2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.4%

Time: 21.9s

Alternatives: 3

Speedup: 1.0×

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: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Python

def code(x, eps):
	t_0 = 1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0))
	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(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))
	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 = 1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0));
	tmp = (eps * t_0) + (((eps ^ 2.0) * (sin(x) * t_0)) / cos(x));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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 60.5%

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

3. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 98.2%

   \[\leadsto \frac{\sin \varepsilon}{\cos \varepsilon} \]
5. 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 2024039 
(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)))