2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.7%

Time: 23.8s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% 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: 99.7% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
        (t_1 (/ (sin x) (cos x)))
        (t_2
         (-
          (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
          (* -0.3333333333333333 t_0))))
   (+
    (+
     (*
      (pow eps 4.0)
      (-
       (/ (* (sin x) (+ 0.3333333333333333 (pow (tan x) 2.0))) (cos x))
       (+
        (* t_1 -0.3333333333333333)
        (-
         (* -0.3333333333333333 (pow (tan x) 3.0))
         (/ (* (sin x) t_2) (cos x))))))
     (+
      (* eps (+ t_0 1.0))
      (* (pow eps 3.0) (+ 0.3333333333333333 (+ t_0 t_2)))))
    (* (pow eps 2.0) (+ t_1 (/ (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);
	double t_1 = sin(x) / cos(x);
	double t_2 = (pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (-0.3333333333333333 * t_0);
	return ((pow(eps, 4.0) * (((sin(x) * (0.3333333333333333 + pow(tan(x), 2.0))) / cos(x)) - ((t_1 * -0.3333333333333333) + ((-0.3333333333333333 * pow(tan(x), 3.0)) - ((sin(x) * t_2) / cos(x)))))) + ((eps * (t_0 + 1.0)) + (pow(eps, 3.0) * (0.3333333333333333 + (t_0 + t_2))))) + (pow(eps, 2.0) * (t_1 + (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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    t_1 = sin(x) / cos(x)
    t_2 = ((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - ((-0.3333333333333333d0) * t_0)
    code = (((eps ** 4.0d0) * (((sin(x) * (0.3333333333333333d0 + (tan(x) ** 2.0d0))) / cos(x)) - ((t_1 * (-0.3333333333333333d0)) + (((-0.3333333333333333d0) * (tan(x) ** 3.0d0)) - ((sin(x) * t_2) / cos(x)))))) + ((eps * (t_0 + 1.0d0)) + ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_0 + t_2))))) + ((eps ** 2.0d0) * (t_1 + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))))
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);
	double t_1 = Math.sin(x) / Math.cos(x);
	double t_2 = (Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (-0.3333333333333333 * t_0);
	return ((Math.pow(eps, 4.0) * (((Math.sin(x) * (0.3333333333333333 + Math.pow(Math.tan(x), 2.0))) / Math.cos(x)) - ((t_1 * -0.3333333333333333) + ((-0.3333333333333333 * Math.pow(Math.tan(x), 3.0)) - ((Math.sin(x) * t_2) / Math.cos(x)))))) + ((eps * (t_0 + 1.0)) + (Math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 + t_2))))) + (Math.pow(eps, 2.0) * (t_1 + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	t_1 = math.sin(x) / math.cos(x)
	t_2 = (math.pow(math.sin(x), 4.0) / math.pow(math.cos(x), 4.0)) - (-0.3333333333333333 * t_0)
	return ((math.pow(eps, 4.0) * (((math.sin(x) * (0.3333333333333333 + math.pow(math.tan(x), 2.0))) / math.cos(x)) - ((t_1 * -0.3333333333333333) + ((-0.3333333333333333 * math.pow(math.tan(x), 3.0)) - ((math.sin(x) * t_2) / math.cos(x)))))) + ((eps * (t_0 + 1.0)) + (math.pow(eps, 3.0) * (0.3333333333333333 + (t_0 + t_2))))) + (math.pow(eps, 2.0) * (t_1 + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0))))

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	t_1 = sin(x) / cos(x);
	t_2 = ((sin(x) ^ 4.0) / (cos(x) ^ 4.0)) - (-0.3333333333333333 * t_0);
	tmp = (((eps ^ 4.0) * (((sin(x) * (0.3333333333333333 + (tan(x) ^ 2.0))) / cos(x)) - ((t_1 * -0.3333333333333333) + ((-0.3333333333333333 * (tan(x) ^ 3.0)) - ((sin(x) * t_2) / cos(x)))))) + ((eps * (t_0 + 1.0)) + ((eps ^ 3.0) * (0.3333333333333333 + (t_0 + t_2))))) + ((eps ^ 2.0) * (t_1 + ((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]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] - N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(0.3333333333333333 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $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 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(t$95$1 + 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 63.8%

