2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.7%

Time: 29.7s

Alternatives: 13

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: 62.1% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \mathsf{fma}\left(-0.5, 1 + t_0, 0.16666666666666666 \cdot t_0\right)\\ t_2 := {\left(\mathsf{hypot}\left(\tan x, t_0\right)\right)}^{2}\\ t_3 := {\cos x}^{2}\\ t_4 := \frac{{\sin x}^{2}}{t_3}\\ \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{t_3}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + t_4}}\right) + \left({\varepsilon}^{3} \cdot \left(t_2 - \left(0.16666666666666666 + t_1\right)\right) + {\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - t_4\right)\right) - \tan x \cdot \left(0.16666666666666666 + \left(t_1 - t_2\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (fma -0.5 (+ 1.0 t_0) (* 0.16666666666666666 t_0)))
        (t_2 (pow (hypot (tan x) t_0) 2.0))
        (t_3 (pow (cos x) 2.0))
        (t_4 (/ (pow (sin x) 2.0) t_3)))
   (+
    (fma
     eps
     (+ 1.0 (/ (- 0.5 (/ (cos (* x 2.0)) 2.0)) t_3))
     (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) (+ 1.0 t_4))))
    (+
     (* (pow eps 3.0) (- t_2 (+ 0.16666666666666666 t_1)))
     (*
      (pow eps 4.0)
      (-
       (* -0.3333333333333333 (* (/ (sin x) (cos x)) (- -1.0 t_4)))
       (* (tan x) (+ 0.16666666666666666 (- t_1 t_2)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = fma(-0.5, (1.0 + t_0), (0.16666666666666666 * t_0));
	double t_2 = pow(hypot(tan(x), t_0), 2.0);
	double t_3 = pow(cos(x), 2.0);
	double t_4 = pow(sin(x), 2.0) / t_3;
	return fma(eps, (1.0 + ((0.5 - (cos((x * 2.0)) / 2.0)) / t_3)), (pow(eps, 2.0) / ((cos(x) / sin(x)) / (1.0 + t_4)))) + ((pow(eps, 3.0) * (t_2 - (0.16666666666666666 + t_1))) + (pow(eps, 4.0) * ((-0.3333333333333333 * ((sin(x) / cos(x)) * (-1.0 - t_4))) - (tan(x) * (0.16666666666666666 + (t_1 - t_2))))));
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = fma(-0.5, Float64(1.0 + t_0), Float64(0.16666666666666666 * t_0))
	t_2 = hypot(tan(x), t_0) ^ 2.0
	t_3 = cos(x) ^ 2.0
	t_4 = Float64((sin(x) ^ 2.0) / t_3)
	return Float64(fma(eps, Float64(1.0 + Float64(Float64(0.5 - Float64(cos(Float64(x * 2.0)) / 2.0)) / t_3)), Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / Float64(1.0 + t_4)))) + Float64(Float64((eps ^ 3.0) * Float64(t_2 - Float64(0.16666666666666666 + t_1))) + Float64((eps ^ 4.0) * Float64(Float64(-0.3333333333333333 * Float64(Float64(sin(x) / cos(x)) * Float64(-1.0 - t_4))) - Float64(tan(x) * Float64(0.16666666666666666 + Float64(t_1 - t_2)))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(1.0 + t$95$0), $MachinePrecision] + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[N[Tan[x], $MachinePrecision] ^ 2 + t$95$0 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]}, N[(N[(eps * N[(1.0 + N[(N[(0.5 - N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(t$95$2 - N[(0.16666666666666666 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(-1.0 - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Tan[x], $MachinePrecision] * N[(0.16666666666666666 + N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right)} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)} \]

7. Applied egg-rr 99.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\left(\tan x \cdot 0.16666666666666666 + \tan x \cdot \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. distribute-lft-out 99.3%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

9. Simplified 99.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
10. Step-by-step derivation

    1. unpow2 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{\sin x \cdot \sin x}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    2. sin-mult 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

11. Applied egg-rr 99.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{\frac{\cos \left(x - x\right) - \cos \left(x + x\right)}{2}}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
12. Step-by-step derivation

