2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.3%

Time: 14.3s

Alternatives: 8

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return eps * (1.0 + (t_0 + (eps * ((sin(x) / cos(x)) * (1.0 + 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 = tan(x) ** 2.0d0
    code = eps * (1.0d0 + (t_0 + (eps * ((sin(x) / cos(x)) * (1.0d0 + t_0)))))
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) / Math.cos(x)) * (1.0 + t_0)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = tan(x) ^ 2.0;
	tmp = eps * (1.0 + (t_0 + (eps * ((sin(x) / cos(x)) * (1.0 + t_0)))));
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[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

    1. expm1-log1p-u 99.8%

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

    2. log1p-define 99.8%

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

    3. expm1-undefine 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in x around 0 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 98.2% 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 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in x around 0 99.3%

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

    1. unpow2 99.3%

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

    2. distribute-rgt-out 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 5: 98.2% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in x around 0 99.3%

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

    1. unpow2 99.3%

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

    2. distribute-rgt-out 99.3%

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

12. Simplified 99.3%

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

    1. +-commutative 99.3%

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

    2. distribute-rgt-in 99.3%

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

    3. +-commutative 99.3%

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

    4. *-un-lft-identity 99.3%

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

14. Applied egg-rr 99.3%

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

15. Final simplification 99.3%

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

Alternative 6: 97.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

   1. associate-+l+ 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. associate-/r/ 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in x around 0 99.0%

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

11. Final simplification 99.0%

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

Alternative 7: 97.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 99.0%

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

   1. unpow2 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 8: 97.7% 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 61.1%

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

3. Taylor expanded in eps around 0 99.8%

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

   1. associate-/l* 99.8%

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

   2. unpow2 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. cancel-sign-sub-inv 99.8%

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

   6. metadata-eval 99.8%

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

   7. *-lft-identity 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 98.9%

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

7. Final simplification 98.9%

   \[\leadsto \varepsilon \]
8. 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 2024046 
(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)))