2tan (problem 3.3.2)

Percentage Accurate: 41.7% → 99.5%

Time: 20.5s

Alternatives: 8

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: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -3.4e-7)
     (fma t_0 (/ 1.0 t_1) (- (tan x)))
     (if (<= eps 2.3e-7)
       (+
        (+ eps (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
        (*
         (* eps eps)
         (+ (/ (pow (sin x) 3.0) (pow (cos x) 3.0)) (/ (sin x) (cos x)))))
       (- (/ t_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 <= -3.4e-7) {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	} else if (eps <= 2.3e-7) {
		tmp = (eps + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + ((eps * eps) * ((pow(sin(x), 3.0) / pow(cos(x), 3.0)) + (sin(x) / cos(x))));
	} else {
		tmp = (t_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 <= -3.4e-7)
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	elseif (eps <= 2.3e-7)
		tmp = Float64(Float64(eps + Float64(eps * Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64(Float64(eps * eps) * Float64(Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)) + Float64(sin(x) / cos(x)))));
	else
		tmp = Float64(Float64(t_0 / t_1) - 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, -3.4e-7], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 2.3e-7], N[(N[(eps + N[(eps * N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -3.39999999999999974e-7

   1. Initial program 58.7%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -3.39999999999999974e-7 < eps < 2.29999999999999995e-7

   1. Initial program 35.9%

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

      1. tan-sum 36.7%

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

      2. div-inv 36.7%

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

      3. fma-neg 36.7%

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

   3. Applied egg-rr 36.7%

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

      1. fma-neg 36.7%

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

      2. associate-*r/ 36.7%

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

      3. *-rgt-identity 36.7%

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

   5. Simplified 36.7%

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

   6. Taylor expanded in eps around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. mul-1-neg 99.6%

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

      3. unsub-neg 99.6%

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

   8. Simplified 99.8%

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

   if 2.29999999999999995e-7 < eps

   1. Initial program 56.0%

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

      1. tan-sum 99.5%

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

      2. div-inv 99.4%

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

      3. fma-neg 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. fma-neg 99.4%

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

      2. associate-*r/ 99.5%

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

      3. *-rgt-identity 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\varepsilon + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := 1 - \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{t_1}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1} - \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- 1.0 (* (tan x) (tan eps)))))
   (if (<= eps -8.6e-9)
     (fma t_0 (/ 1.0 t_1) (- (tan x)))
     (if (<= eps 5.5e-17)
       (fma (pow (tan x) 2.0) eps eps)
       (- (/ t_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 <= -8.6e-9) {
		tmp = fma(t_0, (1.0 / t_1), -tan(x));
	} else if (eps <= 5.5e-17) {
		tmp = fma(pow(tan(x), 2.0), eps, eps);
	} else {
		tmp = (t_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 <= -8.6e-9)
		tmp = fma(t_0, Float64(1.0 / t_1), Float64(-tan(x)));
	elseif (eps <= 5.5e-17)
		tmp = fma((tan(x) ^ 2.0), eps, eps);
	else
		tmp = Float64(Float64(t_0 / t_1) - 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, -8.6e-9], N[(t$95$0 * N[(1.0 / t$95$1), $MachinePrecision] + (-N[Tan[x], $MachinePrecision])), $MachinePrecision], If[LessEqual[eps, 5.5e-17], N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -8.59999999999999925e-9

   1. Initial program 58.7%

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

      1. tan-sum 99.6%

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

      2. div-inv 99.6%

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

      3. fma-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -8.59999999999999925e-9 < eps < 5.50000000000000001e-17

   1. Initial program 34.1%

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

      1. add-cube-cbrt 33.6%

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

      2. pow3 33.6%

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

   3. Applied egg-rr 33.6%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. metadata-eval 97.9%

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

      3. *-lft-identity 97.9%

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

      4. distribute-lft-in 97.7%

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

      5. *-rgt-identity 97.7%

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

   6. Simplified 97.7%

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

      1. rem-cube-cbrt 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. pow2 99.6%

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

      7. sqrt-div 99.7%

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

      8. unpow2 99.7%

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

      9. sqrt-prod 50.2%

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

      10. add-sqr-sqrt 99.7%

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

      11. unpow2 99.7%

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

      12. sqrt-prod 75.0%

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

      13. add-sqr-sqrt 99.7%

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

      14. tan-quot 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 5.50000000000000001e-17 < eps

   1. Initial program 58.1%

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

      1. tan-sum 98.9%

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

      2. div-inv 98.8%

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

      3. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

      1. fma-neg 98.8%

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

      2. associate-*r/ 98.9%

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

      3. *-rgt-identity 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8.6e-9) (not (<= eps 5.5e-17)))
   (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x))
   (fma (pow (tan x) 2.0) eps eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8.6e-9) || !(eps <= 5.5e-17)) {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else {
		tmp = fma(pow(tan(x), 2.0), eps, eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8.6e-9) || !(eps <= 5.5e-17))
		tmp = Float64(Float64(Float64(tan(x) + tan(eps)) / Float64(1.0 - Float64(tan(x) * tan(eps)))) - tan(x));
	else
		tmp = fma((tan(x) ^ 2.0), eps, eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8.6e-9], N[Not[LessEqual[eps, 5.5e-17]], $MachinePrecision]], N[(N[(N[(N[Tan[x], $MachinePrecision] + N[Tan[eps], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.59999999999999925e-9 or 5.50000000000000001e-17 < eps

