2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.7%

Time: 19.8s

Alternatives: 8

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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\sin x}^{2}}{{\cos x}^{2}}\\ \frac{\varepsilon}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(-1 - \left(t\_0 + {\varepsilon}^{2} \cdot \left(0.3333333333333333 + \left(0.3333333333333333 \cdot t\_0 + {\varepsilon}^{2} \cdot \left(0.13333333333333333 + t\_0 \cdot 0.13333333333333333\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
   (*
    (/ eps (+ -1.0 (* (tan x) (tan eps))))
    (-
     -1.0
     (+
      t_0
      (*
       (pow eps 2.0)
       (+
        0.3333333333333333
        (+
         (* 0.3333333333333333 t_0)
         (*
          (pow eps 2.0)
          (+ 0.13333333333333333 (* t_0 0.13333333333333333)))))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0) / pow(cos(x), 2.0);
	return (eps / (-1.0 + (tan(x) * tan(eps)))) * (-1.0 - (t_0 + (pow(eps, 2.0) * (0.3333333333333333 + ((0.3333333333333333 * t_0) + (pow(eps, 2.0) * (0.13333333333333333 + (t_0 * 0.13333333333333333))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = (eps / ((-1.0d0) + (tan(x) * tan(eps)))) * ((-1.0d0) - (t_0 + ((eps ** 2.0d0) * (0.3333333333333333d0 + ((0.3333333333333333d0 * t_0) + ((eps ** 2.0d0) * (0.13333333333333333d0 + (t_0 * 0.13333333333333333d0))))))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0);
	return (eps / (-1.0 + (Math.tan(x) * Math.tan(eps)))) * (-1.0 - (t_0 + (Math.pow(eps, 2.0) * (0.3333333333333333 + ((0.3333333333333333 * t_0) + (Math.pow(eps, 2.0) * (0.13333333333333333 + (t_0 * 0.13333333333333333))))))));
}

Python

def code(x, eps):
	t_0 = math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)
	return (eps / (-1.0 + (math.tan(x) * math.tan(eps)))) * (-1.0 - (t_0 + (math.pow(eps, 2.0) * (0.3333333333333333 + ((0.3333333333333333 * t_0) + (math.pow(eps, 2.0) * (0.13333333333333333 + (t_0 * 0.13333333333333333))))))))

Julia

function code(x, eps)
	t_0 = Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))
	return Float64(Float64(eps / Float64(-1.0 + Float64(tan(x) * tan(eps)))) * Float64(-1.0 - Float64(t_0 + Float64((eps ^ 2.0) * Float64(0.3333333333333333 + Float64(Float64(0.3333333333333333 * t_0) + Float64((eps ^ 2.0) * Float64(0.13333333333333333 + Float64(t_0 * 0.13333333333333333)))))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (sin(x) ^ 2.0) / (cos(x) ^ 2.0);
	tmp = (eps / (-1.0 + (tan(x) * tan(eps)))) * (-1.0 - (t_0 + ((eps ^ 2.0) * (0.3333333333333333 + ((0.3333333333333333 * t_0) + ((eps ^ 2.0) * (0.13333333333333333 + (t_0 * 0.13333333333333333))))))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(eps / N[(-1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(t$95$0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(0.3333333333333333 * t$95$0), $MachinePrecision] + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(0.13333333333333333 + N[(t$95$0 * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

   1. tan-sum 62.8%

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

   2. tan-quot 62.8%

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

   3. frac-sub 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right), 0.3333333333333333 \cdot \cos x\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}, \cos x\right) + \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

8. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \color{blue}{\left(1 + \left({\varepsilon}^{2} \cdot \left(0.3333333333333333 + \left(0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + {\varepsilon}^{2} \cdot \left(0.13333333333333333 + 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]

9. Final simplification 99.7%

   \[\leadsto \frac{\varepsilon}{-1 + \tan x \cdot \tan \varepsilon} \cdot \left(-1 - \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + {\varepsilon}^{2} \cdot \left(0.3333333333333333 + \left(0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + {\varepsilon}^{2} \cdot \left(0.13333333333333333 + \frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot 0.13333333333333333\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. tan-sum 62.8%

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

   2. tan-quot 62.8%

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

   3. frac-sub 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right), 0.3333333333333333 \cdot \cos x\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}, \cos x\right) + \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

8. Taylor expanded in eps around 0 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate--l+ 99.2%

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

   2. associate-/l* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 99.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

   1. tan-sum 62.8%

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

   2. tan-quot 62.8%

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

   3. frac-sub 62.8%

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

4. Applied egg-rr 62.8%

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

5. Taylor expanded in eps around 0 99.7%

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}\right)}}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\left(\cos x + {\varepsilon}^{2} \cdot \left(\left(0.3333333333333333 \cdot \cos x + {\varepsilon}^{2} \cdot \left(0.13333333333333333 \cdot \cos x - -0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{\mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left({\varepsilon}^{2}, \mathsf{fma}\left(\cos x, 0.13333333333333333, 0.13333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}\right), 0.3333333333333333 \cdot \cos x\right) + 0.3333333333333333 \cdot \frac{{\sin x}^{2}}{\cos x}, \cos x\right) + \frac{{\sin x}^{2}}{\cos x}}{\cos x}} \]

8. Taylor expanded in eps around 0 99.0%

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

Alternative 5: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

   1. unpow2 99.0%

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

   2. cos-mult 99.0%

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

7. Applied egg-rr 99.0%

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

   1. +-commutative 99.0%

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

   2. +-inverses 99.0%

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

   3. cos-0 99.0%

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

   4. count-2 99.0%

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

   5. *-commutative 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 6: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (* (pow x 2.0) (+ eps (* 0.6666666666666666 (* eps (* x x)))))))

C

double code(double x, double eps) {
	return eps + (pow(x, 2.0) * (eps + (0.6666666666666666 * (eps * (x * x)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + ((x ** 2.0d0) * (eps + (0.6666666666666666d0 * (eps * (x * x)))))
end function

Java

public static double code(double x, double eps) {
	return eps + (Math.pow(x, 2.0) * (eps + (0.6666666666666666 * (eps * (x * x)))));
}

Python

def code(x, eps):
	return eps + (math.pow(x, 2.0) * (eps + (0.6666666666666666 * (eps * (x * x)))))

Julia

function code(x, eps)
	return Float64(eps + Float64((x ^ 2.0) * Float64(eps + Float64(0.6666666666666666 * Float64(eps * Float64(x * x))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + ((x ^ 2.0) * (eps + (0.6666666666666666 * (eps * (x * x)))));
end

Wolfram

code[x_, eps_] := N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] * N[(eps + N[(0.6666666666666666 * N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 98.5%

   \[\leadsto \color{blue}{\varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. unpow2 98.5%

      \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

8. Applied egg-rr 98.5%

   \[\leadsto \varepsilon + {x}^{2} \cdot \left(\varepsilon + 0.6666666666666666 \cdot \left(\varepsilon \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]
9. Add Preprocessing

Alternative 7: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 8: 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 62.7%

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

3. Taylor expanded in eps around 0 99.0%

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

   1. sub-neg 99.0%

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

   2. mul-1-neg 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto \varepsilon \cdot \color{blue}{1} \]

7. Taylor expanded in eps around 0 98.3%

   \[\leadsto \color{blue}{\varepsilon} \]
8. 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 2024180 
(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)))