2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 23.5s

Alternatives: 10

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

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: 62.5% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := 1 + t\_0\\ t_2 := t\_0 + {\tan x}^{4}\\ t_3 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(1 + \left(t\_3 + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{t\_2}{\varepsilon} + \left(\left(\frac{-0.16666666666666666}{\varepsilon} - \mathsf{fma}\left(-0.5, \frac{t\_1}{\varepsilon}, t\_0 \cdot \frac{0.16666666666666666}{\varepsilon}\right)\right) - \mathsf{fma}\left(\sin x, \frac{\mathsf{fma}\left(t\_0, 0.16666666666666666, \mathsf{fma}\left(t\_0, -0.5, -0.3333333333333333\right)\right) - t\_2}{\cos x}, \frac{t\_1 \cdot \left(\sin x \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + t\_3\right)}{\cos x}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (+ 1.0 t_0))
        (t_2 (+ t_0 (pow (tan x) 4.0)))
        (t_3 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    eps
    (+
     1.0
     (+
      t_3
      (*
       eps
       (+
        (*
         eps
         (*
          eps
          (+
           (/ t_2 eps)
           (-
            (-
             (/ -0.16666666666666666 eps)
             (fma -0.5 (/ t_1 eps) (* t_0 (/ 0.16666666666666666 eps))))
            (fma
             (sin x)
             (/
              (-
               (fma t_0 0.16666666666666666 (fma t_0 -0.5 -0.3333333333333333))
               t_2)
              (cos x))
             (/ (* t_1 (* (sin x) -0.3333333333333333)) (cos x)))))))
        (/ (* (sin x) (+ 1.0 t_3)) (cos x)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = 1.0 + t_0;
	double t_2 = t_0 + pow(tan(x), 4.0);
	double t_3 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (1.0 + (t_3 + (eps * ((eps * (eps * ((t_2 / eps) + (((-0.16666666666666666 / eps) - fma(-0.5, (t_1 / eps), (t_0 * (0.16666666666666666 / eps)))) - fma(sin(x), ((fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333)) - t_2) / cos(x)), ((t_1 * (sin(x) * -0.3333333333333333)) / cos(x))))))) + ((sin(x) * (1.0 + t_3)) / cos(x))))));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(t_0 + (tan(x) ^ 4.0))
	t_3 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(1.0 + Float64(t_3 + Float64(eps * Float64(Float64(eps * Float64(eps * Float64(Float64(t_2 / eps) + Float64(Float64(Float64(-0.16666666666666666 / eps) - fma(-0.5, Float64(t_1 / eps), Float64(t_0 * Float64(0.16666666666666666 / eps)))) - fma(sin(x), Float64(Float64(fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333)) - t_2) / cos(x)), Float64(Float64(t_1 * Float64(sin(x) * -0.3333333333333333)) / cos(x))))))) + Float64(Float64(sin(x) * Float64(1.0 + t_3)) / cos(x)))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(1.0 + N[(t$95$3 + N[(eps * N[(N[(eps * N[(eps * N[(N[(t$95$2 / eps), $MachinePrecision] + N[(N[(N[(-0.16666666666666666 / eps), $MachinePrecision] - N[(-0.5 * N[(t$95$1 / eps), $MachinePrecision] + N[(t$95$0 * N[(0.16666666666666666 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(t$95$0 * 0.16666666666666666 + N[(t$95$0 * -0.5 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.7%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\left(-1 \cdot \left(\varepsilon \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)}{\cos x}\right)\right) + \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right) - \left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

7. Taylor expanded in eps around inf 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\left(-1 \cdot \left(-0.3333333333333333 \cdot \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(\left(0.16666666666666666 + \left(-0.5 \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\varepsilon \cdot {\cos x}^{2}}\right) - \left(-0.5 \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\varepsilon} + \left(0.16666666666666666 \cdot \frac{{\sin x}^{2}}{\varepsilon \cdot {\cos x}^{2}} + 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right)\right)\right)\right)} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

9. Simplified 99.7%

   \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\left({\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\varepsilon} - \mathsf{fma}\left(\sin x, \frac{\left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}}{\cos x}, -0.3333333333333333 \cdot \left(\sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right)\right)\right) - \left(\mathsf{fma}\left(-0.5, \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\varepsilon}, \frac{0.16666666666666666}{\varepsilon} \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + \frac{0.16666666666666666}{\varepsilon}\right)\right)\right)} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{\left(\varepsilon \cdot \left({\tan x}^{2} \cdot \frac{{\tan x}^{2} + 1}{\varepsilon} - \left(\mathsf{fma}\left(\sin x, \frac{\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)}{\cos x}, \left(-0.3333333333333333 \cdot \sin x\right) \cdot \frac{{\tan x}^{2} + 1}{\cos x}\right) + \left(\frac{0.16666666666666666}{\varepsilon} + \mathsf{fma}\left(-0.5, \frac{{\tan x}^{2} + 1}{\varepsilon}, {\tan x}^{2} \cdot \frac{0.16666666666666666}{\varepsilon}\right)\right)\right)\right)\right)}^{1}} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

