2tan (problem 3.3.2)

Percentage Accurate: 42.8% → 99.3%

Time: 18.5s

Alternatives: 16

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.8% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \tan x \cdot \tan \varepsilon\\ t_1 := \tan x + \tan \varepsilon\\ t_2 := \frac{-1}{\tan x}\\ \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + t_1 \cdot t_2}{t_0 \cdot t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan x) (tan eps))))
        (t_1 (+ (tan x) (tan eps)))
        (t_2 (/ -1.0 (tan x))))
   (if (<= eps -2.15e-9)
     (- (- (tan x)) (/ t_1 (fma (tan x) (tan eps) -1.0)))
     (if (<= eps 5.2e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (/ (+ t_0 (* t_1 t_2)) (* t_0 t_2))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (tan(x) * tan(eps));
	double t_1 = tan(x) + tan(eps);
	double t_2 = -1.0 / tan(x);
	double tmp;
	if (eps <= -2.15e-9) {
		tmp = -tan(x) - (t_1 / fma(tan(x), tan(eps), -1.0));
	} else if (eps <= 5.2e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 + (t_1 * t_2)) / (t_0 * t_2);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	t_1 = Float64(tan(x) + tan(eps))
	t_2 = Float64(-1.0 / tan(x))
	tmp = 0.0
	if (eps <= -2.15e-9)
		tmp = Float64(Float64(-tan(x)) - Float64(t_1 / fma(tan(x), tan(eps), -1.0)));
	elseif (eps <= 5.2e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 + Float64(t_1 * t_2)) / Float64(t_0 * t_2));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -2.15e-9], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.2e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.14999999999999981e-9

   1. Initial program 44.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-lft-in 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.5%

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

      2. associate-*l/ 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -2.14999999999999981e-9 < eps < 5.2000000000000002e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 5.2000000000000002e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      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/ 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Simplified 99.3%

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

      1. frac-2neg 99.3%

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

      2. *-commutative 99.3%

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

      3. tan-quot 99.3%

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

      4. tan-quot 99.2%

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

      5. tan-quot 99.1%

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

      6. clear-num 99.1%

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

      7. frac-sub 99.1%

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.15 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \tan x \cdot \tan \varepsilon\right) + \left(\tan x + \tan \varepsilon\right) \cdot \frac{-1}{\tan x}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \frac{-1}{\tan x}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := \frac{\sin \varepsilon}{\cos \varepsilon}\\ \frac{t_1}{1 - t_1 \cdot \frac{\sin x}{\cos x}} - \frac{t_0}{\frac{t_0 + -1}{\tan x}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))) (t_1 (/ (sin eps) (cos eps))))
   (-
    (/ t_1 (- 1.0 (* t_1 (/ (sin x) (cos x)))))
    (/ t_0 (/ (+ t_0 -1.0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = sin(eps) / cos(eps);
	return (t_1 / (1.0 - (t_1 * (sin(x) / cos(x))))) - (t_0 / ((t_0 + -1.0) / tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = tan(x) * tan(eps)
    t_1 = sin(eps) / cos(eps)
    code = (t_1 / (1.0d0 - (t_1 * (sin(x) / cos(x))))) - (t_0 / ((t_0 + (-1.0d0)) / tan(x)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	double t_1 = Math.sin(eps) / Math.cos(eps);
	return (t_1 / (1.0 - (t_1 * (Math.sin(x) / Math.cos(x))))) - (t_0 / ((t_0 + -1.0) / Math.tan(x)));
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	t_1 = math.sin(eps) / math.cos(eps)
	return (t_1 / (1.0 - (t_1 * (math.sin(x) / math.cos(x))))) - (t_0 / ((t_0 + -1.0) / math.tan(x)))

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	t_1 = Float64(sin(eps) / cos(eps))
	return Float64(Float64(t_1 / Float64(1.0 - Float64(t_1 * Float64(sin(x) / cos(x))))) - Float64(t_0 / Float64(Float64(t_0 + -1.0) / tan(x))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) * tan(eps);
	t_1 = sin(eps) / cos(eps);
	tmp = (t_1 / (1.0 - (t_1 * (sin(x) / cos(x))))) - (t_0 / ((t_0 + -1.0) / tan(x)));
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[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 / N[(1.0 - N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 38.5%

