2tan (problem 3.3.2)

Percentage Accurate: 41.9% → 99.4%

Time: 15.6s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- (tan x))))
   (if (<= eps -2.6e-9)
     (fma t_0 (/ -1.0 (fma (tan x) (tan eps) -1.0)) t_1)
     (if (<= eps 6.5e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1)))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = -tan(x);
	double tmp;
	if (eps <= -2.6e-9) {
		tmp = fma(t_0, (-1.0 / fma(tan(x), tan(eps), -1.0)), t_1);
	} else if (eps <= 6.5e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_1);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(-tan(x))
	tmp = 0.0
	if (eps <= -2.6e-9)
		tmp = fma(t_0, Float64(-1.0 / fma(tan(x), tan(eps), -1.0)), t_1);
	elseif (eps <= 6.5e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = fma(t_0, Float64(1.0 / Float64(1.0 - Float64(tan(x) * tan(eps)))), t_1);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Tan[x], $MachinePrecision])}, If[LessEqual[eps, -2.6e-9], N[(t$95$0 * N[(-1.0 / N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[eps, 6.5e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.6000000000000001e-9

   1. Initial program 54.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. /-rgt-identity 99.4%

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

      2. frac-2neg 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. div-inv 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-*r/ 99.5%

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

       2. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

   if -2.6000000000000001e-9 < eps < 6.5000000000000003e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.0%

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

      2. div-inv 28.0%

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

      3. fma-neg 28.0%

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

   3. Applied egg-rr 28.0%

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

      1. fma-neg 28.0%

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

      2. associate-*r/ 28.0%

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

      3. *-rgt-identity 28.0%

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

   5. Simplified 28.0%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. mul-1-neg 99.6%

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

      3. remove-double-neg 99.6%

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

   8. Simplified 99.6%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.7%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.7%

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

       2. fma-def 99.8%

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

   12. Simplified 99.8%

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

   if 6.5000000000000003e-9 < eps

   1. Initial program 45.9%

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

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -4.2e-9)
     (- (/ t_0 t_1) (tan x))
     (if (<= eps 5.7e-9)
       (fma eps (pow (tan x) 2.0) eps)
       (fma t_0 (/ 1.0 t_1) (- (tan x)))))))

C

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = 1.0 - (tan(x) * tan(eps));
	double tmp;
	if (eps <= -4.2e-9) {
		tmp = (t_0 / t_1) - tan(x);
	} else if (eps <= 5.7e-9) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(tan(x) + tan(eps))
	t_1 = Float64(1.0 - Float64(tan(x) * tan(eps)))
	tmp = 0.0
	if (eps <= -4.2e-9)
		tmp = Float64(Float64(t_0 / t_1) - tan(x));
	elseif (eps <= 5.7e-9)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -4.2e-9], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 5.7e-9], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.20000000000000039e-9

   1. Initial program 54.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 -4.20000000000000039e-9 < eps < 5.6999999999999998e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.0%

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

      2. div-inv 28.0%

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

      3. fma-neg 28.0%

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

   3. Applied egg-rr 28.0%

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

      1. fma-neg 28.0%

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

      2. associate-*r/ 28.0%

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

      3. *-rgt-identity 28.0%

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

   5. Simplified 28.0%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. mul-1-neg 99.6%

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

      3. remove-double-neg 99.6%

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

   8. Simplified 99.6%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.7%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.7%

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

       2. fma-def 99.8%

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

   12. Simplified 99.8%

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

   if 5.6999999999999998e-9 < eps

   1. Initial program 45.9%

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

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.7 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.55 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.4e-9) (not (<= eps 3.55e-9)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (fma eps (pow (tan x) 2.0) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.4e-9) || !(eps <= 3.55e-9)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.4e-9) || !(eps <= 3.55e-9))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.4e-9], N[Not[LessEqual[eps, 3.55e-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[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.4e-9 or 3.54999999999999994e-9 < eps

   1. Initial program 50.2%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 -2.4e-9 < eps < 3.54999999999999994e-9

   1. Initial program 28.0%

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

      1. tan-sum 28.0%

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

      2. div-inv 28.0%

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

      3. fma-neg 28.0%

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

   3. Applied egg-rr 28.0%

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

      1. fma-neg 28.0%

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

      2. associate-*r/ 28.0%

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

      3. *-rgt-identity 28.0%

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

   5. Simplified 28.0%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. mul-1-neg 99.6%

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

      3. remove-double-neg 99.6%

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

   8. Simplified 99.6%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. frac-times 99.7%

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

      6. tan-quot 99.7%

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

      7. tan-quot 99.7%

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

      8. pow2 99.7%

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

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.7%

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

       2. fma-def 99.8%

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

   12. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.4 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 3.55 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \end{array} \]

