2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 100.0%

Time: 20.6s

Alternatives: 9

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: 62.6% 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: 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 60.9%

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

   1. tan-quot 60.9%

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

   2. tan-quot 60.9%

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

   3. frac-sub 60.9%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in x around 0 99.8%

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

   1. cos-sum 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. tan-quot 60.9%

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

   2. tan-quot 60.9%

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

   3. frac-sub 60.9%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in x around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \frac{1}{\cos \left(\varepsilon + x\right)} \cdot \frac{\sin \varepsilon}{\cos x} \]
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 60.9%

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

   1. tan-quot 60.9%

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

   2. tan-quot 60.9%

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

   3. frac-sub 60.9%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. tan-quot 60.9%

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

   2. tan-quot 60.9%

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

   3. frac-sub 60.9%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in x around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. times-frac 99.8%

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

   3. +-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in eps around 0 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

   1. tan-quot 60.9%

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

   2. tan-quot 60.9%

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

   3. frac-sub 60.9%

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

4. Applied egg-rr 60.9%

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

5. Taylor expanded in x around 0 99.8%

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

   1. cos-mult 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+l+ 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--l+ 99.8%

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

7. Applied egg-rr 99.8%

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

   1. count-2 99.8%

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

   2. +-inverses 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in eps around 0 98.4%

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

    1. associate-*r/ 98.4%

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

    2. *-commutative 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

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

Alternative 6: 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 60.9%

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

3. Taylor expanded in eps around 0 98.4%

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

   1. sub-neg 98.4%

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

   2. mul-1-neg 98.4%

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

   3. remove-double-neg 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0 97.6%

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

   1. *-commutative 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 7: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (fma x x 1.0)))

C

double code(double x, double eps) {
	return eps * fma(x, x, 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, x, 1.0))
end

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 98.4%

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

   1. sub-neg 98.4%

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

   2. mul-1-neg 98.4%

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

   3. remove-double-neg 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0 97.6%

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

   1. *-commutative 97.6%

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

8. Simplified 97.6%

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

   1. distribute-rgt1-in 97.6%

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

   2. unpow2 97.6%

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

   3. fma-define 97.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right)} \cdot \varepsilon \]

10. Applied egg-rr 97.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 1\right) \cdot \varepsilon} \]

11. Final simplification 97.6%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \]
12. Add Preprocessing

Alternative 8: 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 60.9%

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

3. Taylor expanded in x around 0 97.3%

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

   1. *-un-lft-identity 97.3%

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

   2. quot-tan 97.3%

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

5. Applied egg-rr 97.3%

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

   1. *-lft-identity 97.3%

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

7. Simplified 97.3%

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

8. Final simplification 97.3%

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

Alternative 9: 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 60.9%

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

3. Taylor expanded in x around 0 97.3%

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

4. Taylor expanded in eps around 0 97.3%

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

5. Final simplification 97.3%

   \[\leadsto \varepsilon \]
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 2024079 
(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)))