2tan (problem 3.3.2)

Percentage Accurate: 41.8% → 99.5%

Time: 23.9s

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: 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

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2e-7)
     (- (* t_0 (/ -1.0 (fma (tan x) (tan eps) -1.0))) (tan x))
     (if (<= eps 3e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
       (-
        (/ t_0 (- 1.0 (/ (sin x) (* (cos x) (/ 1.0 (tan eps))))))
        (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -2e-7) {
		tmp = (t_0 * (-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x);
	} else if (eps <= 3e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	} else {
		tmp = (t_0 / (1.0 - (sin(x) / (cos(x) * (1.0 / tan(eps)))))) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -2e-7)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x));
	elseif (eps <= 3e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(sin(x) / Float64(cos(x) * Float64(1.0 / tan(eps)))))) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2e-7], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-7], 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[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Sin[x], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.9999999999999999e-7

   1. Initial program 55.7%

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

      1. log1p-expm1-u 54.8%

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

   3. Applied egg-rr 54.8%

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

      1. log1p-expm1-u 55.7%

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

      2. tan-sum 99.5%

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

      3. div-inv 99.5%

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

      4. frac-2neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-lft-neg-in 99.5%

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

      11. add-sqr-sqrt 56.7%

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

      12. sqrt-unprod 82.5%

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

      13. sqr-neg 82.5%

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

      14. sqrt-unprod 25.8%

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

      15. add-sqr-sqrt 58.4%

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

      16. distribute-lft-neg-in 58.4%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. remove-double-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. frac-2neg 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -1.9999999999999999e-7 < eps < 2.9999999999999999e-7

   1. Initial program 31.2%

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

      1. tan-sum 32.1%

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

      2. div-inv 32.1%

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

      3. fma-neg 32.1%

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

   3. Applied egg-rr 32.1%

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

      1. fma-neg 32.1%

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

      2. associate-*r/ 32.1%

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

      3. *-rgt-identity 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

      4. metadata-eval 99.5%

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

      5. *-lft-identity 99.5%

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

      6. distribute-lft-in 99.6%

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

      7. *-rgt-identity 99.6%

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

   8. Simplified 99.6%

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

   if 2.9999999999999999e-7 < eps

   1. Initial program 53.1%

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

      1. tan-sum 99.4%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.5%

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

   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 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. tan-quot 99.4%

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

      3. clear-num 99.4%

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

      4. tan-quot 99.4%

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

      5. frac-times 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. clear-num 99.5%

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

      8. tan-quot 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4e-9)
     (- (* t_0 (/ -1.0 (fma (tan x) (tan eps) -1.0))) (tan x))
     (if (<= eps 1.9e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4e-9) {
		tmp = (t_0 * (-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x);
	} else if (eps <= 1.9e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4e-9)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / fma(tan(x), tan(eps), -1.0))) - tan(x));
	elseif (eps <= 1.9e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4e-9], N[(N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.9e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.00000000000000025e-9

   1. Initial program 55.7%

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

      1. log1p-expm1-u 54.8%

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

   3. Applied egg-rr 54.8%

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

      1. log1p-expm1-u 55.7%

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

      2. tan-sum 99.5%

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

      3. div-inv 99.5%

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

      4. frac-2neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-lft-neg-in 99.5%

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

      11. add-sqr-sqrt 56.7%

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

      12. sqrt-unprod 82.5%

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

      13. sqr-neg 82.5%

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

      14. sqrt-unprod 25.8%

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

      15. add-sqr-sqrt 58.4%

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

      16. distribute-lft-neg-in 58.4%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. remove-double-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. frac-2neg 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -4.00000000000000025e-9 < eps < 1.90000000000000006e-9

   1. Initial program 31.4%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.6%

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

      5. frac-times 55.6%

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

      6. tan-quot 55.6%

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

      7. tan-quot 55.6%

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

      8. pow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 1.90000000000000006e-9 < eps

