2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.6%

Time: 18.5s

Alternatives: 14

Speedup: 17.3×

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{\sin x + \frac{{\sin x}^{3}}{t\_0}}{\cos x}\\ t_3 := \frac{t\_1}{t\_0}\\ t_4 := \left(\frac{t\_1 + \frac{{\sin x}^{4}}{t\_0}}{t\_0} - \mathsf{fma}\left(0.16666666666666666, t\_3, \mathsf{fma}\left(-0.5, t\_3, -0.5\right)\right)\right) + -0.16666666666666666\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, t\_2, t\_4 \cdot \frac{\sin x}{\cos x}\right), t\_4\right), t\_2\right), t\_3\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \frac{\mathsf{fma}\left(\sin x, t\_0, \sin x\right)}{\cos x}\\ t_2 := 0.5 \cdot \cos \left(x + x\right)\\ t_3 := 0.5 + t\_2\\ t_4 := 0.5 - t\_2\\ t_5 := \frac{\mathsf{fma}\left(t\_4, t\_0, t\_4\right)}{t\_3} - \mathsf{fma}\left(t\_0, -0.3333333333333333, -0.3333333333333333\right)\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\frac{1}{t\_3}, t\_4, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, t\_1, \tan x \cdot t\_5\right), t\_5\right), t\_1\right)\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (/ (fma (sin x) t_0 (sin x)) (cos x)))
        (t_2 (* 0.5 (cos (+ x x))))
        (t_3 (+ 0.5 t_2))
        (t_4 (- 0.5 t_2))
        (t_5
         (-
          (/ (fma t_4 t_0 t_4) t_3)
          (fma t_0 -0.3333333333333333 -0.3333333333333333))))
   (fma
    eps
    (fma
     (/ 1.0 t_3)
     t_4
     (*
      eps
      (fma
       eps
       (fma eps (fma 0.3333333333333333 t_1 (* (tan x) t_5)) t_5)
       t_1)))
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = fma(sin(x), t_0, sin(x)) / cos(x);
	double t_2 = 0.5 * cos((x + x));
	double t_3 = 0.5 + t_2;
	double t_4 = 0.5 - t_2;
	double t_5 = (fma(t_4, t_0, t_4) / t_3) - fma(t_0, -0.3333333333333333, -0.3333333333333333);
	return fma(eps, fma((1.0 / t_3), t_4, (eps * fma(eps, fma(eps, fma(0.3333333333333333, t_1, (tan(x) * t_5)), t_5), t_1))), eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(fma(sin(x), t_0, sin(x)) / cos(x))
	t_2 = Float64(0.5 * cos(Float64(x + x)))
	t_3 = Float64(0.5 + t_2)
	t_4 = Float64(0.5 - t_2)
	t_5 = Float64(Float64(fma(t_4, t_0, t_4) / t_3) - fma(t_0, -0.3333333333333333, -0.3333333333333333))
	return fma(eps, fma(Float64(1.0 / t_3), t_4, Float64(eps * fma(eps, fma(eps, fma(0.3333333333333333, t_1, Float64(tan(x) * t_5)), t_5), t_1))), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * t$95$0 + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 - t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$4 * t$95$0 + t$95$4), $MachinePrecision] / t$95$3), $MachinePrecision] - N[(t$95$0 * -0.3333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(1.0 / t$95$3), $MachinePrecision] * t$95$4 + N[(eps * N[(eps * N[(eps * N[(0.3333333333333333 * t$95$1 + N[(N[Tan[x], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]]]]]]]

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Developer Target 1: 99.9% accurate, 0.6× 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: 62.7% 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 2024230 
(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)))