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

   1. tan-sum 65.3%

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

   2. div-inv 65.3%

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

3. Applied egg-rr 65.3%

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

4. Taylor expanded in x around inf 65.1%

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

   1. associate--l+ 76.8%

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

   2. associate-/r* 76.8%

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

   3. *-commutative 76.8%

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

   4. times-frac 76.9%

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

6. Simplified 76.8%

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

   1. tan-quot 75.4%

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

   2. frac-2neg 75.4%

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

   3. clear-num 73.9%

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

   4. frac-sub 73.4%

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

8. Applied egg-rr 74.6%

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

   1. cancel-sign-sub 74.6%

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

   2. distribute-lft-neg-out 74.6%

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

   3. rgt-mult-inverse 76.9%

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

   4. metadata-eval 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. associate-+r- 99.5%

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

   7. metadata-eval 99.5%

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

   8. neg-sub0 99.5%

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

   9. distribute-rgt-neg-in 99.5%

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

   10. associate-*r/ 99.5%

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

   11. *-rgt-identity 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - t_0} - \frac{t_0}{\frac{t_0 + -1}{\tan x}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps))))
   (-
    (/ (/ (sin eps) (cos eps)) (- 1.0 t_0))
    (/ t_0 (/ (+ t_0 -1.0) (tan x))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	return ((sin(eps) / cos(eps)) / (1.0 - t_0)) - (t_0 / ((t_0 + -1.0) / tan(x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = tan(x) * tan(eps)
    code = ((sin(eps) / cos(eps)) / (1.0d0 - t_0)) - (t_0 / ((t_0 + (-1.0d0)) / tan(x)))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.tan(x) * Math.tan(eps);
	return ((Math.sin(eps) / Math.cos(eps)) / (1.0 - t_0)) - (t_0 / ((t_0 + -1.0) / Math.tan(x)));
}

Python

def code(x, eps):
	t_0 = math.tan(x) * math.tan(eps)
	return ((math.sin(eps) / math.cos(eps)) / (1.0 - t_0)) - (t_0 / ((t_0 + -1.0) / math.tan(x)))

Julia

function code(x, eps)
	t_0 = Float64(tan(x) * tan(eps))
	return Float64(Float64(Float64(sin(eps) / cos(eps)) / Float64(1.0 - t_0)) - Float64(t_0 / Float64(Float64(t_0 + -1.0) / tan(x))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) * tan(eps);
	tmp = ((sin(eps) / cos(eps)) / (1.0 - t_0)) - (t_0 / ((t_0 + -1.0) / tan(x)));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.5%