   1. Initial program 52.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. 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. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5e-9) (not (<= eps 5e-9)))
   (- (- (tan x)) (/ (+ (tan x) (tan eps)) (fma (tan x) (tan eps) -1.0)))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5e-9) || !(eps <= 5e-9)) {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0));
	} else {
		tmp = eps + (eps * pow(tan(x), 2.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5e-9) || !(eps <= 5e-9))
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / fma(tan(x), tan(eps), -1.0)));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5e-9], N[Not[LessEqual[eps, 5e-9]], $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[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5.0000000000000001e-9 or 5.0000000000000001e-9 < eps

   1. Initial program 54.3%

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

      1. log1p-expm1-u 52.9%

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

   3. Applied egg-rr 52.9%

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

      1. log1p-expm1-u 54.3%

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

      2. tan-sum 99.2%

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

      3. div-inv 99.2%

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

      4. frac-2neg 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-*r/ 99.2%

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

      7. sub-neg 99.2%

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

      8. distribute-neg-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. distribute-lft-neg-in 99.2%

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

      11. add-sqr-sqrt 53.8%

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

      12. sqrt-unprod 82.2%

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

      13. sqr-neg 82.2%

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

      14. sqrt-unprod 28.4%

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

      15. add-sqr-sqrt 57.6%

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

      16. distribute-lft-neg-in 57.6%

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

   5. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. neg-mul-1 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.3%

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

   7. Simplified 99.3%

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

   if -5.0000000000000001e-9 < eps < 5.0000000000000001e-9

   1. Initial program 31.4%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.6%

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

      5. frac-times 55.6%

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

      6. tan-quot 55.6%

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

      7. tan-quot 55.6%

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

      8. pow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)\\ \mathbf{if}\;\varepsilon \leq -3.9 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{-1}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (fma (tan x) (tan eps) -1.0)))
   (if (<= eps -3.9e-9)
     (- (* t_0 (/ -1.0 t_1)) (tan x))
     (if (<= eps 3e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (- (tan x)) (/ t_0 t_1))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = fma(tan(x), tan(eps), -1.0);
	double tmp;
	if (eps <= -3.9e-9) {
		tmp = (t_0 * (-1.0 / t_1)) - tan(x);
	} else if (eps <= 3e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = -tan(x) - (t_0 / t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = fma(tan(x), tan(eps), -1.0)
	tmp = 0.0
	if (eps <= -3.9e-9)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / t_1)) - tan(x));
	elseif (eps <= 3e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / t_1));
	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[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -3.9e-9], N[(N[(t$95$0 * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.9000000000000002e-9

   1. Initial program 55.7%

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

      1. log1p-expm1-u 54.8%

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

   3. Applied egg-rr 54.8%

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

      1. log1p-expm1-u 55.7%

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

      2. tan-sum 99.5%

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

      3. div-inv 99.5%

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

      4. frac-2neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-lft-neg-in 99.5%

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

      11. add-sqr-sqrt 56.7%

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

      12. sqrt-unprod 82.5%

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

      13. sqr-neg 82.5%

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

      14. sqrt-unprod 25.8%

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

      15. add-sqr-sqrt 58.4%

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

      16. distribute-lft-neg-in 58.4%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

      3. remove-double-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. frac-2neg 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -3.9000000000000002e-9 < eps < 2.99999999999999998e-9

   1. Initial program 31.4%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.6%

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

      5. frac-times 55.6%

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

      6. tan-quot 55.6%

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

      7. tan-quot 55.6%

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

      8. pow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 2.99999999999999998e-9 < eps