13. Simplified 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{{\tan x}^{2} + {\tan x}^{4}}{\varepsilon} + \left(\left(\frac{-0.16666666666666666}{\varepsilon} - \mathsf{fma}\left(-0.5, \frac{1 + {\tan x}^{2}}{\varepsilon}, {\tan x}^{2} \cdot \frac{0.16666666666666666}{\varepsilon}\right)\right) - \mathsf{fma}\left(\sin x, \frac{\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - \left({\tan x}^{2} + {\tan x}^{4}\right)}{\cos x}, \frac{\left(\sin x \cdot -0.3333333333333333\right) \cdot \left(1 + {\tan x}^{2}\right)}{\cos x}\right)\right)\right)\right)} + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \]

14. Final simplification 99.7%

    \[\leadsto \varepsilon \cdot \left(1 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \varepsilon \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{{\tan x}^{2} + {\tan x}^{4}}{\varepsilon} + \left(\left(\frac{-0.16666666666666666}{\varepsilon} - \mathsf{fma}\left(-0.5, \frac{1 + {\tan x}^{2}}{\varepsilon}, {\tan x}^{2} \cdot \frac{0.16666666666666666}{\varepsilon}\right)\right) - \mathsf{fma}\left(\sin x, \frac{\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - \left({\tan x}^{2} + {\tan x}^{4}\right)}{\cos x}, \frac{\left(1 + {\tan x}^{2}\right) \cdot \left(\sin x \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right)\right) + \frac{\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right)\right) \]
15. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{\sin x}{\cos x}\right)}^{2}\\ t_1 := \sqrt[3]{\tan x}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot t\_0\right) + 0.16666666666666666 \cdot t\_0\right) + t\_0 \cdot \left(-1 - t\_0\right), \sin x \cdot \frac{1 + t\_0}{\cos x}\right), {\left(t\_1 \cdot {\left({\left(\sqrt[3]{t\_1}\right)}^{2}\right)}^{3}\right)}^{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (/ (sin x) (cos x)) 2.0)) (t_1 (cbrt (tan x))))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (fma
       (- eps)
       (+
        (+ (+ -0.3333333333333333 (* -0.5 t_0)) (* 0.16666666666666666 t_0))
        (* t_0 (- -1.0 t_0)))
       (* (sin x) (/ (+ 1.0 t_0) (cos x))))
      (pow (* t_1 (pow (pow (cbrt t_1) 2.0) 3.0)) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((sin(x) / cos(x)), 2.0);
	double t_1 = cbrt(tan(x));
	return eps * (1.0 + fma(eps, fma(-eps, (((-0.3333333333333333 + (-0.5 * t_0)) + (0.16666666666666666 * t_0)) + (t_0 * (-1.0 - t_0))), (sin(x) * ((1.0 + t_0) / cos(x)))), pow((t_1 * pow(pow(cbrt(t_1), 2.0), 3.0)), 2.0)));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x)) ^ 2.0
	t_1 = cbrt(tan(x))
	return Float64(eps * Float64(1.0 + fma(eps, fma(Float64(-eps), Float64(Float64(Float64(-0.3333333333333333 + Float64(-0.5 * t_0)) + Float64(0.16666666666666666 * t_0)) + Float64(t_0 * Float64(-1.0 - t_0))), Float64(sin(x) * Float64(Float64(1.0 + t_0) / cos(x)))), (Float64(t_1 * ((cbrt(t_1) ^ 2.0) ^ 3.0)) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Tan[x], $MachinePrecision], 1/3], $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[((-eps) * N[(N[(N[(-0.3333333333333333 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 * N[Power[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 63.1%

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

   1. tan-quot 63.1%

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

   2. frac-2neg 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left(\left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)}}^{2}\right)\right) \]

   2. pow3 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left({\left(\sqrt[3]{\frac{\sin x}{\cos x}}\right)}^{3}\right)}}^{2}\right)\right) \]

   3. add-cube-cbrt 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}}^{3}\right)}^{2}\right)\right) \]

