2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 99.4%

Time: 17.9s

Alternatives: 10

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

   1. distribute-rgt-in 98.9%

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

   2. *-un-lft-identity 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0 98.9%

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

4. Taylor expanded in eps around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. unpow2 98.9%

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

   3. distribute-rgt-in 98.9%

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

   4. *-commutative 98.9%

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

   5. associate-*r* 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

   8. distribute-lft-in 98.9%

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

   9. unpow2 98.9%

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

   10. +-commutative 98.9%

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

   11. distribute-lft-in 98.9%

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

   12. *-commutative 98.9%

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

   13. unpow2 98.9%

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

   14. associate-*r* 98.9%

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

   15. distribute-rgt-out 98.9%

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

   16. *-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 98.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\left(\varepsilon \cdot \left(x + \varepsilon \cdot \left(0.3333333333333333 - x \cdot \left(x \cdot -0.5 - x \cdot 0.8333333333333334\right)\right)\right) + {x}^{2}\right) + 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (+
    (*
     eps
     (+
      x
      (*
       eps
       (- 0.3333333333333333 (* x (- (* x -0.5) (* x 0.8333333333333334)))))))
    (pow x 2.0))
   1.0)))

C

double code(double x, double eps) {
	return eps * (((eps * (x + (eps * (0.3333333333333333 - (x * ((x * -0.5) - (x * 0.8333333333333334))))))) + pow(x, 2.0)) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((eps * (x + (eps * (0.3333333333333333d0 - (x * ((x * (-0.5d0)) - (x * 0.8333333333333334d0))))))) + (x ** 2.0d0)) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * (((eps * (x + (eps * (0.3333333333333333 - (x * ((x * -0.5) - (x * 0.8333333333333334))))))) + Math.pow(x, 2.0)) + 1.0);
}

Python

def code(x, eps):
	return eps * (((eps * (x + (eps * (0.3333333333333333 - (x * ((x * -0.5) - (x * 0.8333333333333334))))))) + math.pow(x, 2.0)) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(Float64(eps * Float64(x + Float64(eps * Float64(0.3333333333333333 - Float64(x * Float64(Float64(x * -0.5) - Float64(x * 0.8333333333333334))))))) + (x ^ 2.0)) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((eps * (x + (eps * (0.3333333333333333 - (x * ((x * -0.5) - (x * 0.8333333333333334))))))) + (x ^ 2.0)) + 1.0);
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[(eps * N[(x + N[(eps * N[(0.3333333333333333 - N[(x * N[(N[(x * -0.5), $MachinePrecision] - N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around 0 98.9%

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

4. Taylor expanded in eps around 0 98.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \left(x + \varepsilon \cdot \left(0.3333333333333333 + x \cdot \left(0.8333333333333334 \cdot x - -0.5 \cdot x\right)\right)\right) + {x}^{2}\right)\right)} \]

5. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \left(x + \varepsilon \cdot \left(0.3333333333333333 - x \cdot \left(x \cdot -0.5 - x \cdot 0.8333333333333334\right)\right)\right) + {x}^{2}\right) + 1\right) \]
6. Add Preprocessing

Alternative 4: 98.3% 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 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.9%

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

   1. +-commutative 98.9%

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

8. Simplified 98.9%

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

   1. distribute-rgt-in 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. +-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

Alternative 5: 98.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.9%

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

   1. +-commutative 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.9%

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

   1. +-commutative 98.9%

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

8. Simplified 98.9%

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

   1. distribute-rgt-in 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. +-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

11. Taylor expanded in eps around 0 98.8%

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

12. Final simplification 98.8%

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

Alternative 7: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.9%

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

   1. +-commutative 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in x around inf 98.8%

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

10. Final simplification 98.8%

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

Alternative 8: 97.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 9: 97.8% 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 63.0%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. associate--l+ 98.9%

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

   2. associate-/l* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. mul-1-neg 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.6%

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

Alternative 10: 5.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 63.0%

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

3. Taylor expanded in x around inf 5.4%

   \[\leadsto \tan \color{blue}{x} - \tan x \]

4. Taylor expanded in x around 0 5.4%

   \[\leadsto \color{blue}{0} \]
5. 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: 62.4% 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 2024186 
(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)))