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

   1. tan-sum 65.3%

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

   2. div-inv 65.3%

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

3. Applied egg-rr 65.3%

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

4. Taylor expanded in x around inf 65.1%

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

   1. associate--l+ 76.8%

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

   2. associate-/r* 76.8%

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

   3. *-commutative 76.8%

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

   4. times-frac 76.9%

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

6. Simplified 76.8%

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

   1. tan-quot 75.4%

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

   2. frac-2neg 75.4%

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

   3. clear-num 73.9%

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

   4. frac-sub 73.4%

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

8. Applied egg-rr 74.6%

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

   1. cancel-sign-sub 74.6%

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

   2. distribute-lft-neg-out 74.6%

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

   3. rgt-mult-inverse 76.9%

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

   4. metadata-eval 76.9%

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

   5. *-rgt-identity 76.9%

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

   6. associate-+r- 99.5%

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

   7. metadata-eval 99.5%

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

   8. neg-sub0 99.5%

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

   9. distribute-rgt-neg-in 99.5%

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

   10. associate-*r/ 99.5%

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

   11. *-rgt-identity 99.5%

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

10. Simplified 99.5%

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

    1. tan-quot 99.4%

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

    2. tan-quot 99.4%

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

    3. *-commutative 99.4%

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

    4. tan-quot 99.4%

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

    5. clear-num 99.4%

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

    6. tan-quot 99.5%

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

    7. clear-num 99.5%

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

    8. frac-times 99.5%

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

    9. metadata-eval 99.5%

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

    10. clear-num 99.5%

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

    11. tan-quot 99.5%

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

    12. clear-num 99.5%

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

    13. tan-quot 99.4%

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

12. Applied egg-rr 99.4%

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

    1. associate-/r* 99.4%

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

    2. remove-double-div 99.4%

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

    3. associate-/r/ 99.4%

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

    4. /-rgt-identity 99.4%

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

14. Simplified 99.4%

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

15. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;t_0 - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -6.1e-9)
     (- t_0 (/ t_1 (fma (tan x) (tan eps) -1.0)))
     (if (<= eps 4.5e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (fma t_1 (/ 1.0 (- 1.0 (/ (* (sin x) (tan eps)) (cos x)))) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -6.1e-9) {
		tmp = t_0 - (t_1 / fma(tan(x), tan(eps), -1.0));
	} else if (eps <= 4.5e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = fma(t_1, (1.0 / (1.0 - ((sin(x) * tan(eps)) / cos(x)))), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -6.1e-9)
		tmp = Float64(t_0 - Float64(t_1 / fma(tan(x), tan(eps), -1.0)));
	elseif (eps <= 4.5e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = fma(t_1, Float64(1.0 / Float64(1.0 - Float64(Float64(sin(x) * tan(eps)) / cos(x)))), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -6.1e-9], N[(t$95$0 - N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -6.1e-9

   1. Initial program 44.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-lft-in 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.5%

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

      2. associate-*l/ 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -6.1e-9 < eps < 4.49999999999999976e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 4.49999999999999976e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      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. *-commutative 99.4%

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

      2. tan-quot 99.3%

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

      3. associate-*r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.1 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\sin x \cdot \tan \varepsilon}{\cos x}}, -\tan x\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;t_0 - \frac{t_1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -1.9e-9)
     (- t_0 (/ t_1 (fma (tan x) (tan eps) -1.0)))
     (if (<= eps 3.8e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (fma t_1 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -1.9e-9) {
		tmp = t_0 - (t_1 / fma(tan(x), tan(eps), -1.0));
	} else if (eps <= 3.8e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = fma(t_1, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(-tan(x))
	t_1 = Float64(tan(x) + tan(eps))
	tmp = 0.0
	if (eps <= -1.9e-9)
		tmp = Float64(t_0 - Float64(t_1 / fma(tan(x), tan(eps), -1.0)));
	elseif (eps <= 3.8e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = fma(t_1, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[Tan[x], $MachinePrecision])}, Block[{t$95$1 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.9e-9], N[(t$95$0 - N[(t$95$1 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.8e-9], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.90000000000000006e-9

   1. Initial program 44.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-lft-in 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.5%

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

      2. associate-*l/ 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -1.90000000000000006e-9 < eps < 3.80000000000000011e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 3.80000000000000011e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

      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%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{t_0}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.2e-9)
     (- (- (tan x)) (/ t_0 (fma (tan x) (tan eps) -1.0)))
     (if (<= eps 4.5e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (* 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 <= -3.2e-9) {
		tmp = -tan(x) - (t_0 / fma(tan(x), tan(eps), -1.0));
	} else if (eps <= 4.5e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (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 <= -3.2e-9)
		tmp = Float64(Float64(-tan(x)) - Float64(t_0 / fma(tan(x), tan(eps), -1.0)));
	elseif (eps <= 4.5e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(x) * 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, -3.2e-9], N[((-N[Tan[x], $MachinePrecision]) - N[(t$95$0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.5e-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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.20000000000000012e-9

   1. Initial program 44.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. distribute-lft-in 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. distribute-lft-in 99.5%

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

      2. associate-*l/ 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. unsub-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

   if -3.20000000000000012e-9 < eps < 4.49999999999999976e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 4.49999999999999976e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\left(-\tan x\right) - \frac{\tan x + \tan \varepsilon}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 7: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.45e-9) (not (<= eps 3.2e-9)))
   (- (* (+ (tan x) (tan eps)) (/ 1.0 (- 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 <= -3.45e-9) || !(eps <= 3.2e-9)) {
		tmp = ((tan(x) + tan(eps)) * (1.0 / (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 <= (-3.45d-9)) .or. (.not. (eps <= 3.2d-9))) then
        tmp = ((tan(x) + tan(eps)) * (1.0d0 / (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 <= -3.45e-9) || !(eps <= 3.2e-9)) {
		tmp = ((Math.tan(x) + Math.tan(eps)) * (1.0 / (1.0 - (Math.tan(x) * Math.tan(eps))))) - Math.tan(x);
	} else {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.45e-9) or not (eps <= 3.2e-9):
		tmp = ((math.tan(x) + math.tan(eps)) * (1.0 / (1.0 - (math.tan(x) * math.tan(eps))))) - math.tan(x)
	else:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.45e-9) || !(eps <= 3.2e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) * Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps))))) - tan(x));
	else
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.45e-9) || ~((eps <= 3.2e-9)))
		tmp = ((tan(x) + tan(eps)) * (1.0 / (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, -3.45e-9], N[Not[LessEqual[eps, 3.2e-9]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.44999999999999987e-9 or 3.20000000000000012e-9 < eps

