2tan (problem 3.3.2)

Percentage Accurate: 41.3% → 99.5%

Time: 16.2s

Alternatives: 9

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: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1 \cdot \left(\mathsf{fma}\left(\tan x, \tan \varepsilon, {t_0}^{2}\right) + 1\right)}{1 - {t_0}^{3}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 - t_0} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -4.4e-7)
     (-
      (/
       (* t_1 (+ (fma (tan x) (tan eps) (pow t_0 2.0)) 1.0))
       (- 1.0 (pow t_0 3.0)))
      (tan x))
     (if (<= eps 2.9e-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_1 (- 1.0 t_0)) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -4.4e-7) {
		tmp = ((t_1 * (fma(tan(x), tan(eps), pow(t_0, 2.0)) + 1.0)) / (1.0 - pow(t_0, 3.0))) - tan(x);
	} else if (eps <= 2.9e-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_1 / (1.0 - t_0)) - tan(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -4.4e-7)
		tmp = Float64(Float64(Float64(t_1 * Float64(fma(tan(x), tan(eps), (t_0 ^ 2.0)) + 1.0)) / Float64(1.0 - (t_0 ^ 3.0))) - tan(x));
	elseif (eps <= 2.9e-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_1 / Float64(1.0 - t_0)) - tan(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.4e-7], N[(N[(N[(t$95$1 * N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.9e-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$1 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.4000000000000002e-7

   1. Initial program 57.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      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%

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

      1. flip3-- 99.4%

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

      2. associate-/r/ 99.3%

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

      3. metadata-eval 99.3%

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

      4. metadata-eval 99.3%

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

      5. *-un-lft-identity 99.3%

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

      6. +-commutative 99.3%

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

      7. pow2 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. fma-def 99.5%

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

   9. Simplified 99.5%

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

   if -4.4000000000000002e-7 < eps < 2.8999999999999998e-7

   1. Initial program 30.3%

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

      1. tan-sum 31.6%

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

      2. div-inv 31.6%

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

      3. fma-neg 31.6%

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

   3. Applied egg-rr 31.6%

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

      1. fma-neg 31.6%

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

      2. associate-*r/ 31.6%

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

      3. *-rgt-identity 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. tan-quot 31.6%

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

      3. associate-*r/ 31.6%

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

   7. Applied egg-rr 31.6%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   10. Simplified 99.6%

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

   if 2.8999999999999998e-7 < eps

   1. Initial program 53.1%

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

      1. tan-sum 99.6%

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

      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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.6%

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

      3. *-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(\mathsf{fma}\left(\tan x, \tan \varepsilon, {\left(\tan x \cdot \tan \varepsilon\right)}^{2}\right) + 1\right)}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.6e-7) (not (<= eps 3.8e-7)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+
    (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0))
    (*
     (* eps eps)
     (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-7) || !(eps <= 3.8e-7)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = (eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.6d-7)) .or. (.not. (eps <= 3.8d-7))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = (eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + 1.0d0)) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.6e-7) || !(eps <= 3.8e-7)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = (eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + 1.0)) + ((eps * eps) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.6e-7) or not (eps <= 3.8e-7):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = (eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + 1.0)) + ((eps * eps) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.6e-7) || !(eps <= 3.8e-7))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.6e-7) || ~((eps <= 3.8e-7)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = (eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0)) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.6e-7], N[Not[LessEqual[eps, 3.8e-7]], $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[(N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 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]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.59999999999999994e-7 or 3.80000000000000015e-7 < eps

   1. Initial program 55.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      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 -3.59999999999999994e-7 < eps < 3.80000000000000015e-7