   1. Initial program 52.3%

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

      1. log1p-expm1-u 50.1%

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

   3. Applied egg-rr 50.1%

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

      1. log1p-expm1-u 52.3%

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

      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. frac-2neg 98.9%

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

      5. metadata-eval 98.9%

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

      6. associate-*r/ 98.9%

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

      7. sub-neg 98.9%

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

      8. distribute-neg-in 98.9%

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

      9. metadata-eval 98.9%

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

      10. distribute-lft-neg-in 98.9%

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

      11. add-sqr-sqrt 49.8%

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

      12. sqrt-unprod 81.8%

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

      13. sqr-neg 81.8%

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

      14. sqrt-unprod 32.0%

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

      15. add-sqr-sqrt 56.5%

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

      16. distribute-lft-neg-in 56.5%

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

   5. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. neg-mul-1 98.9%

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

      3. +-commutative 98.9%

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

      4. fma-def 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.9 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.1e-9)
     (- (/ t_0 (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (if (<= eps 6e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -3.1e-9) {
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 6e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) + tan(eps)
    if (eps <= (-3.1d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else if (eps <= 6d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - 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 tmp;
	if (eps <= -3.1e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else if (eps <= 6e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.tan(x) + math.tan(eps)
	tmp = 0
	if eps <= -3.1e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	elif eps <= 6e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -3.1e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	elseif (eps <= 6e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = tan(x) + tan(eps);
	tmp = 0.0;
	if (eps <= -3.1e-9)
		tmp = (t_0 / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	elseif (eps <= 6e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - 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]}, If[LessEqual[eps, -3.1e-9], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.10000000000000005e-9

   1. Initial program 55.7%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)} \]
   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 -3.10000000000000005e-9 < eps < 5.99999999999999996e-9

   1. Initial program 31.4%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.6%

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

      5. frac-times 55.6%

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

      6. tan-quot 55.6%

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

      7. tan-quot 55.6%

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

      8. pow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

   if 5.99999999999999996e-9 < eps

   1. Initial program 52.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   2. 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. fma-neg 99.0%

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

   3. Applied egg-rr 99.0%

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

      1. fma-neg 98.9%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

   5. Simplified 98.9%

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

      1. tan-quot 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

      4. clear-num 98.9%

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

      5. tan-quot 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.1e-9) (not (<= eps 3.6e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (pow (tan x) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.1e-9) || !(eps <= 3.6e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * pow(tan(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 <= (-4.1d-9)) .or. (.not. (eps <= 3.6d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.1e-9) || !(eps <= 3.6e-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.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.1e-9) or not (eps <= 3.6e-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.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.1e-9) || !(eps <= 3.6e-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 * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.1e-9) || ~((eps <= 3.6e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * (tan(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.1e-9], N[Not[LessEqual[eps, 3.6e-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[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.1000000000000003e-9 or 3.6e-9 < eps

   1. Initial program 54.3%

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

      1. tan-sum 99.2%

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

      2. div-inv 99.2%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.2%

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

      2. associate-*r/ 99.2%

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

      3. *-rgt-identity 99.2%

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

   5. Simplified 99.2%

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

   if -4.1000000000000003e-9 < eps < 3.6e-9

   1. Initial program 31.4%

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

   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 99.4%

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

      5. *-rgt-identity 99.4%

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

   4. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.6%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.6%

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

      5. frac-times 55.6%

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

      6. tan-quot 55.6%

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

      7. tan-quot 55.6%

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

      8. pow2 55.6%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.6%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.1 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 7: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00033:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.3e-7)
   (tan eps)
   (if (<= eps 0.00033)
     (+ eps (* eps (pow (tan x) 2.0)))
     (- (- (tan x)) (/ (+ (tan x) (tan eps)) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.3e-7) {
		tmp = tan(eps);
	} else if (eps <= 0.00033) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = -tan(x) - ((tan(x) + tan(eps)) / -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 <= (-3.3d-7)) then
        tmp = tan(eps)
    else if (eps <= 0.00033d0) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = -tan(x) - ((tan(x) + tan(eps)) / (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.3e-7) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.00033) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = -Math.tan(x) - ((Math.tan(x) + Math.tan(eps)) / -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.3e-7:
		tmp = math.tan(eps)
	elif eps <= 0.00033:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = -math.tan(x) - ((math.tan(x) + math.tan(eps)) / -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.3e-7)
		tmp = tan(eps);
	elseif (eps <= 0.00033)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(-tan(x)) - Float64(Float64(tan(x) + tan(eps)) / -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.3e-7)
		tmp = tan(eps);
	elseif (eps <= 0.00033)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = -tan(x) - ((tan(x) + tan(eps)) / -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.3e-7], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.00033], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Tan[x], $MachinePrecision]) - N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.3000000000000002e-7

