2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.5%

Time: 20.5s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (/ t_0 (pow (cos x) 2.0))))
   (*
    eps
    (-
     (* t_0 (pow (cos x) -2.0))
     (-
      -1.0
      (*
       eps
       (+
        (*
         eps
         (+
          0.3333333333333333
          (+
           t_1
           (-
            (/ (pow (sin x) 4.0) (pow (cos x) 4.0))
            (* t_1 -0.3333333333333333)))))
        (+ (/ (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);
	double t_1 = t_0 / pow(cos(x), 2.0);
	return eps * ((t_0 * pow(cos(x), -2.0)) - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_1 + ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) - (t_1 * -0.3333333333333333))))) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(x) ** 2.0d0
    t_1 = t_0 / (cos(x) ** 2.0d0)
    code = eps * ((t_0 * (cos(x) ** (-2.0d0))) - ((-1.0d0) - (eps * ((eps * (0.3333333333333333d0 + (t_1 + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (t_1 * (-0.3333333333333333d0)))))) + ((sin(x) / cos(x)) + ((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);
	double t_1 = t_0 / Math.pow(Math.cos(x), 2.0);
	return eps * ((t_0 * Math.pow(Math.cos(x), -2.0)) - (-1.0 - (eps * ((eps * (0.3333333333333333 + (t_1 + ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (t_1 * -0.3333333333333333))))) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

   1. tan-sum 60.7%

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

   2. div-inv 60.7%

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

   3. fmm-def 60.7%

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

4. Applied egg-rr 60.7%

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

   1. fmm-undef 60.7%

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

   2. associate-*r/ 60.7%

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

   3. *-rgt-identity 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in eps around 0 99.8%

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

   1. pow1 99.8%

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

   2. mul-1-neg 99.8%

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

   3. div-inv 99.8%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

    1. unpow1 99.8%

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

    2. distribute-rgt-neg-in 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

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
    t_1 = cos(x) ** 2.0d0
    t_2 = t_0 / t_1
    code = eps * (t_2 + (1.0d0 + (eps * (eps * ((t_2 + (0.3333333333333333d0 + (((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0))) / eps))) + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (t_0 * ((-0.3333333333333333d0) / t_1))))))))
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);
	double t_2 = t_0 / t_1;
	return eps * (t_2 + (1.0 + (eps * (eps * ((t_2 + (0.3333333333333333 + (((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0))) / eps))) + ((Math.pow(Math.sin(x), 4.0) / Math.pow(Math.cos(x), 4.0)) - (t_0 * (-0.3333333333333333 / t_1))))))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

   1. tan-sum 60.7%

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

   2. div-inv 60.7%

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

   3. fmm-def 60.7%

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

4. Applied egg-rr 60.7%

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

   1. fmm-undef 60.7%

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

   2. associate-*r/ 60.7%

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

   3. *-rgt-identity 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in eps around 0 99.8%

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

8. Taylor expanded in eps around inf 99.8%

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

   1. associate--r+ 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

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);
	return eps * ((1.0 + (eps * ((eps * (0.3333333333333333 + (t_1 + (pow(sin(x), 4.0) - (t_1 * -0.3333333333333333))))) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))) + (t_0 * pow(cos(x), -2.0)));
}

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
    t_0 = sin(x) ** 2.0d0
    t_1 = t_0 / (cos(x) ** 2.0d0)
    code = eps * ((1.0d0 + (eps * ((eps * (0.3333333333333333d0 + (t_1 + ((sin(x) ** 4.0d0) - (t_1 * (-0.3333333333333333d0)))))) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))) + (t_0 * (cos(x) ** (-2.0d0))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0);
	double t_1 = t_0 / Math.pow(Math.cos(x), 2.0);
	return eps * ((1.0 + (eps * ((eps * (0.3333333333333333 + (t_1 + (Math.pow(Math.sin(x), 4.0) - (t_1 * -0.3333333333333333))))) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))) + (t_0 * Math.pow(Math.cos(x), -2.0)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.2%

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

   1. tan-sum 60.7%

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

   2. div-inv 60.7%

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

   3. fmm-def 60.7%

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

4. Applied egg-rr 60.7%

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

   1. fmm-undef 60.7%

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

   2. associate-*r/ 60.7%

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

   3. *-rgt-identity 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in eps around 0 99.8%

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

   1. pow1 99.8%

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

   2. mul-1-neg 99.8%

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

   3. div-inv 99.8%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

    1. unpow1 99.8%

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

    2. distribute-rgt-neg-in 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in x around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. tan-sum 60.7%

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

   2. div-inv 60.7%

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

   3. fmm-def 60.7%

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

4. Applied egg-rr 60.7%

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

   1. fmm-undef 60.7%

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

   2. associate-*r/ 60.7%

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

   3. *-rgt-identity 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in eps around 0 99.8%