   1. Initial program 30.3%

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

      1. tan-sum 31.6%

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

      2. div-inv 31.6%

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

      3. fma-neg 31.6%

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

   3. Applied egg-rr 31.6%

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

      1. fma-neg 31.6%

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

      2. associate-*r/ 31.6%

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

      3. *-rgt-identity 31.6%

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

   5. Simplified 31.6%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

      3. sub-neg 99.6%

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

      4. mul-1-neg 99.6%

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

      5. remove-double-neg 99.6%

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

      6. mul-1-neg 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.1e-7) (not (<= eps 3.8e-7)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+
    (+ 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)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-7) || !(eps <= 3.8e-7)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.1d-7)) .or. (.not. (eps <= 3.8d-7))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = (eps + (eps * ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.1e-7) || !(eps <= 3.8e-7)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = (eps + (eps * (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + ((eps * eps) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.1e-7) or not (eps <= 3.8e-7):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = (eps + (eps * (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + ((eps * eps) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.1e-7) || !(eps <= 3.8e-7))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.1e-7) || ~((eps <= 3.8e-7)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = (eps + (eps * ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps * eps) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.1e-7], N[Not[LessEqual[eps, 3.8e-7]], $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[(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]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.1e-7 or 3.80000000000000015e-7 < eps

   1. Initial program 55.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      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 -3.1e-7 < eps < 3.80000000000000015e-7

   1. Initial program 30.3%

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

      1. tan-sum 31.6%

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

      2. div-inv 31.6%

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

      3. fma-neg 31.6%

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

   3. Applied egg-rr 31.6%

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

      1. fma-neg 31.6%

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

      2. associate-*r/ 31.6%

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

      3. *-rgt-identity 31.6%

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

   5. Simplified 31.6%

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

      1. *-commutative 31.6%

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

      2. tan-quot 31.6%

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

      3. associate-*r/ 31.6%

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

   7. Applied egg-rr 31.6%

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

   8. Taylor expanded in eps around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. unsub-neg 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 3.8 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\\ \end{array} \]

Alternative 4: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \mathbf{if}\;\varepsilon \leq -135000000 \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\sin x}{\cos x} \cdot t_0} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin eps) (cos eps))))
   (if (or (<= eps -135000000.0) (not (<= eps 5.2e-9)))
     (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
     (+
      (/ t_0 (- 1.0 (* (/ (sin x) (cos x)) t_0)))
      (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) / cos(eps);
	double tmp;
	if ((eps <= -135000000.0) || !(eps <= 5.2e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = (t_0 / (1.0 - ((sin(x) / cos(x)) * t_0))) + ((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(eps) / cos(eps)
    if ((eps <= (-135000000.0d0)) .or. (.not. (eps <= 5.2d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = (t_0 / (1.0d0 - ((sin(x) / cos(x)) * t_0))) + ((eps * (sin(x) ** 2.0d0)) / (cos(x) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(eps) / Math.cos(eps);
	double tmp;
	if ((eps <= -135000000.0) || !(eps <= 5.2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = (t_0 / (1.0 - ((Math.sin(x) / Math.cos(x)) * t_0))) + ((eps * Math.pow(Math.sin(x), 2.0)) / Math.pow(Math.cos(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin(eps) / math.cos(eps)
	tmp = 0
	if (eps <= -135000000.0) or not (eps <= 5.2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = (t_0 / (1.0 - ((math.sin(x) / math.cos(x)) * t_0))) + ((eps * math.pow(math.sin(x), 2.0)) / math.pow(math.cos(x), 2.0))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) / cos(eps))
	tmp = 0.0
	if ((eps <= -135000000.0) || !(eps <= 5.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(Float64(sin(x) / cos(x)) * t_0))) + Float64(Float64(eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin(eps) / cos(eps);
	tmp = 0.0;
	if ((eps <= -135000000.0) || ~((eps <= 5.2e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = (t_0 / (1.0 - ((sin(x) / cos(x)) * t_0))) + ((eps * (sin(x) ^ 2.0)) / (cos(x) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eps, -135000000.0], N[Not[LessEqual[eps, 5.2e-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[(N[(t$95$0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.35e8 or 5.2000000000000002e-9 < eps

   1. Initial program 54.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

      3. fma-neg 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.5%

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

      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 -1.35e8 < eps < 5.2000000000000002e-9

