2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.6%

Time: 23.1s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{2}\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := 1 + t\_2\\ t_4 := \left(t\_0 \cdot \frac{t\_3}{t\_1} - \mathsf{fma}\left(-0.5, t\_3, t\_2 \cdot 0.16666666666666666\right)\right) - 0.16666666666666666\\ \mathsf{fma}\left(\varepsilon, t\_3, \left(\sin x \cdot {\varepsilon}^{2}\right) \cdot \frac{t\_3}{\cos x}\right) + \left({\varepsilon}^{3} \cdot t\_4 + {\varepsilon}^{4} \cdot \left(-0.3333333333333333 \cdot \left(\sin x \cdot \frac{-1 - t\_2}{\cos x}\right) + \sin x \cdot \frac{t\_4}{\cos x}\right)\right) \end{array} \end{array} \]

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))
        (t_3 (+ 1.0 t_2))
        (t_4
         (-
          (- (* t_0 (/ t_3 t_1)) (fma -0.5 t_3 (* t_2 0.16666666666666666)))
          0.16666666666666666)))
   (+
    (fma eps t_3 (* (* (sin x) (pow eps 2.0)) (/ t_3 (cos x))))
    (+
     (* (pow eps 3.0) t_4)
     (*
      (pow eps 4.0)
      (+
       (* -0.3333333333333333 (* (sin x) (/ (- -1.0 t_2) (cos x))))
       (* (sin x) (/ t_4 (cos x)))))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	double t_2 = t_0 / t_1;
	double t_3 = 1.0 + t_2;
	double t_4 = ((t_0 * (t_3 / t_1)) - fma(-0.5, t_3, (t_2 * 0.16666666666666666))) - 0.16666666666666666;
	return fma(eps, t_3, ((sin(x) * pow(eps, 2.0)) * (t_3 / cos(x)))) + ((pow(eps, 3.0) * t_4) + (pow(eps, 4.0) * ((-0.3333333333333333 * (sin(x) * ((-1.0 - t_2) / cos(x)))) + (sin(x) * (t_4 / cos(x))))));
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(1.0 + t_2)
	t_4 = Float64(Float64(Float64(t_0 * Float64(t_3 / t_1)) - fma(-0.5, t_3, Float64(t_2 * 0.16666666666666666))) - 0.16666666666666666)
	return Float64(fma(eps, t_3, Float64(Float64(sin(x) * (eps ^ 2.0)) * Float64(t_3 / cos(x)))) + Float64(Float64((eps ^ 3.0) * t_4) + Float64((eps ^ 4.0) * Float64(Float64(-0.3333333333333333 * Float64(sin(x) * Float64(Float64(-1.0 - t_2) / cos(x)))) + Float64(sin(x) * Float64(t_4 / cos(x)))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t$95$0 * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * t$95$3 + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]}, N[(N[(eps * t$95$3 + N[(N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[Sin[x], $MachinePrecision] * N[(N[(-1.0 - t$95$2), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(t$95$4 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

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))
        (t_3 (+ 1.0 t_2)))
   (+
    (fma eps t_3 (* (* (sin x) (pow eps 2.0)) (/ t_3 (cos x))))
    (+
     (*
      (pow eps 3.0)
      (-
       (- (* t_0 (/ t_3 t_1)) (fma -0.5 t_3 (* t_2 0.16666666666666666)))
       0.16666666666666666))
     (*
      (pow eps 4.0)
      (-
       (* -0.3333333333333333 (* (sin x) (/ (- -1.0 t_2) (cos x))))
       (* x -0.3333333333333333)))))))

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

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(1.0 + t_2)
	return Float64(fma(eps, t_3, Float64(Float64(sin(x) * (eps ^ 2.0)) * Float64(t_3 / cos(x)))) + Float64(Float64((eps ^ 3.0) * Float64(Float64(Float64(t_0 * Float64(t_3 / t_1)) - fma(-0.5, t_3, Float64(t_2 * 0.16666666666666666))) - 0.16666666666666666)) + Float64((eps ^ 4.0) * Float64(Float64(-0.3333333333333333 * Float64(sin(x) * Float64(Float64(-1.0 - t_2) / cos(x)))) - Float64(x * -0.3333333333333333)))))
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]}, Block[{t$95$3 = N[(1.0 + t$95$2), $MachinePrecision]}, N[(N[(eps * t$95$3 + N[(N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[eps, 3.0], $MachinePrecision] * N[(N[(N[(t$95$0 * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * t$95$3 + N[(t$95$2 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[(-0.3333333333333333 * N[(N[Sin[x], $MachinePrecision] * N[(N[(-1.0 - t$95$2), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)
    code = ((eps ** 2.0d0) * ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))) - ((eps * ((-1.0d0) - t_0)) - ((eps ** 3.0d0) * (0.3333333333333333d0 + (t_0 + (((sin(x) ** 4.0d0) / (cos(x) ** 4.0d0)) - (t_0 * (-0.3333333333333333d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

