Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 5

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. Final simplification 100.0%

   \[\leadsto \frac{2}{e^{x} + e^{-x}} \]

Alternative 2: 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. Taylor expanded in x around 0 79.5%

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

   1. +-commutative 79.5%

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

   2. unpow2 79.5%

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

   3. fma-def 79.5%

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

4. Simplified 79.5%

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

   1. expm1-log1p-u 79.5%

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

   2. expm1-udef 97.7%

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

   3. log1p-udef 97.7%

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

   4. rem-exp-log 97.7%

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

6. Applied egg-rr 97.7%

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

7. Final simplification 97.7%

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

Alternative 3: 87.6% 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. Taylor expanded in x around 0 86.1%

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

      1. +-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. fma-def 86.1%

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

   4. Simplified 86.1%

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

   if 360 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.2%

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

      1. +-commutative 56.2%

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

      2. unpow2 56.2%

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

      3. fma-def 56.2%

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

   4. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

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

      2. expm1-udef 98.3%

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

      3. log1p-udef 98.3%

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

      4. rem-exp-log 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. frac-2neg 98.3%

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

      2. div-inv 98.3%

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

      3. metadata-eval 98.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 98.3%

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

      6. sqr-neg 98.3%

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

      7. sqrt-unprod 98.3%

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

      8. add-sqr-sqrt 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   10. Simplified 98.3%

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

   11. Taylor expanded in x around inf 100.0%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

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

Alternative 4: 75.0% accurate, 67.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 350.0) 1.0 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 350.0) {
		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 <= 350.0d0) 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 <= 350.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 350.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 350.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 67.2%

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

   if 350 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.2%

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

      1. +-commutative 56.2%

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

      2. unpow2 56.2%

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

      3. fma-def 56.2%

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

   4. Simplified 56.2%

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

      1. expm1-log1p-u 56.2%

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

      2. expm1-udef 98.3%

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

      3. log1p-udef 98.3%

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

      4. rem-exp-log 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. frac-2neg 98.3%

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

      2. div-inv 98.3%

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

      3. metadata-eval 98.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 98.3%

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

      6. sqr-neg 98.3%

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

      7. sqrt-unprod 98.3%

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

      8. add-sqr-sqrt 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   10. Simplified 98.3%

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

   11. Taylor expanded in x around inf 100.0%

       \[\leadsto \color{blue}{1} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 51.5% 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. Taylor expanded in x around 0 53.2%

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

3. Final simplification 53.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))