    1. div-sub 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{\frac{\cos \left(x - x\right)}{2} - \frac{\cos \left(x + x\right)}{2}}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    2. +-inverses 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    3. cos-0 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\frac{\color{blue}{1}}{2} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    4. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{0.5} - \frac{\cos \left(x + x\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    5. count-2 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

    6. *-commutative 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

13. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{\color{blue}{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
14. Step-by-step derivation

    1. associate-+r- 99.3%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

15. Applied egg-rr 99.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \color{blue}{\left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)} + {\varepsilon}^{4} \cdot \left(\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

16. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{0.5 - \frac{\cos \left(x \cdot 2\right)}{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) + \left({\varepsilon}^{3} \cdot \left({\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2} - \left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) - \tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)\right)\right) \]
17. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ t_1 := 1 + t_0\\ t_2 := -1 - t_0\\ \mathsf{fma}\left(\varepsilon, t_1, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{t_1}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, t_1, t_0 \cdot 0.16666666666666666\right) + t_0 \cdot t_2\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot -0.3333333333333333 - -0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot t_2\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (+ 1.0 t_0))
        (t_2 (- -1.0 t_0)))
   (-
    (fma eps t_1 (/ (pow eps 2.0) (/ (/ (cos x) (sin x)) t_1)))
    (+
     (*
      (pow eps 3.0)
      (+
       0.16666666666666666
       (+ (fma -0.5 t_1 (* t_0 0.16666666666666666)) (* t_0 t_2))))
     (*
      (pow eps 4.0)
      (-
       (* (tan x) -0.3333333333333333)
       (* -0.3333333333333333 (* (/ (sin x) (cos x)) t_2))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	double t_1 = 1.0 + t_0;
	double t_2 = -1.0 - t_0;
	return fma(eps, t_1, (pow(eps, 2.0) / ((cos(x) / sin(x)) / t_1))) - ((pow(eps, 3.0) * (0.16666666666666666 + (fma(-0.5, t_1, (t_0 * 0.16666666666666666)) + (t_0 * t_2)))) + (pow(eps, 4.0) * ((tan(x) * -0.3333333333333333) - (-0.3333333333333333 * ((sin(x) / cos(x)) * t_2)))));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(-1.0 - t_0)
	return Float64(fma(eps, t_1, Float64((eps ^ 2.0) / Float64(Float64(cos(x) / sin(x)) / t_1))) - Float64(Float64((eps ^ 3.0) * Float64(0.16666666666666666 + Float64(fma(-0.5, t_1, Float64(t_0 * 0.16666666666666666)) + Float64(t_0 * t_2)))) + Float64((eps ^ 4.0) * Float64(Float64(tan(x) * -0.3333333333333333) - Float64(-0.3333333333333333 * Float64(Float64(sin(x) / cos(x)) * t_2))))))
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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - t$95$0), $MachinePrecision]}, N[(N[(eps * t$95$1 + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[(-0.5 * t$95$1 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(N[Tan[x], $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - N[(-0.3333333333333333 * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(-1 \cdot \left({\varepsilon}^{4} \cdot \left(-0.5 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \left(0.16666666666666666 \cdot \frac{\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{\cos x} + \frac{\sin x \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)}{\cos x}\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)\right)} \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)} \]

7. Applied egg-rr 99.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\left(\tan x \cdot 0.16666666666666666 + \tan x \cdot \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. distribute-lft-out 99.3%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

9. Simplified 99.3%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\color{blue}{\tan x \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right) - {\left(\mathsf{hypot}\left(\tan x, {\tan x}^{2}\right)\right)}^{2}\right)\right)} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

10. Taylor expanded in x around 0 99.2%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) - \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot \color{blue}{-0.3333333333333333} + \left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right) \]

11. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{\sin x}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}\right) - \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(\mathsf{fma}\left(-0.5, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.16666666666666666\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + {\varepsilon}^{4} \cdot \left(\tan x \cdot -0.3333333333333333 - -0.3333333333333333 \cdot \left(\frac{\sin x}{\cos x} \cdot \left(-1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	t_1 = Float64(1.0 + t_0)
	return Float64(fma(eps, t_1, Float64((eps ^ 2.0) / Float64(Float64(cos(x) / t_1) / sin(x)))) - Float64((eps ^ 3.0) * Float64(0.16666666666666666 + Float64(fma(-0.5, t_1, Float64(t_0 * 0.16666666666666666)) + Float64(t_0 * Float64(-1.0 - t_0))))))
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]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, N[(N[(eps * t$95$1 + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(N[Cos[x], $MachinePrecision] / t$95$1), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[(-0.5 * t$95$1 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 99.2%