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

   1. tan-sum 64.0%

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

   2. clear-num 63.3%

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

4. Applied egg-rr 63.3%

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

5. Taylor expanded in eps around 0 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

7. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 99.9%

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

9. Simplified 99.9%

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

    1. expm1-log1p-u 99.6%

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

    2. expm1-udef 99.6%

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

    3. cube-div 99.6%

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

    4. tan-quot 99.6%

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

11. Applied egg-rr 99.6%

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

    1. expm1-def 99.6%

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

    2. expm1-log1p 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

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 (+ t_2 1.0))
    (+
     (* (pow eps 2.0) (+ t_0 t_1))
     (*
      (pow eps 3.0)
      (+
       0.3333333333333333
       (-
        t_2
        (/ (* (sin x) (- (* t_0 -0.3333333333333333) t_1)) (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 * (t_2 + 1.0)) + ((pow(eps, 2.0) * (t_0 + t_1)) + (pow(eps, 3.0) * (0.3333333333333333 + (t_2 - ((sin(x) * ((t_0 * -0.3333333333333333) - t_1)) / 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 * (t_2 + 1.0d0)) + (((eps ** 2.0d0) * (t_0 + t_1)) + ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_2 - ((sin(x) * ((t_0 * (-0.3333333333333333d0)) - t_1)) / 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 * (t_2 + 1.0)) + ((Math.pow(eps, 2.0) * (t_0 + t_1)) + (Math.pow(eps, 3.0) * (0.3333333333333333 + (t_2 - ((Math.sin(x) * ((t_0 * -0.3333333333333333) - t_1)) / 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 * (t_2 + 1.0)) + ((math.pow(eps, 2.0) * (t_0 + t_1)) + (math.pow(eps, 3.0) * (0.3333333333333333 + (t_2 - ((math.sin(x) * ((t_0 * -0.3333333333333333) - t_1)) / 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(t_2 + 1.0)) + 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(Float64(t_0 * -0.3333333333333333) - t_1)) / 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 * (t_2 + 1.0)) + (((eps ^ 2.0) * (t_0 + t_1)) + ((eps ^ 3.0) * (0.3333333333333333 + (t_2 - ((sin(x) * ((t_0 * -0.3333333333333333) - t_1)) / 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[(t$95$2 + 1.0), $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[(N[(t$95$0 * -0.3333333333333333), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 63.8%

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

   1. tan-sum 64.0%

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

   2. clear-num 63.3%

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

4. Applied egg-rr 63.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}}} - \tan x \]
5. Step-by-step derivation

   1. associate-/r/ 63.9%

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

   2. fma-neg 63.9%

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

6. Applied egg-rr 63.9%

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

7. Taylor expanded in eps around 0 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ \varepsilon \cdot \left(\frac{t\_0}{e^{\log t\_1}} + 1\right) + \frac{{\varepsilon}^{2} \cdot \left(\sin x \cdot \left(\frac{t\_0}{t\_1} + 1\right)\right)}{\cos x} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (pow (cos x) 2.0)))
   (+
    (* eps (+ (/ t_0 (exp (log t_1))) 1.0))
    (/ (* (pow eps 2.0) (* (sin x) (+ (/ t_0 t_1) 1.0))) (cos x)))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	return (eps * ((t_0 / exp(log(t_1))) + 1.0)) + ((pow(eps, 2.0) * (sin(x) * ((t_0 / t_1) + 1.0))) / 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
    t_0 = sin(x) ** 2.0d0
    t_1 = cos(x) ** 2.0d0
    code = (eps * ((t_0 / exp(log(t_1))) + 1.0d0)) + (((eps ** 2.0d0) * (sin(x) * ((t_0 / t_1) + 1.0d0))) / cos(x))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0)
	t_1 = math.pow(math.cos(x), 2.0)
	return (eps * ((t_0 / math.exp(math.log(t_1))) + 1.0)) + ((math.pow(eps, 2.0) * (math.sin(x) * ((t_0 / t_1) + 1.0))) / math.cos(x))

