2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.6%

Time: 35.6s

Alternatives: 15

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: 62.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\\ t_2 := {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\\ t_3 := 0.16666666666666666 + \mathsf{fma}\left(-0.5, t\_0 + 1, 0.16666666666666666 \cdot t\_0\right)\\ \mathsf{fma}\left(\varepsilon, t\_1, \frac{{\varepsilon}^{2}}{\cos x} \cdot \left(\sin x \cdot t\_1\right)\right) + \left({\varepsilon}^{3} \cdot \left(t\_2 - t\_3\right) - {\varepsilon}^{4} \cdot \left(\left(t\_3 - t\_2\right) \cdot \frac{\sin x}{\cos x} + -0.3333333333333333 \cdot \left(\tan x + {\tan x}^{3}\right)\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.7%

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

5. Simplified 99.8%

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

   1. associate-+r- 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-+r- 99.8%

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

9. Applied egg-rr 99.8%

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

    1. expm1-log1p-u 98.8%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\cos x} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left({\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)\right)}\right)\right) \]

    2. expm1-udef 98.8%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\cos x} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left({\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\frac{\sin x}{\cos x} \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \cdot -0.3333333333333333\right)} - 1\right)}\right)\right) \]

11. Applied egg-rr 98.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\cos x} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left({\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \left(\tan x + {\tan x}^{3}\right)\right)} - 1\right)}\right)\right) \]
12. Step-by-step derivation

    1. expm1-def 98.8%

       \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\cos x} \cdot \left(\sin x \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left({\varepsilon}^{3} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + {\varepsilon}^{4} \cdot \left(\frac{\sin x}{\cos x} \cdot \left(\left(0.16666666666666666 + \mathsf{fma}\left(-0.5, 1 + {\tan x}^{2}, 0.16666666666666666 \cdot {\tan x}^{2}\right)\right) - {\left(\tan x \cdot \mathsf{hypot}\left(1, \tan x\right)\right)}^{2}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \left(\tan x + {\tan x}^{3}\right)\right)\right)}\right)\right) \]

    2. expm1-log1p 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := t\_0 \cdot {\cos x}^{-2}\\ t_2 := t\_1 + 1\\ t_3 := {\cos x}^{2}\\ \mathsf{fma}\left(\varepsilon, t\_2, \frac{{\varepsilon}^{2}}{\frac{\cos x}{\sin x \cdot t\_2}}\right) + {\varepsilon}^{3} \cdot \left(\frac{t\_0}{\frac{t\_3}{t\_2}} - \left(\left(-0.3333333333333333 + -0.5 \cdot t\_1\right) + \frac{0.16666666666666666}{\frac{t\_3}{t\_0}}\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. expm1-log1p-u 61.8%

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

4. Applied egg-rr 61.8%

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

5. Taylor expanded in eps around 0 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \mathsf{fma}\left(\varepsilon, t\_0 + 1, {\varepsilon}^{3} \cdot \left(\left(t\_0 + 0.3333333333333333\right) - \mathsf{fma}\left(-0.3333333333333333, t\_0, \frac{-{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (+
    (fma
     eps
     (+ t_0 1.0)
     (*
      (pow eps 3.0)
      (-
       (+ t_0 0.3333333333333333)
       (fma
        -0.3333333333333333
        t_0
        (/ (- (pow (sin x) 4.0)) (pow (cos x) 4.0))))))
    (*
     (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) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return fma(eps, (t_0 + 1.0), (pow(eps, 3.0) * ((t_0 + 0.3333333333333333) - fma(-0.3333333333333333, t_0, (-pow(sin(x), 4.0) / pow(cos(x), 4.0)))))) + (pow(eps, 2.0) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))));
}

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(fma(eps, Float64(t_0 + 1.0), Float64((eps ^ 3.0) * Float64(Float64(t_0 + 0.3333333333333333) - fma(-0.3333333333333333, t_0, Float64(Float64(-(sin(x) ^ 4.0)) / (cos(x) ^ 4.0)))))) + Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))
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]}, N[(N[(eps * N[(t$95$0 + 1.0), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(t$95$0 + 0.3333333333333333), $MachinePrecision] - N[(-0.3333333333333333 * t$95$0 + N[((-N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision]) / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $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 61.8%