   4. unpow-prod-down 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{3}\right)}}^{2}\right)\right) \]

   5. pow2 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{2}\right)}}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{3}\right)}^{2}\right)\right) \]

   6. tan-quot 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\tan x}}}\right)}^{2}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{3}\right)}^{2}\right)\right) \]

   7. pow3 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left({\left(\sqrt[3]{\sqrt[3]{\tan x}}\right)}^{2}\right)}^{3} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}\right)}^{2}\right)\right) \]

   8. add-cube-cbrt 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left({\left(\sqrt[3]{\sqrt[3]{\tan x}}\right)}^{2}\right)}^{3} \cdot \color{blue}{\sqrt[3]{\frac{\sin x}{\cos x}}}\right)}^{2}\right)\right) \]

   9. tan-quot 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left({\left(\sqrt[3]{\sqrt[3]{\tan x}}\right)}^{2}\right)}^{3} \cdot \sqrt[3]{\color{blue}{\tan x}}\right)}^{2}\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left({\left({\left(\sqrt[3]{\sqrt[3]{\tan x}}\right)}^{2}\right)}^{3} \cdot \sqrt[3]{\tan x}\right)}}^{2}\right)\right) \]

10. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + 0.16666666666666666 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \left(-1 - {\left(\frac{\sin x}{\cos x}\right)}^{2}\right), \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\sqrt[3]{\tan x} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{\tan x}}\right)}^{2}\right)}^{3}\right)}^{2}\right)\right) \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\frac{\sin x}{\cos x}\right)}^{2}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot t\_0\right) + 0.16666666666666666 \cdot t\_0\right) + t\_0 \cdot \left(-1 - t\_0\right), \sin x \cdot \frac{1 + t\_0}{\cos x}\right), {\left({\left(\sqrt[3]{\tan x}\right)}^{3}\right)}^{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (/ (sin x) (cos x)) 2.0)))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (fma
       (- eps)
       (+
        (+ (+ -0.3333333333333333 (* -0.5 t_0)) (* 0.16666666666666666 t_0))
        (* t_0 (- -1.0 t_0)))
       (* (sin x) (/ (+ 1.0 t_0) (cos x))))
      (pow (pow (cbrt (tan x)) 3.0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow((sin(x) / cos(x)), 2.0);
	return eps * (1.0 + fma(eps, fma(-eps, (((-0.3333333333333333 + (-0.5 * t_0)) + (0.16666666666666666 * t_0)) + (t_0 * (-1.0 - t_0))), (sin(x) * ((1.0 + t_0) / cos(x)))), pow(pow(cbrt(tan(x)), 3.0), 2.0)));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x)) ^ 2.0
	return Float64(eps * Float64(1.0 + fma(eps, fma(Float64(-eps), Float64(Float64(Float64(-0.3333333333333333 + Float64(-0.5 * t_0)) + Float64(0.16666666666666666 * t_0)) + Float64(t_0 * Float64(-1.0 - t_0))), Float64(sin(x) * Float64(Float64(1.0 + t_0) / cos(x)))), ((cbrt(tan(x)) ^ 3.0) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[((-eps) * N[(N[(N[(-0.3333333333333333 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[Power[N[Tan[x], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

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

   1. tan-quot 63.1%

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

   2. frac-2neg 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left(\left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)}}^{2}\right)\right) \]

   2. pow3 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left({\left(\sqrt[3]{\frac{\sin x}{\cos x}}\right)}^{3}\right)}}^{2}\right)\right) \]

   3. tan-quot 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left(\sqrt[3]{\color{blue}{\tan x}}\right)}^{3}\right)}^{2}\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\color{blue}{\left({\left(\sqrt[3]{\tan x}\right)}^{3}\right)}}^{2}\right)\right) \]

10. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + 0.16666666666666666 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \left(-1 - {\left(\frac{\sin x}{\cos x}\right)}^{2}\right), \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left({\left(\sqrt[3]{\tan x}\right)}^{3}\right)}^{2}\right)\right) \]
11. Add Preprocessing