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	return Float64(Float64(eps * Float64(Float64(t_0 / exp(log(t_1))) + 1.0)) + Float64(Float64((eps ^ 2.0) * Float64(sin(x) * Float64(Float64(t_0 / t_1) + 1.0))) / cos(x)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) ^ 2.0;
	t_1 = cos(x) ^ 2.0;
	tmp = (eps * ((t_0 / exp(log(t_1))) + 1.0)) + (((eps ^ 2.0) * (sin(x) * ((t_0 / t_1) + 1.0))) / cos(x));
end

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. add-exp-log 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.6% 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 63.8%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. expm1-log1p-u 99.4%

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

   2. expm1-udef 99.4%

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

5. Applied egg-rr 99.4%

   \[\leadsto \varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \frac{{\varepsilon}^{2} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x \cdot \left(1 + {\tan x}^{2}\right)\right)} - 1\right)}}{\cos x} \]
6. Step-by-step derivation

   1. expm1-def 99.4%

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

   2. expm1-log1p 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. fma-def 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-lft-identity 99.8%

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

   5. associate-/l* 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-/r* 99.8%

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

5. Simplified 99.8%

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

   1. unpow2 99.8%

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

   2. div-inv 99.8%

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

   3. times-frac 99.8%

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

   4. +-commutative 99.8%

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

   5. unpow2 99.8%

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

   6. unpow2 99.8%

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

   7. frac-times 99.8%

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

   8. tan-quot 99.8%

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

   9. tan-quot 99.8%

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

   10. unpow2 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-/r/ 99.8%

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

   4. /-rgt-identity 99.8%

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

   5. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 6: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1, \frac{{\varepsilon}^{2}}{x \cdot -1.3333333333333333 + \frac{1}{x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (+ (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) 1.0)
  (/ (pow eps 2.0) (+ (* x -1.3333333333333333) (/ 1.0 x)))))

C

double code(double x, double eps) {
	return fma(eps, ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0), (pow(eps, 2.0) / ((x * -1.3333333333333333) + (1.0 / x))));
}

Julia

function code(x, eps)
	return fma(eps, Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + 1.0), Float64((eps ^ 2.0) / Float64(Float64(x * -1.3333333333333333) + Float64(1.0 / x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] / N[(N[(x * -1.3333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. fma-def 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-lft-identity 99.8%

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

   5. associate-/l* 99.8%

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

   6. *-commutative 99.8%

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

   7. associate-/r* 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, \frac{{\varepsilon}^{2}}{\color{blue}{-1.3333333333333333 \cdot x + \frac{1}{x}}}\right) \]

7. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}} + 1, \frac{{\varepsilon}^{2}}{x \cdot -1.3333333333333333 + \frac{1}{x}}\right) \]
8. Add Preprocessing

Alternative 7: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

   1. log1p-expm1-u 63.8%

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

4. Applied egg-rr 63.8%

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

5. Taylor expanded in eps around 0 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. *-lft-identity 99.4%

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

   4. +-commutative 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 8: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 99.4%

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

   1. cancel-sign-sub-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. *-lft-identity 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 9: 98.2% 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 63.8%

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

3. Taylor expanded in x around 0 98.2%

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

   1. tan-quot 98.2%

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

   2. expm1-log1p-u 98.2%

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

   3. expm1-udef 6.6%

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

5. Applied egg-rr 6.6%

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

   1. expm1-def 98.2%

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

   2. expm1-log1p 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 10: 98.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in x around 0 98.2%

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

4. Taylor expanded in eps around 0 98.2%

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

5. Final simplification 98.2%

   \[\leadsto \varepsilon \]
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 2024031 
(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)))