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

   1. tan-sum 62.0%

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

   2. div-inv 61.9%

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

   3. *-un-lft-identity 61.9%

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

   4. prod-diff 61.9%

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

   5. *-commutative 61.9%

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

   6. *-un-lft-identity 61.9%

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

   7. *-commutative 61.9%

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

   8. *-un-lft-identity 61.9%

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

4. Applied egg-rr 61.9%

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

   1. +-commutative 61.9%

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

   2. fma-udef 61.9%

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

   3. associate-+r+ 61.9%

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

   4. unsub-neg 61.9%

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

6. Simplified 62.0%

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

7. Taylor expanded in eps around 0 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \left(\varepsilon \cdot \left(t\_0 + 1\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(t\_0 - \left(t\_0 \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (+
    (*
     (pow eps 2.0)
     (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0))))
    (+
     (* eps (+ t_0 1.0))
     (*
      (pow eps 3.0)
      (+
       0.3333333333333333
       (-
        t_0
        (-
         (* t_0 -0.3333333333333333)
         (/ (pow (sin x) 4.0) (pow (cos x) 4.0))))))))))

C

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

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 ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))) + ((eps * (t_0 + 1.0d0)) + ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_0 - ((t_0 * (-0.3333333333333333d0)) - ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)))))))
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 (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)))) + ((eps * (t_0 + 1.0)) + (Math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 - ((t_0 * -0.3333333333333333) - (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return (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)))) + ((eps * (t_0 + 1.0)) + (math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 - ((t_0 * -0.3333333333333333) - (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)))))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(Float64((eps ^ 2.0) * Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))) + Float64(Float64(eps * Float64(t_0 + 1.0)) + Float64((eps ^ 3.0) * Float64(0.3333333333333333 + Float64(t_0 - Float64(Float64(t_0 * -0.3333333333333333) - Float64((sin(x) ^ 4.0) / (cos(x) ^ 4.0))))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = ((eps ^ 2.0) * ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))) + ((eps * (t_0 + 1.0)) + ((eps ^ 3.0) * (0.3333333333333333 + (t_0 - ((t_0 * -0.3333333333333333) - ((sin(x) ^ 4.0) / (cos(x) ^ 4.0)))))));
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]}, N[(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] + N[(N[(eps * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 3.0], $MachinePrecision] * N[(0.3333333333333333 + N[(t$95$0 - N[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.8%

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

   1. tan-sum 62.0%

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

   2. div-inv 61.9%

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

   3. *-un-lft-identity 61.9%

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

   4. prod-diff 61.9%

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

   5. *-commutative 61.9%

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

   6. *-un-lft-identity 61.9%

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

   7. *-commutative 61.9%

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

   8. *-un-lft-identity 61.9%

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

4. Applied egg-rr 61.9%

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

   1. +-commutative 61.9%

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

   2. fma-udef 61.9%

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

   3. associate-+r+ 61.9%

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

   4. unsub-neg 61.9%

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

6. Simplified 62.0%

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

7. Taylor expanded in eps around 0 99.7%

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

8. Final simplification 99.7%

   \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \left(\varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + 1\right) + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot -0.3333333333333333 - \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 5: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. fma-def 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-lft-identity 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)) + ((pow(eps, 2.0) * (sin(x) * (pow(tan(x), 2.0) + 1.0))) / cos(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 ** 2.0d0) * (sin(x) * ((tan(x) ** 2.0d0) + 1.0d0))) / cos(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)) + ((Math.pow(eps, 2.0) * (Math.sin(x) * (Math.pow(Math.tan(x), 2.0) + 1.0))) / Math.cos(x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. sqr-neg 99.4%

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

   4. mul-1-neg 99.4%

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

   5. mul-1-neg 99.4%

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

   6. sqrt-unprod 43.0%

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

   7. add-sqr-sqrt 98.9%

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

   8. *-commutative 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. neg-mul-1 98.9%

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

7. Simplified 98.9%

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

   1. associate-*r* 98.9%

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

   2. cancel-sign-sub-inv 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-un-lft-identity 98.9%

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

   5. distribute-rgt-in 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. add-sqr-sqrt 43.0%