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{1 + t\_0}{\cos x} + \varepsilon \cdot \left(\left(t\_0 + {\tan x}^{4}\right) - \mathsf{fma}\left(t\_0, 0.16666666666666666, \mathsf{fma}\left(t\_0, -0.5, -0.3333333333333333\right)\right)\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (+
       (* (sin x) (/ (+ 1.0 t_0) (cos x)))
       (*
        eps
        (-
         (+ t_0 (pow (tan x) 4.0))
         (fma t_0 0.16666666666666666 (fma t_0 -0.5 -0.3333333333333333)))))
      (pow (/ (sin x) (cos x)) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return eps * (1.0 + fma(eps, ((sin(x) * ((1.0 + t_0) / cos(x))) + (eps * ((t_0 + pow(tan(x), 4.0)) - fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333))))), pow((sin(x) / cos(x)), 2.0)));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(eps * Float64(1.0 + fma(eps, Float64(Float64(sin(x) * Float64(Float64(1.0 + t_0) / cos(x))) + Float64(eps * Float64(Float64(t_0 + (tan(x) ^ 4.0)) - fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333))))), (Float64(sin(x) / cos(x)) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(t$95$0 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * 0.16666666666666666 + N[(t$95$0 * -0.5 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

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

   1. tan-quot 63.1%

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

   2. frac-2neg 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. fma-undefine 99.6%

      \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\left(-\varepsilon\right) \cdot \left(\left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}\right) + \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]

9. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\left(-\varepsilon\right) \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)\right) + \sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x}}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]
10. Step-by-step derivation

    1. +-commutative 99.6%

       \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x} + \left(-\varepsilon\right) \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]

    2. distribute-lft-neg-out 99.6%

       \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x} + \color{blue}{\left(-\varepsilon \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)\right)\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]

    3. unsub-neg 99.6%

       \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x} - \varepsilon \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right)\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]

11. Simplified 99.6%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \color{blue}{\sin x \cdot \frac{1 + {\tan x}^{2}}{\cos x} - \varepsilon \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - \left({\tan x}^{2} + {\tan x}^{4}\right)\right)}, {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]

12. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{1 + {\tan x}^{2}}{\cos x} + \varepsilon \cdot \left(\left({\tan x}^{2} + {\tan x}^{4}\right) - \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right)\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right) \]
13. Add Preprocessing

Alternative 5: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{1 + t\_0}{\cos x} + \varepsilon \cdot \left(\left(t\_0 + {\tan x}^{4}\right) - \mathsf{fma}\left(t\_0, 0.16666666666666666, \mathsf{fma}\left(t\_0, -0.5, -0.3333333333333333\right)\right)\right), t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (+
       (* (sin x) (/ (+ 1.0 t_0) (cos x)))
       (*
        eps
        (-
         (+ t_0 (pow (tan x) 4.0))
         (fma t_0 0.16666666666666666 (fma t_0 -0.5 -0.3333333333333333)))))
      t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return eps * (1.0 + fma(eps, ((sin(x) * ((1.0 + t_0) / cos(x))) + (eps * ((t_0 + pow(tan(x), 4.0)) - fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333))))), t_0));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return Float64(eps * Float64(1.0 + fma(eps, Float64(Float64(sin(x) * Float64(Float64(1.0 + t_0) / cos(x))) + Float64(eps * Float64(Float64(t_0 + (tan(x) ^ 4.0)) - fma(t_0, 0.16666666666666666, fma(t_0, -0.5, -0.3333333333333333))))), t_0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(t$95$0 + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * 0.16666666666666666 + N[(t$95$0 * -0.5 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

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

   1. tan-quot 63.1%

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

   2. frac-2neg 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. *-un-lft-identity 99.6%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(1 \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \left(\left(-0.3333333333333333 + -0.5 \cdot {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) + {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot 0.16666666666666666\right) - {\left(\frac{\sin x}{\cos x}\right)}^{2} \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{1}, \sin x \cdot \frac{1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}}{\cos x}\right), {\left(\frac{\sin x}{\cos x}\right)}^{2}\right)\right)\right)} \]

9. Applied egg-rr 99.6%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right), \sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x}\right), {\tan x}^{2}\right)\right)\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 99.6%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - {\tan x}^{2} \cdot \left({\tan x}^{2} + 1\right), \sin x \cdot \frac{{\tan x}^{2} + 1}{\cos x}\right), {\tan x}^{2}\right)\right)} \]

11. Simplified 99.6%

    \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{1 + {\tan x}^{2}}{\cos x} - \varepsilon \cdot \left(\mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right) - \left({\tan x}^{2} + {\tan x}^{4}\right)\right), {\tan x}^{2}\right)\right)} \]

12. Final simplification 99.6%

    \[\leadsto \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \sin x \cdot \frac{1 + {\tan x}^{2}}{\cos x} + \varepsilon \cdot \left(\left({\tan x}^{2} + {\tan x}^{4}\right) - \mathsf{fma}\left({\tan x}^{2}, 0.16666666666666666, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.3333333333333333\right)\right)\right), {\tan x}^{2}\right)\right) \]
13. Add Preprocessing

