2tan (problem 3.3.2)

Percentage Accurate: 62.8% → 99.9%

Time: 37.6s

Alternatives: 6

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: 62.8% 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, 0.7× speedup

Math

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

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(sin(eps) / Float64(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[Sin[eps], $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.2%

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

   1. tan-quot 64.1%

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

   2. tan-quot 64.2%

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

   3. frac-sub 64.2%

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

4. Applied egg-rr 64.2%

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

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%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

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 64.2%

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

   1. tan-quot 64.1%

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

   2. tan-quot 64.2%

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

   3. frac-sub 64.2%

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

4. Applied egg-rr 64.2%

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

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 98.8%

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

   1. clear-num 98.5%

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

   2. associate-/r/ 98.8%

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

   3. pow-flip 98.8%

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

   4. metadata-eval 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.2%

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

   1. tan-quot 64.1%

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

   2. tan-quot 64.2%

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

   3. frac-sub 64.2%

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

4. Applied egg-rr 64.2%

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

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 98.8%

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

   1. unpow2 98.8%

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

   2. cos-mult 98.8%

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

8. Applied egg-rr 98.8%

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

   1. +-commutative 98.8%

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

   2. +-inverses 98.8%

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

   3. cos-0 98.8%

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

   4. count-2 98.8%

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

   5. *-commutative 98.8%

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

10. Simplified 98.8%

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

Alternative 4: 98.3% 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 64.2%

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

   1. tan-quot 64.1%

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

   2. tan-quot 64.2%

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

   3. frac-sub 64.2%

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

4. Applied egg-rr 64.2%

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

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 98.8%

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

7. Taylor expanded in x around 0 98.1%

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

Alternative 5: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return tan(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Tan[eps], $MachinePrecision]

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in x around 0 97.8%

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

   1. *-un-lft-identity 97.8%

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

   2. quot-tan 97.8%

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

5. Applied egg-rr 97.8%

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

   1. *-lft-identity 97.8%

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

7. Simplified 97.8%

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

Alternative 6: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 64.2%

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

3. Taylor expanded in x around 0 97.8%

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

4. Taylor expanded in eps around 0 97.7%

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

Reproduce

herbie shell --seed 2024143 
(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)))))

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