   1. tan-sum 60.6%

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

   2. div-inv 60.6%

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

   3. fma-neg 60.6%

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

4. Applied egg-rr 60.6%

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

   1. fma-neg 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*l/ 60.6%

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

   4. *-lft-identity 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in eps around 0 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps + N[(eps * N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.5%

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

   1. tan-sum 60.6%

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

   2. div-inv 60.6%

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

   3. fma-neg 60.6%

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

4. Applied egg-rr 60.6%

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

   1. fma-neg 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*l/ 60.6%

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

   4. *-lft-identity 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in eps around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. unsub-neg 99.4%

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

   4. sub-neg 99.4%

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

   5. mul-1-neg 99.4%

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

   6. remove-double-neg 99.4%

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

9. Simplified 99.4%

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

    1. +-commutative 99.1%

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

    2. distribute-lft-in 99.1%

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

    3. unpow2 99.1%

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

    4. unpow2 99.1%

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

    5. frac-times 99.1%

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

    6. pow2 99.1%

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

    7. quot-tan 99.1%

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

    8. *-rgt-identity 99.1%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (pow(eps, 2.0) * (tan(x) + pow(tan(x), 3.0))) - (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 ** 2.0d0) * (tan(x) + (tan(x) ** 3.0d0))) - (eps * ((-1.0d0) - ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]

Derivation

1. Initial program 60.5%

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

   1. tan-sum 60.6%

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

   2. div-inv 60.6%

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

   3. fma-neg 60.6%

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

4. Applied egg-rr 60.6%

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

   1. fma-neg 60.6%

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

   2. *-commutative 60.6%

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

   3. associate-*l/ 60.6%

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

   4. *-lft-identity 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in eps around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. unsub-neg 99.4%

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

   4. sub-neg 99.4%

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

   5. mul-1-neg 99.4%

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

   6. remove-double-neg 99.4%

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

9. Simplified 99.4%

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

    1. sub-neg 99.4%

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

    2. distribute-frac-neg 99.4%

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

    3. cube-div 99.4%

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

    4. pow3 99.4%

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

    5. distribute-neg-out 99.4%

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

    6. pow3 99.4%

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

    7. quot-tan 99.4%

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

    8. quot-tan 99.4%

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

11. Applied egg-rr 99.4%

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

    1. +-commutative 99.4%

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

    2. distribute-neg-in 99.4%

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

    3. sub-neg 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. *-lft-identity 99.1%

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

5. Simplified 99.1%

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

   1. add-cbrt-cube 99.1%

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

   2. add-cbrt-cube 99.1%

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

   3. cbrt-undiv 99.1%

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

   4. pow3 99.1%

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

   5. unpow2 99.1%

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

   6. pow-prod-down 99.1%

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

   7. sqr-neg 99.1%

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

   8. pow3 99.1%

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

   9. unpow2 99.1%

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

   10. pow-prod-down 99.1%

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

   11. frac-times 99.1%

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

   12. cbrt-unprod 99.1%

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

7. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. *-lft-identity 99.1%

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

5. Simplified 99.1%

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

   1. +-commutative 99.1%

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

   2. distribute-lft-in 99.1%

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

   3. unpow2 99.1%

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

   4. unpow2 99.1%

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

   5. frac-times 99.1%

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

   6. pow2 99.1%

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

   7. quot-tan 99.1%

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

   8. *-rgt-identity 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
9. Add Preprocessing

Alternative 8: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. *-lft-identity 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in x around 0 98.7%

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

7. Final simplification 98.7%

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

Alternative 9: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in x around 0 98.4%

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

   1. tan-quot 98.4%

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

   2. *-un-lft-identity 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{1 \cdot \tan \varepsilon} \]
6. Step-by-step derivation

   1. *-lft-identity 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

   \[\leadsto \tan \varepsilon \]
9. 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 60.5%

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

3. Taylor expanded in x around 0 98.4%

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

4. Taylor expanded in eps around 0 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024048 
(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
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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