Alternative 6: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(t\_0 + \left(1 + \varepsilon \cdot \left(\sin x \cdot \frac{1 + t\_0}{\cos x}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (* eps (+ t_0 (+ 1.0 (* eps (* (sin x) (/ (+ 1.0 t_0) (cos x)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return eps * (t_0 + (1.0 + (eps * (sin(x) * ((1.0 + t_0) / cos(x))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = eps * (t_0 + (1.0d0 + (eps * (sin(x) * ((1.0d0 + t_0) / cos(x))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return eps * (t_0 + (1.0 + (eps * (Math.sin(x) * ((1.0 + t_0) / Math.cos(x))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return eps * (t_0 + (1.0 + (eps * (math.sin(x) * ((1.0 + t_0) / math.cos(x))))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(eps * Float64(t_0 + Float64(1.0 + Float64(eps * Float64(sin(x) * Float64(Float64(1.0 + t_0) / cos(x)))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = eps * (t_0 + (1.0 + (eps * (sin(x) * ((1.0 + t_0) / cos(x))))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(t$95$0 + N[(1.0 + N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.5%

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

   1. associate-+r+ 99.5%

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

   2. remove-double-neg 99.5%

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

   3. mul-1-neg 99.5%

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

   4. sub-neg 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\sin x \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{\cos x}\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) (+ 1.0 (* eps x)))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (1.0 + (eps * x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + (1.0d0 + (eps * x)))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + (1.0 + (eps * x)));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + (1.0 + (eps * x)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(1.0 + Float64(eps * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + (1.0 + (eps * x)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.5%

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

   1. associate-+r+ 99.5%

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

   2. remove-double-neg 99.5%

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

   3. mul-1-neg 99.5%

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

   4. sub-neg 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in x around 0 99.3%

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

10. Final simplification 99.3%

    \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot x\right)\right) \]
11. Add Preprocessing

Alternative 8: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + {\left(\frac{\sin x}{\cos x}\right)}^{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (pow (/ (sin x) (cos x)) 2.0))))

C

double code(double x, double eps) {
	return eps * (1.0 + pow((sin(x) / cos(x)), 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((sin(x) / cos(x)) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + Math.pow((Math.sin(x) / Math.cos(x)), 2.0));
}

Python

def code(x, eps):
	return eps * (1.0 + math.pow((math.sin(x) / math.cos(x)), 2.0))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + (Float64(sin(x) / cos(x)) ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((sin(x) / cos(x)) ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[Power[N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

   1. tan-quot 63.1%

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

   2. frac-2neg 63.1%

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0 99.2%

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

   1. sub-neg 99.2%

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

   2. mul-1-neg 99.2%

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

   3. remove-double-neg 99.2%

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

   4. unpow2 99.2%

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

   5. unpow2 99.2%

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

   6. times-frac 99.2%

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

   7. remove-double-neg 99.2%

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

   8. distribute-frac-neg2 99.2%

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

   9. distribute-frac-neg 99.2%

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

   10. remove-double-neg 99.2%

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

   11. distribute-frac-neg2 99.2%

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

   12. distribute-frac-neg 99.2%

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

   13. unpow2 99.2%

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

   14. distribute-frac-neg 99.2%

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

   15. distribute-frac-neg2 99.2%

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

   16. remove-double-neg 99.2%

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

7. Simplified 99.2%

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

Alternative 9: 98.4% accurate, 22.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* x (* eps (+ eps x)))))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps + (x * (eps * (eps + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(-1 \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) - \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) - -1 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-\varepsilon, \sin x \cdot \frac{0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right)}{\cos x} + \left(\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right) \cdot -0.3333333333333333, -0.16666666666666666\right) - \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) - {\sin x}^{2} \cdot \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}\right), \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

6. Taylor expanded in eps around 0 99.5%

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

   1. associate-+r+ 99.5%

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

   2. remove-double-neg 99.5%

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

   3. mul-1-neg 99.5%

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

   4. sub-neg 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in x around 0 98.7%

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

    1. +-commutative 98.7%

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

    2. unpow2 98.7%

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

    3. distribute-lft-in 98.7%

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

11. Simplified 98.7%

    \[\leadsto \color{blue}{\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)} \]
12. Add Preprocessing

Alternative 10: 98.0% 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 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.2%

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

4. Taylor expanded in eps around 0 98.2%

   \[\leadsto \color{blue}{\varepsilon} \]
5. Add Preprocessing

Developer Target 1: 99.9% 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]

Developer Target 2: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := 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]

Developer Target 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform default (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))