   1. Initial program 48.3%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. Applied egg-rr 99.4%

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

   if -3.44999999999999987e-9 < eps < 3.20000000000000012e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.45 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 8: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -3.4e-9)
     (- (/ t_0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps))))) (tan x))
     (if (<= eps 5.2e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (* 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 <= -3.4e-9) {
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) * 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 <= (-3.4d-9)) then
        tmp = (t_0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps))))) - tan(x)
    else if (eps <= 5.2d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - (tan(x) * 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 <= -3.4e-9) {
		tmp = (t_0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps))))) - Math.tan(x);
	} else if (eps <= 5.2e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (Math.tan(x) * 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 <= -3.4e-9:
		tmp = (t_0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps))))) - math.tan(x)
	elif eps <= 5.2e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 * (1.0 / (1.0 - (math.tan(x) * 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 <= -3.4e-9)
		tmp = Float64(Float64(t_0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps))))) - tan(x));
	elseif (eps <= 5.2e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(x) * 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 <= -3.4e-9)
		tmp = (t_0 / (1.0 - (tan(x) / (1.0 / tan(eps))))) - tan(x);
	elseif (eps <= 5.2e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) * 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, -3.4e-9], 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], If[LessEqual[eps, 5.2e-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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.3999999999999998e-9

   1. Initial program 44.9%

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

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

      1. tan-quot 99.3%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

      4. clear-num 99.3%

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

      5. tan-quot 99.5%

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

   7. Applied egg-rr 99.5%

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

   if -3.3999999999999998e-9 < eps < 5.2000000000000002e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 5.2000000000000002e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 9: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -4.6e-9)
     (- (* t_0 (/ 1.0 (- 1.0 (/ (tan x) (/ 1.0 (tan eps)))))) (tan x))
     (if (<= eps 3.6e-9)
       (+ eps (* eps (pow (tan x) 2.0)))
       (- (* 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 <= -4.6e-9) {
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) / (1.0 / tan(eps)))))) - tan(x);
	} else if (eps <= 3.6e-9) {
		tmp = eps + (eps * pow(tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) * 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 <= (-4.6d-9)) then
        tmp = (t_0 * (1.0d0 / (1.0d0 - (tan(x) / (1.0d0 / tan(eps)))))) - tan(x)
    else if (eps <= 3.6d-9) then
        tmp = eps + (eps * (tan(x) ** 2.0d0))
    else
        tmp = (t_0 * (1.0d0 / (1.0d0 - (tan(x) * 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 <= -4.6e-9) {
		tmp = (t_0 * (1.0 / (1.0 - (Math.tan(x) / (1.0 / Math.tan(eps)))))) - Math.tan(x);
	} else if (eps <= 3.6e-9) {
		tmp = eps + (eps * Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = (t_0 * (1.0 / (1.0 - (Math.tan(x) * 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 <= -4.6e-9:
		tmp = (t_0 * (1.0 / (1.0 - (math.tan(x) / (1.0 / math.tan(eps)))))) - math.tan(x)
	elif eps <= 3.6e-9:
		tmp = eps + (eps * math.pow(math.tan(x), 2.0))
	else:
		tmp = (t_0 * (1.0 / (1.0 - (math.tan(x) * 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 <= -4.6e-9)
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(x) / Float64(1.0 / tan(eps)))))) - tan(x));
	elseif (eps <= 3.6e-9)
		tmp = Float64(eps + Float64(eps * (tan(x) ^ 2.0)));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(x) * 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 <= -4.6e-9)
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) / (1.0 / tan(eps)))))) - tan(x);
	elseif (eps <= 3.6e-9)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = (t_0 * (1.0 / (1.0 - (tan(x) * 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, -4.6e-9], N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] / N[(1.0 / N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.6e-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[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.5999999999999998e-9