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{3} \cdot \left(0.16666666666666666 + \left(-1 \cdot \frac{{\sin x}^{2} \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}{{\cos x}^{2}} + \left(-0.5 \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right) + \left(\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)}{\cos x}\right)} \]
4. Step-by-step derivation

   1. +-commutative 99.2%

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

   2. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ t_1 := \frac{{\sin x}^{3}}{{\cos x}^{3}}\\ t_2 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \varepsilon \cdot \left(1 + t_2\right) + \left({\varepsilon}^{2} \cdot \left(t_0 + t_1\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t_2 + \frac{\sin x \cdot \left(t_1 - t_0 \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))
        (t_2 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (+
    (* eps (+ 1.0 t_2))
    (+
     (* (pow eps 2.0) (+ t_0 t_1))
     (*
      (pow eps 3.0)
      (+
       0.3333333333333333
       (+
        t_2
        (/ (* (sin x) (- t_1 (* t_0 -0.3333333333333333))) (cos x)))))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = pow(sin(x), 3.0) / pow(cos(x), 3.0);
	double t_2 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return (eps * (1.0 + t_2)) + ((pow(eps, 2.0) * (t_0 + t_1)) + (pow(eps, 3.0) * (0.3333333333333333 + (t_2 + ((sin(x) * (t_1 - (t_0 * -0.3333333333333333))) / cos(x))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = sin(x) / cos(x)
    t_1 = (sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)
    t_2 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = (eps * (1.0d0 + t_2)) + (((eps ** 2.0d0) * (t_0 + t_1)) + ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_2 + ((sin(x) * (t_1 - (t_0 * (-0.3333333333333333d0)))) / cos(x))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) / Math.cos(x);
	double t_1 = Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0);
	double t_2 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return (eps * (1.0 + t_2)) + ((Math.pow(eps, 2.0) * (t_0 + t_1)) + (Math.pow(eps, 3.0) * (0.3333333333333333 + (t_2 + ((Math.sin(x) * (t_1 - (t_0 * -0.3333333333333333))) / Math.cos(x))))));
}

Python

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	t_1 = math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)
	t_2 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return (eps * (1.0 + t_2)) + ((math.pow(eps, 2.0) * (t_0 + t_1)) + (math.pow(eps, 3.0) * (0.3333333333333333 + (t_2 + ((math.sin(x) * (t_1 - (t_0 * -0.3333333333333333))) / math.cos(x))))))

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	t_1 = Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))
	t_2 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(Float64(eps * Float64(1.0 + t_2)) + Float64(Float64((eps ^ 2.0) * Float64(t_0 + t_1)) + Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_2 + Float64(Float64(sin(x) * Float64(t_1 - Float64(t_0 * -0.3333333333333333))) / cos(x)))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) / cos(x);
	t_1 = (sin(x) ^ 3.0) / (cos(x) ^ 3.0);
	t_2 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = (eps * (1.0 + t_2)) + (((eps ^ 2.0) * (t_0 + t_1)) + ((eps ^ 3.0) * (0.3333333333333333 + (t_2 + ((sin(x) * (t_1 - (t_0 * -0.3333333333333333))) / cos(x))))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$2 + N[(N[(N[Sin[x], $MachinePrecision] * N[(t$95$1 - N[(t$95$0 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 64.1%

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

   1. *-un-lft-identity 64.1%

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

   2. *-commutative 64.1%

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

   3. tan-quot 64.1%

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

   4. div-inv 64.0%

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

   5. prod-diff 64.0%

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

4. Applied egg-rr 64.0%

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

   1. +-commutative 64.0%

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

   2. fma-udef 64.0%

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

   3. *-rgt-identity 64.0%

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

   4. associate-+r+ 64.0%

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

   5. unsub-neg 64.0%

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

6. Simplified 64.1%

   \[\leadsto \color{blue}{\left(0 + \tan \left(\varepsilon + x\right)\right) - \frac{\sin x}{\cos x}} \]
7. Step-by-step derivation

