2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 98.6%

Time: 13.6s

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: 62.2% 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: 98.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = (sin(eps) ** 2.0d0) / (cos(eps) ** 2.0d0)
    code = (x * (t_0 + ((x * (sin(eps) * (1.0d0 + t_0))) / cos(eps)))) + (sin(eps) / cos(eps))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in x around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in x around 0 99.6%

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

4. Taylor expanded in eps around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in x around 0 99.6%

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

4. Taylor expanded in eps around 0 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. quot-tan 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-lft-identity 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 98.6% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in x around 0 99.6%

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

4. Taylor expanded in eps around 0 99.6%

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

5. Taylor expanded in eps around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 98.5% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in x around 0 99.6%

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

4. Taylor expanded in eps around 0 99.3%

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

5. Taylor expanded in eps around 0 99.3%

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

6. Final simplification 99.3%

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

Alternative 6: 98.2% 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 60.8%

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

3. Taylor expanded in x around 0 98.3%

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

4. Taylor expanded in eps around 0 98.3%

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

5. Final simplification 98.3%

   \[\leadsto \varepsilon \]
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 2024071 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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