   1. Initial program 44.9%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. tan-quot 99.3%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

      4. clear-num 99.3%

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

      5. tan-quot 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -4.5999999999999998e-9 < eps < 3.6e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 3.6e-9 < eps

   1. Initial program 51.7%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.4%

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

   3. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.6 \cdot 10^{-9}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \frac{\tan x}{\frac{1}{\tan \varepsilon}}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 10: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.4e-9) (not (<= eps 5.8e-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 <= -1.4e-9) || !(eps <= 5.8e-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 <= (-1.4d-9)) .or. (.not. (eps <= 5.8d-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 <= -1.4e-9) || !(eps <= 5.8e-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 <= -1.4e-9) or not (eps <= 5.8e-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 <= -1.4e-9) || !(eps <= 5.8e-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 <= -1.4e-9) || ~((eps <= 5.8e-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, -1.4e-9], N[Not[LessEqual[eps, 5.8e-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 < -1.39999999999999992e-9 or 5.79999999999999982e-9 < eps

   1. Initial program 48.3%

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

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

   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/ 99.4%

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

      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%

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

   if -1.39999999999999992e-9 < eps < 5.79999999999999982e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.5%

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

      2. div-inv 28.4%

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

      3. fma-neg 28.4%

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

   3. Applied egg-rr 28.4%

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

      1. fma-neg 28.4%

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

      2. associate-*r/ 28.5%

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

      3. *-rgt-identity 28.5%

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

   5. Simplified 28.5%

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

   6. Taylor expanded in eps around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

   8. Simplified 99.5%

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

      1. distribute-rgt-in 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.6%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \end{array} \]

Alternative 11: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.118:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0105)
   (tan eps)
   (if (<= eps 0.118) (* eps (+ 1.0 (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0105) {
		tmp = tan(eps);
	} else if (eps <= 0.118) {
		tmp = eps * (1.0 + 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 <= (-0.0105d0)) then
        tmp = tan(eps)
    else if (eps <= 0.118d0) then
        tmp = eps * (1.0d0 + (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 <= -0.0105) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.118) {
		tmp = eps * (1.0 + Math.pow(Math.tan(x), 2.0));
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0105:
		tmp = math.tan(eps)
	elif eps <= 0.118:
		tmp = eps * (1.0 + math.pow(math.tan(x), 2.0))
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0105)
		tmp = tan(eps);
	elseif (eps <= 0.118)
		tmp = Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0105)
		tmp = tan(eps);
	elseif (eps <= 0.118)
		tmp = eps * (1.0 + (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0105], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.118], N[(eps * N[(1.0 + 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 < -0.0105000000000000007 or 0.11799999999999999 < eps

   1. Initial program 49.3%

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

   2. Taylor expanded in x around 0 52.8%

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

      1. tan-quot 53.0%

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

      2. expm1-log1p-u 40.8%

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

      3. expm1-udef 40.4%

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

   4. Applied egg-rr 40.4%

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

      1. expm1-def 40.8%

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

      2. expm1-log1p 53.0%

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

   6. Simplified 53.0%

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

   if -0.0105000000000000007 < eps < 0.11799999999999999

   1. Initial program 27.4%

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

      1. tan-sum 30.0%

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

      2. div-inv 30.0%

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

      3. fma-neg 30.0%

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

   3. Applied egg-rr 30.0%

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

      1. fma-neg 30.0%

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

      2. associate-*r/ 30.0%

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

      3. *-rgt-identity 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in eps around 0 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. metadata-eval 97.9%

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

      3. *-lft-identity 97.9%

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

   8. Simplified 97.9%

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

      1. unpow2 97.9%

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

      2. unpow2 97.9%

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

      3. frac-times 97.8%

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

      4. tan-quot 97.9%

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

      5. tan-quot 97.9%

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

   10. Applied egg-rr 97.9%

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

       1. unpow2 97.9%

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

   12. Simplified 97.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.118:\\ \;\;\;\;\varepsilon \cdot \left(1 + {\tan x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 12: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.118:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0105)
   (tan eps)
   (if (<= eps 0.118) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0105) {
		tmp = tan(eps);
	} else if (eps <= 0.118) {
		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 <= (-0.0105d0)) then
        tmp = tan(eps)
    else if (eps <= 0.118d0) 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 <= -0.0105) {
		tmp = Math.tan(eps);
	} else if (eps <= 0.118) {
		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 <= -0.0105:
		tmp = math.tan(eps)
	elif eps <= 0.118:
		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 <= -0.0105)
		tmp = tan(eps);
	elseif (eps <= 0.118)
		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 <= -0.0105)
		tmp = tan(eps);
	elseif (eps <= 0.118)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0105], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 0.118], 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 < -0.0105000000000000007 or 0.11799999999999999 < eps