   1. +-lft-identity 64.1%

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

   2. +-commutative 64.1%

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

   3. tan-sum 64.2%

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

   4. div-inv 64.2%

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

   5. tan-quot 64.2%

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

   6. fma-neg 64.3%

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

8. Applied egg-rr 64.3%

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

9. Taylor expanded in eps around 0 99.2%

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

10. Final simplification 99.2%

    \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \left({\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \frac{\sin x \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} - \frac{\sin x}{\cos x} \cdot -0.3333333333333333\right)}{\cos x}\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. fma-def 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-lft-identity 98.9%

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

   5. associate-/l* 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\frac{\cos x}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}}{\sin x}}\right) \]
7. Add Preprocessing

Alternative 6: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
  (*
   (pow eps 2.0)
   (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))) + ((eps ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + (Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)))) + (Math.pow(eps, 2.0) * ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
}

Python

def code(x, eps):
	return (eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))) + (math.pow(eps, 2.0) * ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))) + ((eps ^ 2.0) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0))));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

   1. *-un-lft-identity 64.1%

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

   2. *-commutative 64.1%

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

   3. tan-quot 64.1%

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

   4. div-inv 64.0%

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

   5. prod-diff 64.0%

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

4. Applied egg-rr 64.0%

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

   1. +-commutative 64.0%

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

   2. fma-udef 64.0%

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

   3. *-rgt-identity 64.0%

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

   4. associate-+r+ 64.0%

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

   5. unsub-neg 64.0%

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

6. Simplified 64.1%

   \[\leadsto \color{blue}{\left(0 + \tan \left(\varepsilon + x\right)\right) - \frac{\sin x}{\cos x}} \]
7. Step-by-step derivation

   1. +-lft-identity 64.1%

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

   2. +-commutative 64.1%

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

   3. tan-sum 64.2%

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

   4. div-inv 64.2%

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

   5. tan-quot 64.2%

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

   6. fma-neg 64.3%

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

8. Applied egg-rr 64.3%

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

9. Taylor expanded in eps around 0 98.9%

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

10. Final simplification 98.9%

    \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \]
11. Add Preprocessing

Alternative 7: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + \left(-1.5 \cdot {x}^{2} + \left(-0.25416666666666665 \cdot {x}^{6} + 0.875 \cdot {x}^{4}\right)\right)}{\sin x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
  (/
   (pow eps 2.0)
   (/
    (+
     1.0
     (+
      (* -1.5 (pow x 2.0))
      (+ (* -0.25416666666666665 (pow x 6.0)) (* 0.875 (pow x 4.0)))))
    (sin x)))))

C

double code(double x, double eps) {
	return fma(eps, (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), (pow(eps, 2.0) / ((1.0 + ((-1.5 * pow(x, 2.0)) + ((-0.25416666666666665 * pow(x, 6.0)) + (0.875 * pow(x, 4.0))))) / sin(x))));
}

Julia

function code(x, eps)
	return fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), Float64((eps ^ 2.0) / Float64(Float64(1.0 + Float64(Float64(-1.5 * (x ^ 2.0)) + Float64(Float64(-0.25416666666666665 * (x ^ 6.0)) + Float64(0.875 * (x ^ 4.0))))) / sin(x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(1.0 + N[(N[(-1.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25416666666666665 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.875 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. fma-def 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-lft-identity 98.9%

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

   5. associate-/l* 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\color{blue}{1 + \left(-1.5 \cdot {x}^{2} + \left(-0.25416666666666665 \cdot {x}^{6} + 0.875 \cdot {x}^{4}\right)\right)}}{\sin x}}\right) \]

7. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + \left(-1.5 \cdot {x}^{2} + \left(-0.25416666666666665 \cdot {x}^{6} + 0.875 \cdot {x}^{4}\right)\right)}{\sin x}}\right) \]
8. Add Preprocessing

Alternative 8: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + \left(-1.5 \cdot {x}^{2} + 0.875 \cdot {x}^{4}\right)}{\sin x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
  (/
   (pow eps 2.0)
   (/ (+ 1.0 (+ (* -1.5 (pow x 2.0)) (* 0.875 (pow x 4.0)))) (sin x)))))