   1. Initial program 58.5%

      \[\tan \left(x + \varepsilon\right) - \tan x \]
   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 99.2%

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

      3. fma-neg 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. fma-neg 99.2%

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

      2. associate-*r/ 99.3%

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

      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%

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

   if -8.59999999999999925e-9 < eps < 5.50000000000000001e-17

   1. Initial program 34.1%

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

      1. add-cube-cbrt 33.6%

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

      2. pow3 33.6%

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

   3. Applied egg-rr 33.6%

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

   4. Taylor expanded in eps around 0 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. metadata-eval 97.9%

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

      3. *-lft-identity 97.9%

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

      4. distribute-lft-in 97.7%

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

      5. *-rgt-identity 97.7%

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

   6. Simplified 97.7%

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

      1. rem-cube-cbrt 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. pow2 99.6%

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

      7. sqrt-div 99.7%

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

      8. unpow2 99.7%

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

      9. sqrt-prod 50.2%

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

      10. add-sqr-sqrt 99.7%

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

      11. unpow2 99.7%

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

      12. sqrt-prod 75.0%

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

      13. add-sqr-sqrt 99.7%

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

      14. tan-quot 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.6 \cdot 10^{-9} \lor \neg \left(\varepsilon \leq 5.5 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \end{array} \]

Alternative 4: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.2e-6)
   (tan eps)
   (if (<= eps 1.8) (fma (pow (tan x) 2.0) eps eps) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.2e-6) {
		tmp = tan(eps);
	} else if (eps <= 1.8) {
		tmp = fma(pow(tan(x), 2.0), eps, eps);
	} else {
		tmp = tan(eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.2e-6)
		tmp = tan(eps);
	elseif (eps <= 1.8)
		tmp = fma((tan(x) ^ 2.0), eps, eps);
	else
		tmp = tan(eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.2e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.8], N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision], N[Tan[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.1999999999999999e-6 or 1.80000000000000004 < eps

   1. Initial program 58.0%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. tan-quot 60.6%

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

      2. expm1-log1p-u 45.2%

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

      3. expm1-udef 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. expm1-def 45.2%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if -3.1999999999999999e-6 < eps < 1.80000000000000004

   1. Initial program 35.7%

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

      1. add-cube-cbrt 35.1%

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

      2. pow3 35.1%

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

   3. Applied egg-rr 35.1%

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

   4. Taylor expanded in eps around 0 97.0%

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

      1. cancel-sign-sub-inv 97.0%

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

      2. metadata-eval 97.0%

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

      3. *-lft-identity 97.0%

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

      4. distribute-lft-in 96.8%

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

      5. *-rgt-identity 96.8%

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

   6. Simplified 96.8%

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

      1. rem-cube-cbrt 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-def 98.7%

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

      5. add-sqr-sqrt 98.7%

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

      6. pow2 98.7%

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

      7. sqrt-div 98.8%

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

      8. unpow2 98.8%

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

      9. sqrt-prod 51.5%

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

      10. add-sqr-sqrt 98.8%

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

      11. unpow2 98.8%

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

      12. sqrt-prod 75.1%

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

      13. add-sqr-sqrt 98.8%

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

      14. tan-quot 98.8%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 5: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\varepsilon \cdot \left({\tan x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.5e-6)
   (tan eps)
   (if (<= eps 1.8) (* eps (+ (pow (tan x) 2.0) 1.0)) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.5e-6) {
		tmp = tan(eps);
	} else if (eps <= 1.8) {
		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 <= (-8.5d-6)) then
        tmp = tan(eps)
    else if (eps <= 1.8d0) 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 <= -8.5e-6) {
		tmp = Math.tan(eps);
	} else if (eps <= 1.8) {
		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 <= -8.5e-6:
		tmp = math.tan(eps)
	elif eps <= 1.8:
		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 <= -8.5e-6)
		tmp = tan(eps);
	elseif (eps <= 1.8)
		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 <= -8.5e-6)
		tmp = tan(eps);
	elseif (eps <= 1.8)
		tmp = eps * ((tan(x) ^ 2.0) + 1.0);
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.5e-6], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.8], 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 < -8.4999999999999999e-6 or 1.80000000000000004 < eps

   1. Initial program 58.0%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. tan-quot 60.6%

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

      2. expm1-log1p-u 45.2%

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

      3. expm1-udef 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. expm1-def 45.2%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if -8.4999999999999999e-6 < eps < 1.80000000000000004