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

   8. sqrt-unprod 99.4%

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

   9. sqr-neg 99.4%

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

   10. sqrt-unprod 99.4%

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

   11. add-sqr-sqrt 99.4%

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

9. Applied egg-rr 99.4%

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

    1. *-lft-identity 99.4%

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

    2. distribute-rgt-in 99.4%

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

    3. associate-*l* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 7: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. *-un-lft-identity 98.5%

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

   2. add-sqr-sqrt 98.5%

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

   3. pow2 98.5%

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

   4. sqrt-div 98.5%

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

   5. unpow2 98.5%

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

   6. sqrt-prod 47.9%

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

   7. add-sqr-sqrt 98.5%

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

   8. unpow2 98.5%

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

   9. sqrt-prod 97.8%

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

   10. add-sqr-sqrt 98.5%

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

   11. tan-quot 98.5%

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

5. Applied egg-rr 99.4%

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

   1. *-lft-identity 98.5%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 8: 99.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return (eps * (t_0 + 1.0)) + ((pow(eps, 2.0) * (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 = tan(x) ** 2.0d0
    code = (eps * (t_0 + 1.0d0)) + (((eps ** 2.0d0) * (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.tan(x), 2.0);
	return (eps * (t_0 + 1.0)) + ((Math.pow(eps, 2.0) * (Math.sin(x) * (1.0 - t_0))) / Math.cos(x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.4%

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

   3. sqr-neg 99.4%

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

   4. mul-1-neg 99.4%

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

   5. mul-1-neg 99.4%

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

   6. sqrt-unprod 43.0%

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

   7. add-sqr-sqrt 98.9%

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

   8. *-commutative 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-commutative 98.9%

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

   2. neg-mul-1 98.9%

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

7. Simplified 98.9%

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

   1. *-un-lft-identity 98.5%

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

   2. add-sqr-sqrt 98.5%

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

   3. pow2 98.5%

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

   4. sqrt-div 98.5%

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

   5. unpow2 98.5%

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

   6. sqrt-prod 47.9%

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

   7. add-sqr-sqrt 98.5%

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

   8. unpow2 98.5%

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

   9. sqrt-prod 97.8%

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

   10. add-sqr-sqrt 98.5%

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

   11. tan-quot 98.5%

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

9. Applied egg-rr 98.9%

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

    1. *-lft-identity 98.5%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 9: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0)) + ((x * pow(eps, 2.0)) / cos(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)) + ((x * (eps ** 2.0d0)) / cos(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)) + ((x * Math.pow(eps, 2.0)) / Math.cos(x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Taylor expanded in x around 0 98.9%

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

5. Final simplification 98.9%

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

Alternative 10: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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) / 2.0d0)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

   1. unpow2 98.5%

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

   2. cos-mult 98.5%

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

7. Applied egg-rr 98.5%

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

   1. +-commutative 98.5%

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

   2. +-inverses 98.5%

      \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{\frac{\cos \color{blue}{0} + \cos \left(x + x\right)}{2}}\right) \]

   3. cos-0 98.5%

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

   4. count-2 98.5%

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

   5. *-commutative 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 11: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. add-sqr-sqrt 98.5%

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

   3. pow2 98.5%

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

   4. sqrt-div 98.5%

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

   5. unpow2 98.5%

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

   6. sqrt-prod 47.9%

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

   7. add-sqr-sqrt 98.5%

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

   8. unpow2 98.5%

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

   9. sqrt-prod 97.8%

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

   10. add-sqr-sqrt 98.5%

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

   11. tan-quot 98.5%

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

7. Applied egg-rr 98.5%

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

   1. *-lft-identity 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 12: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in x around 0 97.3%

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

7. Final simplification 97.3%

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

Alternative 13: 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 61.8%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in x around 0 97.3%

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

   1. *-commutative 97.3%

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

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 14: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in x around 0 96.5%

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

   1. expm1-log1p-u 96.5%

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

   2. expm1-udef 7.3%

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

   3. quot-tan 7.3%

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

5. Applied egg-rr 7.3%

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

   1. expm1-def 96.5%

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

   2. expm1-log1p 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 15: 97.8% 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 61.8%

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

3. Taylor expanded in x around 0 96.5%

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

4. Taylor expanded in eps around 0 96.5%

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

5. Final simplification 96.5%

   \[\leadsto \varepsilon \]
6. 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 2024075 
(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)))