5.2000000000000001e-11

   1. Initial program 28.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan \varepsilon, \tan x, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -6.2e-9)
     (- (* t_0 (/ -1.0 (fma (tan eps) (tan x) -1.0))) (tan x))
     (if (<= eps 5.2e-11)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -6.2e-9) {
		tmp = (t_0 * (-1.0 / fma(tan(eps), tan(x), -1.0))) - tan(x);
	} else if (eps <= 5.2e-11) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -6.2e-9)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / fma(tan(eps), tan(x), -1.0))) - tan(x));
	elseif (eps <= 5.2e-11)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.2e-9], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[eps], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-11], 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[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.2000000000000001e-9

   1. Initial program 51.5%

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

      1. log1p-expm1-u 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. log1p-expm1-u 51.5%

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

      2. tan-sum 98.9%

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

      3. div-inv 98.9%

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

      4. flip-+ 98.7%

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

      5. frac-2neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. frac-times 98.7%

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

      8. pow2 98.7%

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

      9. pow2 98.7%

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

      10. sub-neg 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. metadata-eval 98.7%

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

      13. distribute-lft-neg-in 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. *-commutative 98.7%

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

      3. times-frac 98.7%

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

      4. +-commutative 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.8%

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

      7. unpow2 98.8%

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

      8. unpow2 98.8%

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

      9. flip-+ 99.0%

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

      10. +-commutative 99.0%

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

   8. Applied egg-rr 99.0%

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

   if -6.2000000000000001e-9 < eps < 5.2000000000000001e-11

   1. Initial program 28.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

   if 5.2000000000000001e-11 < eps

   1. Initial program 52.2%

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

      1. log1p-expm1-u 50.6%

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

   4. Applied egg-rr 50.6%

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

      1. log1p-expm1-u 52.2%

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

      2. tan-sum 99.4%

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

      3. frac-2neg 99.4%

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

      4. sub-neg 99.4%

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

      5. distribute-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. distribute-lft-neg-in 99.4%

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

      8. add-sqr-sqrt 53.7%

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

      9. sqrt-unprod 78.2%

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

      10. sqr-neg 78.2%

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

      11. sqrt-unprod 24.4%

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

      12. add-sqr-sqrt 54.8%

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

      13. distribute-lft-neg-in 54.8%

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

      14. add-sqr-sqrt 30.4%

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

      15. sqrt-unprod 76.0%

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

      16. sqr-neg 76.0%

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

      17. sqrt-unprod 45.5%

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

      18. add-sqr-sqrt 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. neg-sub0 99.4%

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

      2. +-commutative 99.4%

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

      3. associate--r+ 99.4%

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

      4. neg-sub0 99.4%

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

      5. +-commutative 99.4%

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

      6. fma-def 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan \varepsilon, \tan x, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -2.6e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 5.2e-11)
       (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (- (* t_0 (/ 1.0 t_1)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -2.6e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 5.2e-11) {
		tmp = eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    t_1 = 1.0d0 - (tan(x) * tan(eps))
    if (eps <= (-2.6d-9)) then
        tmp = (t_0 / t_1) - tan(x)
    else if (eps <= 5.2d-11) then
        tmp = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
    else
        tmp = (t_0 * (1.0d0 / t_1)) - tan(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) + Math.tan(eps);
	double t_1 = 1.0 - (Math.tan(x) * Math.tan(eps));
	double tmp;
	if (eps <= -2.6e-9) {
		tmp = (t_0 / t_1) - Math.tan(x);
	} else if (eps <= 5.2e-11) {
		tmp = eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)));
	} else {
		tmp = (t_0 * (1.0 / t_1)) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	t_1 = 1.0 - (math.tan(x) * math.tan(eps))
	tmp = 0
	if eps <= -2.6e-9:
		tmp = (t_0 / t_1) - math.tan(x)
	elif eps <= 5.2e-11:
		tmp = eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))
	else:
		tmp = (t_0 * (1.0 / t_1)) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -2.6e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 5.2e-11)
		tmp = Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_1)) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	t_1 = 1.0 - (tan(x) * tan(eps));
	tmp = 0.0;
	if (eps <= -2.6e-9)
		tmp = (t_0 / t_1) - tan(x);
	elseif (eps <= 5.2e-11)
		tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
	else
		tmp = (t_0 * (1.0 / t_1)) - tan(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.6e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-11], 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 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.6000000000000001e-9

   1. Initial program 51.5%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. prod-diff 98.9%

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

      5. *-commutative 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. *-un-lft-identity 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. +-commutative 98.9%