   1. Initial program 30.8%

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

      1. tan-sum 32.2%

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

      2. div-inv 32.2%

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

      3. fma-neg 32.2%

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

   3. Applied egg-rr 32.2%

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

      1. fma-neg 32.2%

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

      2. associate-*r/ 32.2%

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

      3. *-rgt-identity 32.2%

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

   5. Simplified 32.2%

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

   6. Taylor expanded in x around inf 32.2%

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

      1. associate--l+ 54.5%

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

   8. Simplified 54.5%

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

   9. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -135000000 \lor \neg \left(\varepsilon \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{4}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.5e-9) (not (<= eps 4.5e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (+ eps (* eps (sqrt (pow (tan x) 4.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.5e-9) || !(eps <= 4.5e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = eps + (eps * sqrt(pow(tan(x), 4.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-2.5d-9)) .or. (.not. (eps <= 4.5d-9))) then
        tmp = ((tan(x) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
    else
        tmp = eps + (eps * sqrt((tan(x) ** 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.5e-9) || !(eps <= 4.5e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) / (1.0 - (Math.tan(x) * Math.tan(eps)))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.sqrt(Math.pow(Math.tan(x), 4.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.5e-9) or not (eps <= 4.5e-9):
		tmp = ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)
	else:
		tmp = eps + (eps * math.sqrt(math.pow(math.tan(x), 4.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.5e-9) || !(eps <= 4.5e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * sqrt((tan(x) ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -2.5e-9) || ~((eps <= 4.5e-9)))
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	else
		tmp = eps + (eps * sqrt((tan(x) ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.5e-9], N[Not[LessEqual[eps, 4.5e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Sqrt[N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.5000000000000001e-9 or 4.49999999999999976e-9 < eps

   1. Initial program 55.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. fma-neg 99.5%

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

      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 -2.5000000000000001e-9 < eps < 4.49999999999999976e-9

   1. Initial program 30.3%

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

   2. Taylor expanded in eps around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. *-lft-identity 98.9%

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

      4. distribute-lft-in 98.9%

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

      5. *-rgt-identity 98.9%

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

   4. Simplified 98.9%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 98.9%

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

      3. pow2 98.9%

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

      4. unpow2 98.9%

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

      5. unpow2 98.9%

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

      6. frac-times 98.9%

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

      7. pow2 98.9%

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

      8. tan-quot 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. unpow2 99.0%

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

      2. pow-sqr 99.1%

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

      3. metadata-eval 99.1%

         \[\leadsto \varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{\color{blue}{4}}} \]

   8. Simplified 99.1%

      \[\leadsto \varepsilon + \varepsilon \cdot \color{blue}{\sqrt{{\tan x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.5 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{4}}\\ \end{array} \]

Alternative 6: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00074:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.6e-5)
   (tan eps)
   (if (<= eps 0.00074) (+ eps (* eps (sqrt (pow (tan x) 4.0)))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.6e-5) {
		tmp = tan(eps);
	} else if (eps <= 0.00074) {
		tmp = eps + (eps * sqrt(pow(tan(x), 4.0)));
	} 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 <= (-3.6d-5)) then
        tmp = tan(eps)
    else if (eps <= 0.00074d0) then
        tmp = eps + (eps * sqrt((tan(x) ** 4.0d0)))
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.6e-5) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.00074) {
		tmp = eps + (eps * Math.sqrt(Math.pow(Math.tan(x), 4.0)));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.6e-5:
		tmp = math.tan(eps)
	elif eps <= 0.00074:
		tmp = eps + (eps * math.sqrt(math.pow(math.tan(x), 4.0)))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.6e-5)
		tmp = tan(eps);
	elseif (eps <= 0.00074)
		tmp = Float64(eps + Float64(eps * sqrt((tan(x) ^ 4.0))));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.6e-5)
		tmp = tan(eps);
	elseif (eps <= 0.00074)
		tmp = eps + (eps * sqrt((tan(x) ^ 4.0)));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.6e-5], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.00074], N[(eps + N[(eps * N[Sqrt[N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.60000000000000009e-5 or 7.3999999999999999e-4 < eps