C

double code(double x, double eps) {
	return fma(eps, (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), (pow(eps, 2.0) / ((1.0 + ((-1.5 * pow(x, 2.0)) + (0.875 * pow(x, 4.0)))) / sin(x))));
}

Julia

function code(x, eps)
	return fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), Float64((eps ^ 2.0) / Float64(Float64(1.0 + Float64(Float64(-1.5 * (x ^ 2.0)) + Float64(0.875 * (x ^ 4.0)))) / sin(x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(1.0 + N[(N[(-1.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.875 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. fma-def 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-lft-identity 98.9%

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

   5. associate-/l* 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\color{blue}{1 + \left(-1.5 \cdot {x}^{2} + 0.875 \cdot {x}^{4}\right)}}{\sin x}}\right) \]

7. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + \left(-1.5 \cdot {x}^{2} + 0.875 \cdot {x}^{4}\right)}{\sin x}}\right) \]
8. Add Preprocessing

Alternative 9: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + -1.5 \cdot {x}^{2}}{\sin x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
  (/ (pow eps 2.0) (/ (+ 1.0 (* -1.5 (pow x 2.0))) (sin x)))))

C

double code(double x, double eps) {
	return fma(eps, (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0))), (pow(eps, 2.0) / ((1.0 + (-1.5 * pow(x, 2.0))) / sin(x))));
}

Julia

function code(x, eps)
	return fma(eps, Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))), Float64((eps ^ 2.0) / Float64(Float64(1.0 + Float64(-1.5 * (x ^ 2.0))) / sin(x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(1.0 + N[(-1.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. fma-def 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-lft-identity 98.9%

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

   5. associate-/l* 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\color{blue}{1 + -1.5 \cdot {x}^{2}}}{\sin x}}\right) \]
7. Step-by-step derivation

   1. *-commutative 98.6%

      \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + \color{blue}{{x}^{2} \cdot -1.5}}{\sin x}}\right) \]

8. Simplified 98.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{\color{blue}{1 + {x}^{2} \cdot -1.5}}{\sin x}}\right) \]

9. Final simplification 98.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\frac{1 + -1.5 \cdot {x}^{2}}{\sin x}}\right) \]
10. Add Preprocessing

Alternative 10: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (pow (tan x) 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. *-lft-identity 98.4%

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

5. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

   2. expm1-udef 98.4%

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

   3. unpow2 98.4%

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

   4. unpow2 98.4%

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

   5. frac-times 98.4%

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

   6. tan-quot 98.4%

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

   7. tan-quot 98.4%

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

   8. pow2 98.4%

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

7. Applied egg-rr 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 11: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow (tan x) 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(tan(x), 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (tan(x) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(Math.tan(x), 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(math.tan(x), 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (tan(x) ^ 2.0));
end

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. *-lft-identity 98.4%

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

5. Simplified 98.4%

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

   1. +-commutative 98.4%

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

   2. distribute-lft-in 98.4%

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

   3. unpow2 98.4%

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

   4. unpow2 98.4%

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

   5. frac-times 98.4%

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

   6. tan-quot 98.4%

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

   7. tan-quot 98.4%

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

   8. pow2 98.4%

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

   9. *-rgt-identity 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

   \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
9. Add Preprocessing

Alternative 12: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* eps (pow x 2.0))))

C

double code(double x, double eps) {
	return eps + (eps * pow(x, 2.0));
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps + (eps * math.pow(x, 2.0))

Julia

function code(x, eps)
	return Float64(eps + Float64(eps * (x ^ 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.4%

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

   1. cancel-sign-sub-inv 98.4%

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

   2. metadata-eval 98.4%

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

   3. *-lft-identity 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0 97.3%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

   \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
10. Add Preprocessing

Alternative 13: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. fma-def 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-lft-identity 98.9%

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

   5. associate-/l* 98.9%

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

   6. *-commutative 98.9%

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

   7. associate-/r* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 96.7%

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

7. Final simplification 96.7%

   \[\leadsto \varepsilon \]
8. Add Preprocessing

Developer target: 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]

Reproduce

herbie shell --seed 2024024 
(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)))

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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