2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.9%

Time: 19.4s

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: 62.4% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin \varepsilon \cdot \sin x\\ \frac{\frac{\sin \varepsilon}{\cos x} \cdot \left({t\_0}^{2} + t\_1 \cdot \cos \left(x - \varepsilon\right)\right)}{{t\_0}^{3} - {t\_1}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin eps) (sin x))))
   (/
    (* (/ (sin eps) (cos x)) (+ (pow t_0 2.0) (* t_1 (cos (- x eps)))))
    (- (pow t_0 3.0) (pow t_1 3.0)))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(eps) * sin(x);
	return ((sin(eps) / cos(x)) * (pow(t_0, 2.0) + (t_1 * cos((x - eps))))) / (pow(t_0, 3.0) - pow(t_1, 3.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    t_0 = cos(x) * cos(eps)
    t_1 = sin(eps) * sin(x)
    code = ((sin(eps) / cos(x)) * ((t_0 ** 2.0d0) + (t_1 * cos((x - eps))))) / ((t_0 ** 3.0d0) - (t_1 ** 3.0d0))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(eps) * Math.sin(x);
	return ((Math.sin(eps) / Math.cos(x)) * (Math.pow(t_0, 2.0) + (t_1 * Math.cos((x - eps))))) / (Math.pow(t_0, 3.0) - Math.pow(t_1, 3.0));
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(eps) * math.sin(x)
	return ((math.sin(eps) / math.cos(x)) * (math.pow(t_0, 2.0) + (t_1 * math.cos((x - eps))))) / (math.pow(t_0, 3.0) - math.pow(t_1, 3.0))

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(eps) * sin(x))
	return Float64(Float64(Float64(sin(eps) / cos(x)) * Float64((t_0 ^ 2.0) + Float64(t_1 * cos(Float64(x - eps))))) / Float64((t_0 ^ 3.0) - (t_1 ^ 3.0)))
end

MATLAB

function tmp = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(eps) * sin(x);
	tmp = ((sin(eps) / cos(x)) * ((t_0 ^ 2.0) + (t_1 * cos((x - eps))))) / ((t_0 ^ 3.0) - (t_1 ^ 3.0));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$1 * N[Cos[N[(x - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

   1. cos-sum 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. cos-sum 99.9%

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

   3. *-commutative 99.9%

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

   4. frac-times 99.9%

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

   5. associate-*r/ 99.9%

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

   6. cos-sum 99.9%

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

   7. flip3-- 100.0%

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

   8. associate-/r/ 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-*l/ 100.0%

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

    2. *-commutative 100.0%

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

    3. *-commutative 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

   1. cos-sum 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. *-commutative 99.9%

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

   3. times-frac 99.9%

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

   4. +-commutative 99.9%

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

7. Applied egg-rr 99.9%

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

8. Taylor expanded in x around inf 99.9%

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

   1. associate-/r* 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. *-commutative 99.9%

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

   3. times-frac 99.9%

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

   4. +-commutative 99.9%

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

7. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. *-un-lft-identity 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. tan-quot 62.3%

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

   2. tan-quot 62.3%

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

   3. frac-sub 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in eps around 0 98.5%

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

7. Final simplification 98.5%

   \[\leadsto \frac{\varepsilon}{{\cos x}^{2}} \]
8. Add Preprocessing

Alternative 7: 97.9% 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 62.4%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in x around 0 97.4%

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

7. Final simplification 97.4%

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

Alternative 8: 97.9% 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 62.4%

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

3. Taylor expanded in eps around 0 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. metadata-eval 98.5%

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

   3. *-lft-identity 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in x around 0 97.4%

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

   1. *-commutative 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 9: 97.5% 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 62.4%

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

3. Taylor expanded in x around 0 97.0%

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

   1. *-un-lft-identity 97.0%

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

   2. quot-tan 97.0%

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

5. Applied egg-rr 97.0%

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

   1. *-lft-identity 97.0%

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

7. Simplified 97.0%

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

8. Final simplification 97.0%

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

Alternative 10: 97.5% 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 62.4%

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

3. Taylor expanded in x around 0 97.0%

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

4. Taylor expanded in eps around 0 97.0%

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

5. Final simplification 97.0%

   \[\leadsto \varepsilon \]
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 2024044 
(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)))