Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

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: 75.3% 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 76.2%

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

   1. mul-1-neg 76.2%

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

   2. unsub-neg 76.2%

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

5. Simplified 76.2%

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

Alternative 3: 71.2% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $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 74.8%

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

4. Taylor expanded in x around 0 52.0%

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

   1. *-lft-identity 52.0%

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

   2. *-lft-identity 52.0%

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

   3. *-commutative 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in x around 0 73.2%

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

   1. *-commutative 73.2%

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

9. Simplified 73.2%

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

10. Final simplification 73.2%

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

Alternative 4: 83.7% 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 76.2%

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

   1. mul-1-neg 76.2%

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

   2. unsub-neg 76.2%

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

5. Simplified 76.2%

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

6. Taylor expanded in x around 0 85.8%

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

   1. *-lft-identity 52.0%

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

   2. *-lft-identity 52.0%

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

   3. *-commutative 52.0%

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

8. Simplified 85.8%

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

9. Final simplification 85.8%

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

Alternative 5: 75.6% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $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 76.2%

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

   1. mul-1-neg 76.2%

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

   2. unsub-neg 76.2%

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

5. Simplified 76.2%

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

6. Taylor expanded in x around 0 77.4%

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

   1. *-commutative 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in x around inf 77.4%

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

10. Final simplification 77.4%

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

Alternative 6: 75.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - x), $MachinePrecision] + -1.0), $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 76.2%

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

   1. mul-1-neg 76.2%

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

   2. unsub-neg 76.2%

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

5. Simplified 76.2%

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

6. Taylor expanded in x around 0 77.4%

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

   1. *-commutative 77.4%

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

8. Simplified 77.4%

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

   1. expm1-log1p-u 64.9%

      \[\leadsto \frac{2}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - x\right)\right)}} \]

10. Applied egg-rr 64.9%

    \[\leadsto \frac{2}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - x\right)\right)}} \]
11. Step-by-step derivation

    1. expm1-undefine 64.9%

       \[\leadsto \frac{2}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(1 - x\right)} - 1\right)}} \]

    2. sub-neg 64.9%

       \[\leadsto \frac{2}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(1 - x\right)} + \left(-1\right)\right)}} \]

    3. log1p-undefine 64.9%

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

    4. rem-exp-log 77.4%

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

    5. associate-+r- 77.4%

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

    6. metadata-eval 77.4%

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

    7. metadata-eval 77.4%

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

12. Simplified 77.4%

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

Alternative 7: 75.6% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $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 74.8%

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

4. Taylor expanded in x around 0 52.0%

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

   1. *-lft-identity 52.0%

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

   2. *-lft-identity 52.0%

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

   3. *-commutative 52.0%

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

6. Simplified 52.0%

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

7. Taylor expanded in x around 0 73.2%

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

   1. *-commutative 73.2%

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

9. Simplified 73.2%

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

10. Taylor expanded in x around 0 77.4%

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

11. Final simplification 77.4%

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

Alternative 8: 75.1% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 / N[(N[(1.0 - x), $MachinePrecision] + N[(1.0 + N[(x * 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 76.2%

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

   1. mul-1-neg 76.2%

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

   2. unsub-neg 76.2%

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

5. Simplified 76.2%

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

6. Taylor expanded in x around 0 77.4%

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

   1. *-commutative 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in x around inf 77.4%

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

10. Taylor expanded in x around inf 76.5%

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

    1. *-commutative 76.5%

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

12. Simplified 76.5%

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

13. Final simplification 76.5%

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

Alternative 9: 50.4% 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 52.9%

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

Reproduce

herbie shell --seed 2024119 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))