Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 7

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: 87.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) (/ 2.0 (fma x x 2.0)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 2.0 / fma(x, x, 2.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(2.0 / fma(x, x, 2.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. unpow2 83.0%

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

      3. fma-define 83.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   5. Simplified 83.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

   if 360 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -1 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 75.9%

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

   1. +-commutative 75.9%

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

   2. unpow2 75.9%

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

   3. fma-define 75.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Simplified 75.9%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 75.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right)\right)} \]

   2. expm1-undefine 97.4%

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

   3. log1p-undefine 97.4%

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

   4. rem-exp-log 97.4%

      \[\leadsto \color{blue}{\left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right)} - 1 \]

7. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{\left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) - 1} \]

8. Final simplification 97.4%

   \[\leadsto \left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -1 \]
9. Add Preprocessing

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(2 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -2 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 75.9%

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

   1. +-commutative 75.9%

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

   2. unpow2 75.9%

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

   3. fma-define 75.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]

5. Simplified 75.9%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 75.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right)\right)} \]

   2. expm1-undefine 97.4%

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

   3. log1p-undefine 97.4%

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

   4. rem-exp-log 97.4%

      \[\leadsto \color{blue}{\left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right)} - 1 \]

7. Applied egg-rr 97.4%

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

   1. expm1-log1p-u 95.8%

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

   2. expm1-undefine 95.8%

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

   3. log1p-undefine 96.3%

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

   4. +-commutative 96.3%

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

   5. add-exp-log 97.4%

      \[\leadsto \left(\color{blue}{\left(\left(1 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + 1\right)} - 1\right) - 1 \]

   6. +-commutative 97.4%

      \[\leadsto \left(\left(\color{blue}{\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + 1\right)} + 1\right) - 1\right) - 1 \]

   7. associate-+l+ 97.4%

      \[\leadsto \left(\color{blue}{\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + \left(1 + 1\right)\right)} - 1\right) - 1 \]

   8. metadata-eval 97.4%

      \[\leadsto \left(\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + \color{blue}{2}\right) - 1\right) - 1 \]

9. Applied egg-rr 97.4%

   \[\leadsto \color{blue}{\left(\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + 2\right) - 1\right)} - 1 \]
10. Step-by-step derivation

    1. associate--l- 97.4%

       \[\leadsto \color{blue}{\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + 2\right) - \left(1 + 1\right)} \]

    2. metadata-eval 97.4%

       \[\leadsto \left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + 2\right) - \color{blue}{2} \]

    3. sub-neg 97.4%

       \[\leadsto \color{blue}{\left(\frac{2}{\mathsf{fma}\left(x, x, 2\right)} + 2\right) + \left(-2\right)} \]

    4. +-commutative 97.4%

       \[\leadsto \color{blue}{\left(2 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right)} + \left(-2\right) \]

    5. metadata-eval 97.4%

       \[\leadsto \left(2 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + \color{blue}{-2} \]

11. Applied egg-rr 97.4%

    \[\leadsto \color{blue}{\left(2 + \frac{2}{\mathsf{fma}\left(x, x, 2\right)}\right) + -2} \]
12. Add Preprocessing

Alternative 5: 75.8% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{1} \]

   if 360 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 35.1% accurate, 34.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 360.0) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 360.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 360.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 360.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 360.0], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 100.0%

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

   4. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{0.5} \]

   if 360 < x

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 53.8%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))