   1. Initial program 55.7%

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

   2. Taylor expanded in x around 0 58.4%

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

      1. tan-quot 58.6%

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

      2. expm1-log1p-u 39.7%

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

      3. expm1-udef 39.0%

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

   4. Applied egg-rr 39.0%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 58.6%

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

   6. Simplified 58.6%

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

   if -3.3000000000000002e-7 < eps < 3.3e-4

   1. Initial program 30.9%

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

   2. Taylor expanded in eps around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. *-lft-identity 98.5%

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

      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%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.5%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.5%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.5%

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

      5. frac-times 55.5%

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

      6. tan-quot 55.5%

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

      7. tan-quot 55.5%

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

      8. pow2 55.5%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.5%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 98.7%

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

   8. Simplified 98.7%

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

   if 3.3e-4 < eps

   1. Initial program 53.9%

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

      1. log1p-expm1-u 51.7%

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

   3. Applied egg-rr 51.7%

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

      1. log1p-expm1-u 53.9%

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

      2. tan-sum 99.5%

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

      3. div-inv 99.5%

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

      4. frac-2neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r/ 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-lft-neg-in 99.5%

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

      11. add-sqr-sqrt 49.8%

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

      12. sqrt-unprod 82.6%

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

      13. sqr-neg 82.6%

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

      14. sqrt-unprod 32.8%

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

      15. add-sqr-sqrt 57.7%

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

      16. distribute-lft-neg-in 57.7%

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

   5. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. neg-mul-1 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 58.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00033:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{-1}\\ \end{array} \]

Alternative 8: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00033:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.3e-7)
   (tan eps)
   (if (<= eps 0.00033) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.3e-7) {
		tmp = tan(eps);
	} else if (eps <= 0.00033) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-3.3d-7)) then
        tmp = tan(eps)
    else if (eps <= 0.00033d0) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.3e-7) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.00033) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.3e-7:
		tmp = math.tan(eps)
	elif eps <= 0.00033:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.3e-7)
		tmp = tan(eps);
	elseif (eps <= 0.00033)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.3e-7)
		tmp = tan(eps);
	elseif (eps <= 0.00033)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.3e-7], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.00033], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.3000000000000002e-7 or 3.3e-4 < eps

   1. Initial program 55.0%

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

   2. Taylor expanded in x around 0 58.4%

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

      1. tan-quot 58.6%

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

      2. expm1-log1p-u 40.2%

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

      3. expm1-udef 39.7%

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

   4. Applied egg-rr 39.7%

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

      1. expm1-def 40.2%

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

      2. expm1-log1p 58.6%

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

   6. Simplified 58.6%

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

   if -3.3000000000000002e-7 < eps < 3.3e-4

   1. Initial program 30.9%

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

   2. Taylor expanded in eps around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. *-lft-identity 98.5%

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

      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%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

      2. expm1-udef 55.5%

         \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} - 1\right)} \]

      3. unpow2 55.5%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}\right)} - 1\right) \]

      4. unpow2 55.5%

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

      5. frac-times 55.5%

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

      6. tan-quot 55.5%

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

      7. tan-quot 55.5%

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

      8. pow2 55.5%

         \[\leadsto \varepsilon + \left(e^{\mathsf{log1p}\left(\varepsilon \cdot \color{blue}{{\tan x}^{2}}\right)} - 1\right) \]

   6. Applied egg-rr 55.5%

      \[\leadsto \varepsilon + \color{blue}{\left(e^{\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \varepsilon + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot {\tan x}^{2}\right)\right)} \]

      2. expm1-log1p 98.7%

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

   8. Simplified 98.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00033:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 9: 58.2% 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 44.6%

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

2. Taylor expanded in x around 0 56.9%

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

   1. tan-quot 57.1%

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

   2. expm1-log1p-u 46.5%

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

   3. expm1-udef 25.3%

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

4. Applied egg-rr 25.3%

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

   1. expm1-def 46.5%

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

   2. expm1-log1p 57.1%

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

6. Simplified 57.1%

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

7. Final simplification 57.1%

   \[\leadsto \tan \varepsilon \]

Alternative 10: 31.5% 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 44.6%

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

2. Taylor expanded in x around 0 56.9%

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

3. Taylor expanded in eps around 0 26.2%

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

4. Final simplification 26.2%

   \[\leadsto \varepsilon \]

Developer target: 76.3% 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 2023279 
(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)))