Alternative 4: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \frac{\sin x}{\cos x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0072)
   (tan eps)
   (if (<= eps 2.25e-5)
     (fma eps (pow (tan x) 2.0) eps)
     (- (+ (tan x) (tan eps)) (/ (sin x) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0072) {
		tmp = tan(eps);
	} else if (eps <= 2.25e-5) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = (tan(x) + tan(eps)) - (sin(x) / cos(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0072)
		tmp = tan(eps);
	elseif (eps <= 2.25e-5)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = Float64(Float64(tan(x) + tan(eps)) - Float64(sin(x) / cos(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0072], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 2.25e-5], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0071999999999999998

   1. Initial program 54.9%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. tan-quot 58.6%

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

      2. expm1-log1p-u 41.4%

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

      3. expm1-udef 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 58.6%

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

   6. Simplified 58.6%

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

   if -0.0071999999999999998 < eps < 2.25000000000000014e-5

   1. Initial program 27.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 28.5%

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

      3. fma-neg 28.5%

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

   3. Applied egg-rr 28.5%

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

      1. fma-neg 28.5%

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

      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 99.1%

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

      1. sub-neg 99.1%

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

      2. mul-1-neg 99.1%

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

      3. remove-double-neg 99.1%

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

   8. Simplified 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. unpow2 99.1%

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

      4. unpow2 99.1%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.2%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.2%

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

       2. fma-def 99.3%

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

   12. Simplified 99.3%

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

   if 2.25000000000000014e-5 < eps

   1. Initial program 45.9%

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

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

      4. div-inv 99.3%

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

      5. prod-diff 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-udef 99.3%

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

      3. associate-+r+ 99.3%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 48.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan x + \tan \varepsilon\right) - \frac{\sin x}{\cos x}\\ \end{array} \]

Alternative 5: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0072)
   (tan eps)
   (if (<= eps 6.3e-6)
     (fma eps (pow (tan x) 2.0) eps)
     (fma (+ (tan x) (tan eps)) 1.0 (- (tan x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0072) {
		tmp = tan(eps);
	} else if (eps <= 6.3e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = fma((tan(x) + tan(eps)), 1.0, -tan(x));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0072)
		tmp = tan(eps);
	elseif (eps <= 6.3e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = fma(Float64(tan(x) + tan(eps)), 1.0, Float64(-tan(x)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0072], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 6.3e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] * 1.0 + (-N[Tan[x], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.0071999999999999998

   1. Initial program 54.9%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. tan-quot 58.6%

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

      2. expm1-log1p-u 41.4%

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

      3. expm1-udef 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. expm1-def 41.4%

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

      2. expm1-log1p 58.6%

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

   6. Simplified 58.6%

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

   if -0.0071999999999999998 < eps < 6.29999999999999982e-6

   1. Initial program 27.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 28.5%

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

      3. fma-neg 28.5%

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

   3. Applied egg-rr 28.5%

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

      1. fma-neg 28.5%

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

      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 99.1%

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

      1. sub-neg 99.1%

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

      2. mul-1-neg 99.1%

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

      3. remove-double-neg 99.1%

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

   8. Simplified 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. unpow2 99.1%

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

      4. unpow2 99.1%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.2%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.2%

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

       2. fma-def 99.3%

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

   12. Simplified 99.3%

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

   if 6.29999999999999982e-6 < eps

   1. Initial program 45.9%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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. /-rgt-identity 99.4%

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

      2. frac-2neg 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 48.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 6.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, 1, -\tan x\right)\\ \end{array} \]

Alternative 6: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0072)
   (tan eps)
   (if (<= eps 3.75e-6) (fma eps (pow (tan x) 2.0) eps) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0072) {
		tmp = tan(eps);
	} else if (eps <= 3.75e-6) {
		tmp = fma(eps, pow(tan(x), 2.0), eps);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0072)
		tmp = tan(eps);
	elseif (eps <= 3.75e-6)
		tmp = fma(eps, (tan(x) ^ 2.0), eps);
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0072], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 3.75e-6], N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + eps), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0071999999999999998 or 3.7500000000000001e-6 < eps

   1. Initial program 50.6%

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

   2. Taylor expanded in x around 0 53.7%

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

      1. tan-quot 53.9%

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

      2. expm1-log1p-u 41.7%

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

      3. expm1-udef 41.4%

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

   4. Applied egg-rr 41.4%

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

      1. expm1-def 41.7%

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

      2. expm1-log1p 53.9%

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

   6. Simplified 53.9%

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

   if -0.0071999999999999998 < eps < 3.7500000000000001e-6

   1. Initial program 27.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 28.5%

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

      3. fma-neg 28.5%

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

   3. Applied egg-rr 28.5%

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

      1. fma-neg 28.5%

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

      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 99.1%

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

      1. sub-neg 99.1%

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

      2. mul-1-neg 99.1%

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

      3. remove-double-neg 99.1%

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

   8. Simplified 99.1%

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

      1. distribute-lft-in 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. unpow2 99.1%

