Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + e^{-x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (exp (- x)))))

C

double code(double x) {
	return 2.0 / (exp(x) + exp(-x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + exp(-x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + Math.exp(-x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + math.exp(-x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + exp(Float64(-x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + exp(-x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + \left(1 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) (- 1.0 x))))

C

double code(double x) {
	return 2.0 / (exp(x) + (1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (exp(x) + (1.0d0 - x))
end function

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + (1.0 - x));
}

Python

def code(x):
	return 2.0 / (math.exp(x) + (1.0 - x))

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + (1.0 - x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]
6. Add Preprocessing

Alternative 3: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x} + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ (exp x) 1.0)))

C

double code(double x) {
	return 2.0 / (exp(x) + 1.0);
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (Math.exp(x) + 1.0);
}

Python

def code(x):
	return 2.0 / (math.exp(x) + 1.0)

Julia

function code(x)
	return Float64(2.0 / Float64(exp(x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (exp(x) + 1.0);
end

Wolfram

code[x_] := N[(2.0 / N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{1}} \]
7. Add Preprocessing

Alternative 4: 83.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(1 - x\right) + \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (+ (- 1.0 x) (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x) {
	return 2.0 / ((1.0 - x) + (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((1.0d0 - x) + (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))))
end function

Java

public static double code(double x) {
	return 2.0 / ((1.0 - x) + (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
}

Python

def code(x):
	return 2.0 / ((1.0 - x) + (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(1.0 - x) + Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((1.0 - x) + (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 - x), $MachinePrecision] + N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 82.1%

   \[\leadsto \frac{2}{\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} + \left(1 - x\right)} \]
7. Step-by-step derivation

   1. *-lft-identity 82.1%

      \[\leadsto \frac{2}{\left(1 + x \cdot \color{blue}{\left(1 \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)}\right) + \left(1 - x\right)} \]

   2. *-lft-identity 82.1%

      \[\leadsto \frac{2}{\left(1 + x \cdot \color{blue}{\left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}\right) + \left(1 - x\right)} \]

   3. *-commutative 82.1%

      \[\leadsto \frac{2}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) + \left(1 - x\right)} \]

8. Simplified 82.1%

   \[\leadsto \frac{2}{\color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} + \left(1 - x\right)} \]

9. Final simplification 82.1%

   \[\leadsto \frac{2}{\left(1 - x\right) + \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \]
10. Add Preprocessing

Alternative 5: 83.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (+ 2.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))))

C

double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{1}} \]

7. Taylor expanded in x around 0 81.8%

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}} \]

8. Final simplification 81.8%

   \[\leadsto \frac{2}{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
9. Add Preprocessing

Alternative 6: 83.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot \left(1 + x \cdot \left(x \cdot 0.16666666666666666\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (+ 2.0 (* x (+ 1.0 (* x (* x 0.16666666666666666)))))))

C

double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * (x * 0.16666666666666666)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + (x * (1.0d0 + (x * (x * 0.16666666666666666d0)))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * (x * 0.16666666666666666)))));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * (1.0 + (x * (x * 0.16666666666666666)))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.16666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + (x * (1.0 + (x * (x * 0.16666666666666666)))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * N[(1.0 + N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{1}} \]

7. Taylor expanded in x around 0 81.8%

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}} \]

8. Taylor expanded in x around inf 81.8%

   \[\leadsto \frac{2}{2 + x \cdot \left(1 + x \cdot \color{blue}{\left(0.16666666666666666 \cdot x\right)}\right)} \]
9. Step-by-step derivation

   1. *-commutative 81.8%

      \[\leadsto \frac{2}{2 + x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)}\right)} \]

10. Simplified 81.8%

    \[\leadsto \frac{2}{2 + x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)}\right)} \]
11. Add Preprocessing

Alternative 7: 75.6% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x \cdot \left(1 + x \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ 2.0 (* x (+ 1.0 (* x 0.5))))))

C

double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * 0.5))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (2.0d0 + (x * (1.0d0 + (x * 0.5d0))))
end function

Java

public static double code(double x) {
	return 2.0 / (2.0 + (x * (1.0 + (x * 0.5))));
}

Python

def code(x):
	return 2.0 / (2.0 + (x * (1.0 + (x * 0.5))))

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + (x * (1.0 + (x * 0.5))));
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{1}} \]

7. Taylor expanded in x around 0 74.8%

   \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(1 + 0.5 \cdot x\right)}} \]

8. Final simplification 74.8%

   \[\leadsto \frac{2}{2 + x \cdot \left(1 + x \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 8: 51.9% accurate, 41.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2 + x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (+ 2.0 x)))

C

double code(double x) {
	return 2.0 / (2.0 + x);
}

Fortran

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

Java

public static double code(double x) {
	return 2.0 / (2.0 + x);
}

Python

def code(x):
	return 2.0 / (2.0 + x)

Julia

function code(x)
	return Float64(2.0 / Float64(2.0 + x))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (2.0 + x);
end

Wolfram

code[x_] := N[(2.0 / N[(2.0 + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 69.5%

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

   1. mul-1-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \left(1 + \color{blue}{\left(-x\right)}\right)} \]

   2. unsub-neg 69.5%

      \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

5. Simplified 69.5%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{\left(1 - x\right)}} \]

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{2}{e^{x} + \color{blue}{1}} \]

7. Taylor expanded in x around 0 45.6%

   \[\leadsto \frac{2}{\color{blue}{2 + x}} \]
8. Step-by-step derivation

   1. +-commutative 45.6%

      \[\leadsto \frac{2}{\color{blue}{x + 2}} \]

9. Simplified 45.6%

   \[\leadsto \frac{2}{\color{blue}{x + 2}} \]

10. Final simplification 45.6%

    \[\leadsto \frac{2}{2 + x} \]
11. Add Preprocessing

Alternative 9: 51.2% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 44.5%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))