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

      2. fma-udef 98.9%

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

      3. associate-+r+ 98.9%

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

      4. unsub-neg 98.9%

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

   6. Simplified 98.9%

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

   if -2.6000000000000001e-9 < eps < 5.2000000000000001e-11

   1. Initial program 28.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

   if 5.2000000000000001e-11 < eps

   1. Initial program 52.2%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.3e-9) (not (<= eps 5.2e-11)))
   (- (/ (+ (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 <= -4.3e-9) || !(eps <= 5.2e-11)) {
		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 <= (-4.3d-9)) .or. (.not. (eps <= 5.2d-11))) 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 <= -4.3e-9) || !(eps <= 5.2e-11)) {
		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 <= -4.3e-9) or not (eps <= 5.2e-11):
		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 <= -4.3e-9) || !(eps <= 5.2e-11))
		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 <= -4.3e-9) || ~((eps <= 5.2e-11)))
		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, -4.3e-9], N[Not[LessEqual[eps, 5.2e-11]], $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 < -4.29999999999999963e-9 or 5.2000000000000001e-11 < eps

   1. Initial program 51.9%

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

      1. tan-sum 99.1%

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

      2. div-inv 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.2%

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

      5. *-commutative 99.2%

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

      6. *-un-lft-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. *-un-lft-identity 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-udef 99.1%

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

      3. associate-+r+ 99.1%

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

      4. unsub-neg 99.1%

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

   6. Simplified 99.1%

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

   if -4.29999999999999963e-9 < eps < 5.2000000000000001e-11

   1. Initial program 28.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-lft-identity 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.3 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.6e-5) (not (<= eps 5.2e-11)))
   (tan eps)
   (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.6e-5) || !(eps <= 5.2e-11)) {
		tmp = tan(eps);
	} 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 <= (-1.6d-5)) .or. (.not. (eps <= 5.2d-11))) then
        tmp = tan(eps)
    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 <= -1.6e-5) || !(eps <= 5.2e-11)) {
		tmp = Math.tan(eps);
	} 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 <= -1.6e-5) or not (eps <= 5.2e-11):
		tmp = math.tan(eps)
	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 <= -1.6e-5) || !(eps <= 5.2e-11))
		tmp = tan(eps);
	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 <= -1.6e-5) || ~((eps <= 5.2e-11)))
		tmp = tan(eps);
	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, -1.6e-5], N[Not[LessEqual[eps, 5.2e-11]], $MachinePrecision]], N[Tan[eps], $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 < -1.59999999999999993e-5 or 5.2000000000000001e-11 < eps

   1. Initial program 51.9%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. tan-quot 54.6%

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

      2. expm1-log1p-u 41.5%

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

      3. expm1-udef 40.2%

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

   5. Applied egg-rr 40.2%

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

      1. expm1-def 41.5%

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

      2. expm1-log1p 54.6%

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

   7. Simplified 54.6%

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

   if -1.59999999999999993e-5 < eps < 5.2000000000000001e-11

   1. Initial program 28.9%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0 99.3%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
   4. 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%

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

   5. 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 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 57.6%

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

   1. tan-quot 57.7%

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

   2. expm1-log1p-u 50.7%

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

   3. expm1-udef 24.5%

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

5. Applied egg-rr 24.5%

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

   1. expm1-def 50.7%

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

   2. expm1-log1p 57.7%

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

7. Simplified 57.7%

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

8. Final simplification 57.7%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 9: 31.5% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ 1.0 (+ (* eps -0.3333333333333333) (/ 1.0 eps))))

C

double code(double x, double eps) {
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 / ((eps * (-0.3333333333333333d0)) + (1.0d0 / eps))
end function

Java

public static double code(double x, double eps) {
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
}

Python

def code(x, eps):
	return 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps))

Julia

function code(x, eps)
	return Float64(1.0 / Float64(Float64(eps * -0.3333333333333333) + Float64(1.0 / eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 / ((eps * -0.3333333333333333) + (1.0 / eps));
end

Wolfram

code[x_, eps_] := N[(1.0 / N[(N[(eps * -0.3333333333333333), $MachinePrecision] + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 57.6%

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

   1. tan-quot 57.7%

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

   2. add-sqr-sqrt 29.3%

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

   3. pow2 29.3%

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

5. Applied egg-rr 29.3%

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

   1. unpow2 29.3%

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

   2. add-sqr-sqrt 57.7%

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

   3. tan-quot 57.6%

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

   4. clear-num 57.5%

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

   5. clear-num 57.5%

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

   6. tan-quot 57.6%

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

7. Applied egg-rr 57.6%

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

8. Taylor expanded in eps around 0 32.1%

   \[\leadsto \frac{1}{\color{blue}{-0.3333333333333333 \cdot \varepsilon + \frac{1}{\varepsilon}}} \]

9. Final simplification 32.1%

   \[\leadsto \frac{1}{\varepsilon \cdot -0.3333333333333333 + \frac{1}{\varepsilon}} \]
10. Add Preprocessing

Alternative 10: 31.1% 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 41.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 57.6%

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

4. Taylor expanded in eps around 0 31.3%

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

5. Final simplification 31.3%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 76.6% 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 2024024 
(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)))