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

      4. unpow2 99.1%

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

      5. frac-times 99.1%

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

      6. tan-quot 99.2%

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

      7. tan-quot 99.2%

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

      8. pow2 99.2%

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

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot {\tan x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative 99.2%

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

       2. fma-def 99.3%

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

   12. Simplified 99.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left({\tan x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0072)
   (tan eps)
   (if (<= eps 1.48e-5) (* eps (+ (pow (tan x) 2.0) 1.0)) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0072) {
		tmp = tan(eps);
	} else if (eps <= 1.48e-5) {
		tmp = eps * (pow(tan(x), 2.0) + 1.0);
	} 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 <= (-0.0072d0)) then
        tmp = tan(eps)
    else if (eps <= 1.48d-5) then
        tmp = eps * ((tan(x) ** 2.0d0) + 1.0d0)
    else
        tmp = tan(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0072) {
		tmp = Math.tan(eps);
	} else if (eps <= 1.48e-5) {
		tmp = eps * (Math.pow(Math.tan(x), 2.0) + 1.0);
	} else {
		tmp = Math.tan(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0072:
		tmp = math.tan(eps)
	elif eps <= 1.48e-5:
		tmp = eps * (math.pow(math.tan(x), 2.0) + 1.0)
	else:
		tmp = math.tan(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0072)
		tmp = tan(eps);
	elseif (eps <= 1.48e-5)
		tmp = Float64(eps * Float64((tan(x) ^ 2.0) + 1.0));
	else
		tmp = tan(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0072)
		tmp = tan(eps);
	elseif (eps <= 1.48e-5)
		tmp = eps * ((tan(x) ^ 2.0) + 1.0);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0072], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.48e-5], N[(eps * N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0071999999999999998 or 1.4800000000000001e-5 < eps

   1. Initial program 50.6%

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

   2. Taylor expanded in x around 0 53.7%

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

      1. tan-quot 53.9%

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

      2. expm1-log1p-u 41.7%

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

      3. expm1-udef 41.4%

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

   4. Applied egg-rr 41.4%

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

      1. expm1-def 41.7%

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

      2. expm1-log1p 53.9%

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

   6. Simplified 53.9%

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

   if -0.0071999999999999998 < eps < 1.4800000000000001e-5

   1. Initial program 27.8%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 28.5%

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

      3. fma-neg 28.5%

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

   3. Applied egg-rr 28.5%

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

      1. fma-neg 28.5%

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

      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 99.1%

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

      1. sub-neg 99.1%

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

      2. mul-1-neg 99.1%

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

      3. remove-double-neg 99.1%

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

   8. Simplified 99.1%

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

      1. unpow2 99.1%

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

      2. unpow2 99.1%

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

      3. frac-times 99.2%

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

      4. tan-quot 99.2%

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

      5. tan-quot 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. unpow2 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0072:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.48 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \left({\tan x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 8: 57.6% 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 39.1%

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

2. Taylor expanded in x around 0 57.1%

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

   1. tan-quot 57.2%

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

   2. expm1-log1p-u 51.1%

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

   3. expm1-udef 23.6%

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

4. Applied egg-rr 23.6%

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

   1. expm1-def 51.1%

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

   2. expm1-log1p 57.2%

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

6. Simplified 57.2%

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

7. Final simplification 57.2%

   \[\leadsto \tan \varepsilon \]

Alternative 9: 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 39.1%

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

   1. add-cube-cbrt 38.2%

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

   2. pow3 38.3%

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

3. Applied egg-rr 38.3%

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

4. Taylor expanded in eps around 0 4.3%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \frac{\sin x}{\cos x} - \frac{\sin x}{\cos x}} \]
5. Step-by-step derivation

   1. pow-base-1 4.3%

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

   2. *-lft-identity 4.3%

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

   3. +-inverses 4.3%

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

6. Simplified 4.3%

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

7. Final simplification 4.3%

   \[\leadsto 0 \]

Alternative 10: 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 39.1%

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

2. Taylor expanded in x around 0 57.1%

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

3. Taylor expanded in eps around 0 31.9%

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

4. Final simplification 31.9%

   \[\leadsto \varepsilon \]

Developer target: 75.8% 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 2023293 
(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)))