   1. Initial program 35.7%

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

      1. add-cube-cbrt 35.1%

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

      2. pow3 35.1%

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

   3. Applied egg-rr 35.1%

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

   4. Taylor expanded in eps around 0 97.0%

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

      1. cancel-sign-sub-inv 97.0%

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

      2. metadata-eval 97.0%

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

      3. *-lft-identity 97.0%

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

      4. distribute-lft-in 96.8%

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

      5. *-rgt-identity 96.8%

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

   6. Simplified 96.8%

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

      1. rem-cube-cbrt 98.8%

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

      2. add-cube-cbrt 97.3%

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

      3. fma-def 97.2%

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

      4. pow2 97.2%

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

      5. add-sqr-sqrt 97.1%

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

      6. pow2 97.1%

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

      7. sqrt-div 97.2%

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

      8. unpow2 97.2%

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

      9. sqrt-prod 50.8%

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

      10. add-sqr-sqrt 97.2%

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

      11. unpow2 97.2%

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

      12. sqrt-prod 73.8%

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

      13. add-sqr-sqrt 97.2%

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

      14. tan-quot 97.2%

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

   8. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\varepsilon}\right)}^{2}, \sqrt[3]{\varepsilon}, \varepsilon \cdot {\tan x}^{2}\right)} \]
   9. Step-by-step derivation

      1. fma-udef 97.3%

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

      2. unpow2 97.3%

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

      3. add-cube-cbrt 98.8%

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

      4. *-commutative 98.8%

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

      5. distribute-rgt1-in 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\varepsilon \cdot \left({\tan x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 6: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.8e-5)
   (tan eps)
   (if (<= eps 1.8) (+ eps (* eps (pow (tan x) 2.0))) (tan eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.8e-5) {
		tmp = tan(eps);
	} else if (eps <= 1.8) {
		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 <= (-1.8d-5)) then
        tmp = tan(eps)
    else if (eps <= 1.8d0) 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 <= -1.8e-5) {
		tmp = Math.tan(eps);
	} else if (eps <= 1.8) {
		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 <= -1.8e-5:
		tmp = math.tan(eps)
	elif eps <= 1.8:
		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 <= -1.8e-5)
		tmp = tan(eps);
	elseif (eps <= 1.8)
		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 <= -1.8e-5)
		tmp = tan(eps);
	elseif (eps <= 1.8)
		tmp = eps + (eps * (tan(x) ^ 2.0));
	else
		tmp = tan(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.8e-5], N[Tan[eps], $MachinePrecision], If[LessEqual[eps, 1.8], 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 < -1.80000000000000005e-5 or 1.80000000000000004 < eps

   1. Initial program 58.0%

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

   2. Taylor expanded in x around 0 60.3%

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

      1. tan-quot 60.6%

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

      2. expm1-log1p-u 45.2%

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

      3. expm1-udef 44.9%

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

   4. Applied egg-rr 44.9%

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

      1. expm1-def 45.2%

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

      2. expm1-log1p 60.6%

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

   6. Simplified 60.6%

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

   if -1.80000000000000005e-5 < eps < 1.80000000000000004

   1. Initial program 35.7%

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

      1. add-cube-cbrt 35.1%

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

      2. pow3 35.1%

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

   3. Applied egg-rr 35.1%

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

   4. Taylor expanded in eps around 0 97.0%

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

      1. cancel-sign-sub-inv 97.0%

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

      2. metadata-eval 97.0%

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

      3. *-lft-identity 97.0%

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

      4. distribute-lft-in 96.8%

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

      5. *-rgt-identity 96.8%

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

   6. Simplified 96.8%

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

      1. rem-cube-cbrt 98.8%

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

      2. +-commutative 98.8%

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

      3. add-sqr-sqrt 98.7%

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

      4. pow2 98.7%

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

      5. sqrt-div 98.7%

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

      6. unpow2 98.7%

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

      7. sqrt-prod 51.5%

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

      8. add-sqr-sqrt 98.7%

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

      9. unpow2 98.7%

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

      10. sqrt-prod 75.0%

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

      11. add-sqr-sqrt 98.7%

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

      12. tan-quot 98.8%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\tan \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8:\\ \;\;\;\;\varepsilon + \varepsilon \cdot {\tan x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\tan \varepsilon\\ \end{array} \]

Alternative 7: 57.1% 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 46.6%

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

2. Taylor expanded in x around 0 61.4%

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

   1. tan-quot 61.5%

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

   2. expm1-log1p-u 54.0%

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

   3. expm1-udef 25.5%

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

4. Applied egg-rr 25.5%

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

   1. expm1-def 54.0%

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

   2. expm1-log1p 61.5%

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

6. Simplified 61.5%

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

7. Final simplification 61.5%

   \[\leadsto \tan \varepsilon \]

Alternative 8: 30.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 46.6%

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

2. Taylor expanded in x around 0 61.4%

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

3. Taylor expanded in eps around 0 33.4%

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

4. Final simplification 33.4%

   \[\leadsto \varepsilon \]

Developer target: 75.5% 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 2023275 
(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)))