   1. Initial program 49.3%

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

   2. Taylor expanded in x around 0 52.8%

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

      1. tan-quot 53.0%

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

      2. expm1-log1p-u 40.8%

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

      3. expm1-udef 40.4%

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

   4. Applied egg-rr 40.4%

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

      1. expm1-def 40.8%

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

      2. expm1-log1p 53.0%

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

   6. Simplified 53.0%

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

   if -0.0105000000000000007 < eps < 0.11799999999999999

   1. Initial program 27.4%

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

      1. tan-sum 30.0%

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

      2. div-inv 30.0%

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

      3. fma-neg 30.0%

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

   3. Applied egg-rr 30.0%

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

      1. fma-neg 30.0%

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

      2. associate-*r/ 30.0%

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

      3. *-rgt-identity 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in eps around 0 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. metadata-eval 97.9%

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

      3. *-lft-identity 97.9%

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

   8. Simplified 97.9%

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

      1. distribute-rgt-in 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. unpow2 98.0%

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

      4. unpow2 98.0%

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

      5. frac-times 98.0%

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

      6. tan-quot 98.0%

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

      7. tan-quot 98.0%

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

      8. pow2 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0105:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.118:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 13: 34.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

2. Taylor expanded in x around 0 52.4%

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

   1. add-sqr-sqrt 28.7%

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

   2. sqrt-unprod 25.5%

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

   3. pow2 25.5%

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

4. Applied egg-rr 25.5%

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

   1. unpow2 25.5%

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

   2. rem-sqrt-square 30.6%

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

6. Simplified 30.6%

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

7. Taylor expanded in eps around 0 21.3%

   \[\leadsto \color{blue}{\left|\sin \varepsilon\right|} \]
8. Step-by-step derivation

   1. rem-square-sqrt 17.5%

      \[\leadsto \left|\color{blue}{\sqrt{\sin \varepsilon} \cdot \sqrt{\sin \varepsilon}}\right| \]

   2. fabs-sqr 17.5%

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

   3. rem-square-sqrt 30.9%

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

9. Simplified 30.9%

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

10. Final simplification 30.9%

    \[\leadsto \sin \varepsilon \]

Alternative 14: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

2. Taylor expanded in x around 0 52.4%

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

   1. tan-quot 52.5%

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

   2. expm1-log1p-u 46.3%

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

   3. expm1-udef 23.5%

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

4. Applied egg-rr 23.5%

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

   1. expm1-def 46.3%

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

   2. expm1-log1p 52.5%

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

6. Simplified 52.5%

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

7. Final simplification 52.5%

   \[\leadsto \tan \varepsilon \]

Alternative 15: 4.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 38.5%

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

   1. sub-neg 38.5%

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

   2. +-commutative 38.5%

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

   3. add-log-exp 38.0%

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

   4. add-log-exp 27.5%

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

   5. sum-log 27.6%

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

3. Applied egg-rr 27.6%

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

4. Taylor expanded in eps around 0 4.2%

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

   1. exp-neg 4.1%

      \[\leadsto \log \left(e^{\frac{\sin x}{\cos x}} \cdot \color{blue}{\frac{1}{e^{\frac{\sin x}{\cos x}}}}\right) \]

   2. rgt-mult-inverse 4.1%

      \[\leadsto \log \color{blue}{1} \]

   3. metadata-eval 4.1%

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

6. Simplified 4.1%

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

7. Final simplification 4.1%

   \[\leadsto 0 \]

Alternative 16: 30.9% 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 38.5%

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

2. Taylor expanded in x around 0 52.4%

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

3. Taylor expanded in eps around 0 27.3%

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

4. Final simplification 27.3%

   \[\leadsto \varepsilon \]

Developer target: 76.2% 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 2023201 
(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)))