2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 98.9%

Time: 17.8s

Alternatives: 11

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: 62.7% 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: 98.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x)))
        (t_1 (- 1.0 (* (sin x) (/ (sin eps) (* (cos eps) (cos x)))))))
   (+ (/ (/ (sin eps) (cos eps)) t_1) (- (/ t_0 t_1) t_0))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	double t_1 = 1.0 - (sin(x) * (sin(eps) / (cos(eps) * cos(x))));
	return ((sin(eps) / cos(eps)) / t_1) + ((t_0 / t_1) - t_0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(x) / cos(x)
    t_1 = 1.0d0 - (sin(x) * (sin(eps) / (cos(eps) * cos(x))))
    code = ((sin(eps) / cos(eps)) / t_1) + ((t_0 / t_1) - t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) / cos(x);
	t_1 = 1.0 - (sin(x) * (sin(eps) / (cos(eps) * cos(x))));
	tmp = ((sin(eps) / cos(eps)) / t_1) + ((t_0 / t_1) - t_0);
end

Wolfram

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

Derivation

1. Initial program 60.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 60.5%

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

   3. fmm-def 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. Taylor expanded in x around inf 60.6%

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

   1. associate--l+ 99.1%

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

   2. associate-/r* 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. associate-/l* 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 60.5%

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

   3. fmm-def 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. Taylor expanded in x around inf 60.6%

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

   1. associate--l+ 99.1%

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

   2. associate-/r* 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. associate-/l* 99.1%

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

7. Simplified 99.1%

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

   1. tan-quot 99.0%

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

   2. associate-+r- 60.5%

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

9. Applied egg-rr 60.6%

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

    1. associate-+r- 99.1%

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

    2. +-commutative 99.1%

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

11. Simplified 99.1%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

   1. sub-neg 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 4: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in eps around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

   1. add-log-exp 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot e^{\mathsf{log1p}\left({\sin x}^{2} \cdot {\cos x}^{-2}\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (exp (log1p (* (pow (sin x) 2.0) (pow (cos x) -2.0))))))

C

double code(double x, double eps) {
	return eps * exp(log1p((pow(sin(x), 2.0) * pow(cos(x), -2.0))));
}

Java

public static double code(double x, double eps) {
	return eps * Math.exp(Math.log1p((Math.pow(Math.sin(x), 2.0) * Math.pow(Math.cos(x), -2.0))));
}

Python

def code(x, eps):
	return eps * math.exp(math.log1p((math.pow(math.sin(x), 2.0) * math.pow(math.cos(x), -2.0))))

Julia

function code(x, eps)
	return Float64(eps * exp(log1p(Float64((sin(x) ^ 2.0) * (cos(x) ^ -2.0)))))
end

Wolfram

code[x_, eps_] := N[(eps * N[Exp[N[Log[1 + 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 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in eps around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

   1. add-exp-log 98.3%

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

   2. log1p-define 98.3%

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

   3. pow1 98.3%

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

   4. div-inv 98.3%

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

   5. pow1 98.3%

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

   6. pow-flip 98.3%

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

   7. metadata-eval 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

    \[\leadsto \varepsilon \cdot e^{\mathsf{log1p}\left({\sin x}^{2} \cdot {\cos x}^{-2}\right)} \]
12. Add Preprocessing

Alternative 6: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in eps around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

   1. unpow2 98.3%

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

   2. cos-mult 98.3%

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

10. Applied egg-rr 98.3%

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

    1. +-commutative 98.3%

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

    2. +-inverses 98.3%

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

    3. cos-0 98.3%

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

    4. count-2 98.3%

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

    5. *-commutative 98.3%

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

12. Simplified 98.3%

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

13. Final simplification 98.3%

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

Alternative 7: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in eps around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in x around 0 97.4%

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

    1. mul-1-neg 97.4%

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

    2. unsub-neg 97.4%

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

11. Simplified 97.4%

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

    1. expm1-log1p-u 97.4%

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

    2. expm1-undefine 97.4%

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

13. Applied egg-rr 97.4%

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

    1. sub-neg 97.4%

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

    2. log1p-undefine 97.4%

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

    3. rem-exp-log 97.4%

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

    4. associate-+r+ 97.4%

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

    5. metadata-eval 97.4%

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

    6. metadata-eval 97.4%

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

15. Simplified 97.4%

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

16. Final simplification 97.4%

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

Alternative 8: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in eps around 0 98.3%

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

   1. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in x around 0 97.4%

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

    1. mul-1-neg 97.4%

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

    2. unsub-neg 97.4%

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

11. Simplified 97.4%

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

    1. unpow2 97.4%

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

13. Applied egg-rr 97.4%

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

14. Final simplification 97.4%

    \[\leadsto \varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{1 - x \cdot x}\right) \]
15. 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 60.3%

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

3. Taylor expanded in x around 0 97.4%

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

   1. tan-quot 97.4%

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

   2. *-un-lft-identity 97.4%

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

5. Applied egg-rr 97.4%

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

   1. *-lft-identity 97.4%

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

7. Simplified 97.4%

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

Alternative 10: 98.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (1.0 + (x * (eps + x)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 97.3%

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

   1. +-commutative 97.3%

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

8. Simplified 97.3%

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

9. Final simplification 97.3%

   \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(\varepsilon + x\right)\right) \]
10. Add Preprocessing

Alternative 11: 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 60.3%

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

3. Taylor expanded in eps around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. associate-/l* 98.6%

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

   3. mul-1-neg 98.6%

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

   4. mul-1-neg 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 97.3%

   \[\leadsto \color{blue}{\varepsilon} \]
7. 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: 62.9% 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 2024169 
(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)))