2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.3%

Time: 18.0s

Alternatives: 7

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(eps * t$95$0), $MachinePrecision] + N[(N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[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-*r* 99.8%

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

   2. cancel-sign-sub-inv 99.8%

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

   3. metadata-eval 99.8%

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

   4. *-un-lft-identity 99.8%

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

   5. distribute-rgt-in 99.8%

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

   6. *-un-lft-identity 99.8%

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

   7. unpow2 99.8%

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

   8. unpow2 99.8%

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

   9. frac-times 99.8%

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

   10. tan-quot 99.8%

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

   11. tan-quot 99.8%

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

   12. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

   4. distribute-lft-out 99.8%

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

   5. *-rgt-identity 99.8%

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

   6. distribute-lft-in 99.8%

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

   7. +-commutative 99.8%

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

7. Simplified 99.8%

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

   1. unpow2 99.8%

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

   2. unpow2 99.8%

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

   3. frac-times 99.8%

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

   4. tan-quot 99.8%

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

   5. tan-quot 99.8%

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

   6. *-un-lft-identity 99.8%

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

   7. pow2 99.8%

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

9. Applied egg-rr 99.8%

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

    1. *-lft-identity 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[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. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in eps around 0 99.7%

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

   1. cancel-sign-sub-inv 99.7%

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

   2. metadata-eval 99.7%

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

   3. *-lft-identity 99.7%

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

5. Simplified 99.7%

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

   1. distribute-lft-in 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. +-commutative 99.7%

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

   4. unpow2 99.7%

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

   5. unpow2 99.7%

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

   6. frac-times 99.7%

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

   7. tan-quot 99.7%

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

   8. tan-quot 99.7%

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

   9. unpow2 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps * N[(N[Power[x, 2.0], $MachinePrecision] + 1.0), $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 99.7%

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

   1. cancel-sign-sub-inv 99.7%

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

   2. metadata-eval 99.7%

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

   3. *-lft-identity 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(eps + N[(eps * N[Power[x, 2.0], $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 99.7%

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

   1. cancel-sign-sub-inv 99.7%

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

   2. metadata-eval 99.7%

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

   3. *-lft-identity 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.2%

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

7. Final simplification 99.2%

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

Alternative 6: 97.8% 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 61.1%

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

3. Taylor expanded in x around 0 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. quot-tan 98.9%

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

5. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

   \[\leadsto \tan \varepsilon \]
9. Add Preprocessing

Alternative 7: 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 x around 0 98.9%

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

4. Taylor expanded in eps around 0 98.9%

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

5. Final simplification 98.9%

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

Developer target: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 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)))

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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