   1. Initial program 55.6%

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

   2. Taylor expanded in x around 0 59.7%

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

      1. tan-quot 59.9%

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

      2. expm1-log1p-u 47.4%

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

      3. expm1-udef 47.1%

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

   4. Applied egg-rr 47.1%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 59.9%

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

   6. Simplified 59.9%

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

   if -3.60000000000000009e-5 < eps < 7.3999999999999999e-4

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-lft-identity 98.4%

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

      4. distribute-lft-in 98.4%

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

      5. *-rgt-identity 98.4%

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

   4. Simplified 98.4%

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

      1. add-sqr-sqrt 98.4%

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

      2. sqrt-unprod 98.4%

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

      3. pow2 98.4%

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

      4. unpow2 98.4%

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

      5. unpow2 98.4%

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

      6. frac-times 98.4%

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

      7. pow2 98.4%

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

      8. tan-quot 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. unpow2 98.5%

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

      2. pow-sqr 98.6%

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

      3. metadata-eval 98.6%

         \[\leadsto \varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{\color{blue}{4}}} \]

   8. Simplified 98.6%

      \[\leadsto \varepsilon + \varepsilon \cdot \color{blue}{\sqrt{{\tan x}^{4}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00074:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \sqrt{{\tan x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00074:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.8e-6)
   (tan eps)
   (if (<= eps 0.00074) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.8e-6) {
		tmp = tan(eps);
	} else if (eps <= 0.00074) {
		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 <= (-2.8d-6)) then
        tmp = tan(eps)
    else if (eps <= 0.00074d0) 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 <= -2.8e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.00074) {
		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 <= -2.8e-6:
		tmp = math.tan(eps)
	elif eps <= 0.00074:
		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 <= -2.8e-6)
		tmp = tan(eps);
	elseif (eps <= 0.00074)
		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 <= -2.8e-6)
		tmp = tan(eps);
	elseif (eps <= 0.00074)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.8e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.00074], 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 < -2.79999999999999987e-6 or 7.3999999999999999e-4 < eps

   1. Initial program 55.6%

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

   2. Taylor expanded in x around 0 59.7%

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

      1. tan-quot 59.9%

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

      2. expm1-log1p-u 47.4%

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

      3. expm1-udef 47.1%

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

   4. Applied egg-rr 47.1%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 59.9%

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

   6. Simplified 59.9%

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

   if -2.79999999999999987e-6 < eps < 7.3999999999999999e-4

   1. Initial program 30.1%

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

   2. Taylor expanded in eps around 0 98.4%

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

      1. cancel-sign-sub-inv 98.4%

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

      2. metadata-eval 98.4%

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

      3. *-lft-identity 98.4%

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

      4. distribute-lft-in 98.4%

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

      5. *-rgt-identity 98.4%

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

   4. Simplified 98.4%

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

      1. expm1-log1p-u 98.4%

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

      2. expm1-udef 54.0%

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

      3. unpow2 54.0%

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

      4. unpow2 54.0%

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

      5. frac-times 54.0%

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

      6. pow2 54.0%

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

      7. tan-quot 54.0%

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

   6. Applied egg-rr 54.0%

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

      1. expm1-def 98.5%

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

      2. expm1-log1p 98.5%

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

   8. Simplified 98.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00074:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8: 57.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.8%

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

2. Taylor expanded in x around 0 56.4%

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

   1. tan-quot 56.5%

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

   2. expm1-log1p-u 50.3%

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

   3. expm1-udef 27.1%

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

4. Applied egg-rr 27.1%

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

   1. expm1-def 50.3%

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

   2. expm1-log1p 56.5%

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

6. Simplified 56.5%

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

7. Final simplification 56.5%

   \[\leadsto \tan \varepsilon \]

Alternative 9: 30.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 42.8%

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

2. Taylor expanded in x around 0 56.4%

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

3. Taylor expanded in eps around 0 28.0%

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

4. Final simplification 28.0%

   \[\leadsto \varepsilon \]

Developer target: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023222 
(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)))