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

   1. pow1 99.8%

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

   2. mul-1-neg 99.8%

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

   3. div-inv 99.8%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

9. Applied egg-rr 99.8%

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

    1. unpow1 99.8%

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

    2. distribute-rgt-neg-in 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in eps around 0 99.7%

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

    1. mul-1-neg 99.7%

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

    2. distribute-rgt-neg-in 99.7%

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

    3. distribute-lft-out 99.7%

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

    4. distribute-lft-neg-in 99.7%

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

    5. metadata-eval 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 5: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 98.8%

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

   1. sub-neg 98.8%

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

   2. mul-1-neg 98.8%

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

   3. remove-double-neg 98.8%

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

5. Simplified 98.8%

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

Alternative 6: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* eps (+ 1.0 (* 0.3333333333333333 (pow eps 2.0))))
  (* x (* eps (+ eps x)))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * pow(eps, 2.0)))) + (x * (eps * (eps + x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (1.0d0 + (0.3333333333333333d0 * (eps ** 2.0d0)))) + (x * (eps * (eps + x)))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * Math.pow(eps, 2.0)))) + (x * (eps * (eps + x)));
}

Python

def code(x, eps):
	return (eps * (1.0 + (0.3333333333333333 * math.pow(eps, 2.0)))) + (x * (eps * (eps + x)))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * (eps ^ 2.0)))) + Float64(x * Float64(eps * Float64(eps + x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + (0.3333333333333333 * (eps ^ 2.0)))) + (x * (eps * (eps + x)));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2}\right)} \]

5. Taylor expanded in eps around 0 97.6%

   \[\leadsto \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \color{blue}{\left(\varepsilon \cdot \left(\varepsilon + x\right)\right)} \]
6. Add Preprocessing

Alternative 7: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* eps (+ 1.0 (* 0.3333333333333333 (pow eps 2.0)))) (* x (* eps x))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * pow(eps, 2.0)))) + (x * (eps * x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (1.0d0 + (0.3333333333333333d0 * (eps ** 2.0d0)))) + (x * (eps * x))
end function

Java

public static double code(double x, double eps) {
	return (eps * (1.0 + (0.3333333333333333 * Math.pow(eps, 2.0)))) + (x * (eps * x));
}

Python

def code(x, eps):
	return (eps * (1.0 + (0.3333333333333333 * math.pow(eps, 2.0)))) + (x * (eps * x))

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * (eps ^ 2.0)))) + Float64(x * Float64(eps * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (1.0 + (0.3333333333333333 * (eps ^ 2.0)))) + (x * (eps * x));
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + 1.3333333333333333 \cdot {\varepsilon}^{2}\right)\right) + {\varepsilon}^{2}\right)} \]

5. Taylor expanded in eps around 0 97.5%

   \[\leadsto \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) + x \cdot \color{blue}{\left(\varepsilon \cdot x\right)} \]
6. Add Preprocessing

Alternative 8: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (+ (* 0.3333333333333333 (pow eps 2.0)) (* eps x)))))

C

double code(double x, double eps) {
	return eps * (1.0 + ((0.3333333333333333 * pow(eps, 2.0)) + (eps * x)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((0.3333333333333333d0 * (eps ** 2.0d0)) + (eps * x)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + ((0.3333333333333333 * Math.pow(eps, 2.0)) + (eps * x)));
}

Python

def code(x, eps):
	return eps * (1.0 + ((0.3333333333333333 * math.pow(eps, 2.0)) + (eps * x)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(Float64(0.3333333333333333 * (eps ^ 2.0)) + Float64(eps * x))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((0.3333333333333333 * (eps ^ 2.0)) + (eps * x)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 97.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(0.3333333333333333 \cdot {\varepsilon}^{2} + \varepsilon \cdot x\right)\right)} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (* 0.3333333333333333 (pow eps 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (0.3333333333333333 * pow(eps, 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (0.3333333333333333d0 * (eps ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (0.3333333333333333 * Math.pow(eps, 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + (0.3333333333333333 * math.pow(eps, 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(0.3333333333333333 * (eps ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (0.3333333333333333 * (eps ^ 2.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(0.3333333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 97.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 97.3%

      \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot 0.3333333333333333}\right) \]

6. Simplified 97.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot 0.3333333333333333\right)} \]

7. Final simplification 97.3%

   \[\leadsto \varepsilon \cdot \left(1 + 0.3333333333333333 \cdot {\varepsilon}^{2}\right) \]
8. Add Preprocessing

Alternative 10: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in x around 0 7.9%

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

4. Taylor expanded in eps around 0 7.9%

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

5. Taylor expanded in eps around inf 97.3%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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

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

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