Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

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. Final simplification 100.0%

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

Alternative 2: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 135000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot {x}^{-2}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 135000000.0)
   (/ 2.0 (fma x x 2.0))
   (+ (+ 1.0 (* 2.0 (pow x -2.0))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 135000000.0) {
		tmp = 2.0 / fma(x, x, 2.0);
	} else {
		tmp = (1.0 + (2.0 * pow(x, -2.0))) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 135000000.0)
		tmp = Float64(2.0 / fma(x, x, 2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(2.0 * (x ^ -2.0))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 135000000.0], N[(2.0 / N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(2.0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. fma-define 78.2%

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

   5. Simplified 78.2%

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

   if 1.35e8 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.8%

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

      1. +-commutative 55.8%

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

      2. unpow2 55.8%

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

      3. fma-define 55.8%

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

   5. Simplified 55.8%

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

   6. Taylor expanded in x around inf 55.8%

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

      1. add-sqr-sqrt 55.8%

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

      2. sqrt-unprod 79.9%

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

      3. frac-times 79.9%

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

      4. metadata-eval 79.9%

         \[\leadsto \sqrt{\frac{\color{blue}{4}}{{x}^{2} \cdot {x}^{2}}} \]

      5. pow-sqr 79.9%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{x}^{\left(2 \cdot 2\right)}}}} \]

      6. metadata-eval 79.9%

         \[\leadsto \sqrt{\frac{4}{{x}^{\color{blue}{4}}}} \]

   8. Applied egg-rr 79.9%

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

      1. pow1/2 79.9%

         \[\leadsto \color{blue}{{\left(\frac{4}{{x}^{4}}\right)}^{0.5}} \]

      2. div-inv 79.9%

         \[\leadsto {\color{blue}{\left(4 \cdot \frac{1}{{x}^{4}}\right)}}^{0.5} \]

      3. unpow-prod-down 79.9%

         \[\leadsto \color{blue}{{4}^{0.5} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{0.5}} \]

      4. pow-flip 79.9%

         \[\leadsto {4}^{0.5} \cdot {\color{blue}{\left({x}^{\left(-4\right)}\right)}}^{0.5} \]

      5. pow-pow 53.6%

         \[\leadsto {4}^{0.5} \cdot \color{blue}{{x}^{\left(\left(-4\right) \cdot 0.5\right)}} \]

      6. metadata-eval 53.6%

         \[\leadsto {4}^{0.5} \cdot {x}^{\left(\color{blue}{-4} \cdot 0.5\right)} \]

      7. metadata-eval 53.6%

         \[\leadsto {4}^{0.5} \cdot {x}^{\color{blue}{-2}} \]

      8. unpow1/2 53.6%

         \[\leadsto \color{blue}{\sqrt{4}} \cdot {x}^{-2} \]

      9. metadata-eval 53.6%

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

      10. expm1-log1p-u 53.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-2}\right)\right)} \]

      11. *-commutative 53.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{x}^{-2} \cdot 2}\right)\right) \]

      12. expm1-undefine 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(1 + {x}^{-2} \cdot 2\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 135000000:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + 2 \cdot {x}^{-2}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (* 2.0 (pow x -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 * pow(x, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 * (x ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 * Math.pow(x, -2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = 2.0 * math.pow(x, -2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(2.0 * (x ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = 2.0 * (x ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(2.0 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.5%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. unpow2 54.9%

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

      3. fma-define 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 54.9%

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

      1. clear-num 54.9%

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

      2. associate-/r/ 54.9%

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

      3. pow-flip 52.8%

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

      4. metadata-eval 52.8%

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

   8. Applied egg-rr 52.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.4) 1.0 (/ 2.0 (pow x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / pow(x, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = 2.0d0 / (x ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = 2.0 / Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = 2.0 / math.pow(x, 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(2.0 / (x ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = 2.0 / (x ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.5%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. unpow2 54.9%

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

      3. fma-define 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 54.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 72.8%

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

   1. +-commutative 72.8%

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

   2. unpow2 72.8%

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

   3. fma-define 72.8%

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

5. Simplified 72.8%

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

6. Final simplification 72.8%

   \[\leadsto \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \]
7. Add Preprocessing

Alternative 6: 63.6% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(2 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4) 1.0 (* (/ 1.0 x) (* 2.0 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / x) * (2.0 * (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / x) * (2.0d0 * (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / x) * (2.0 * (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = 1.0
	else:
		tmp = (1.0 / x) * (2.0 * (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(2.0 * Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = (1.0 / x) * (2.0 * (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(N[(1.0 / x), $MachinePrecision] * N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.5%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. +-commutative 54.9%

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

      2. unpow2 54.9%

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

      3. fma-define 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 54.9%

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

      1. add-sqr-sqrt 54.9%

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

      2. sqrt-unprod 78.7%

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

      3. frac-times 78.7%

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

      4. metadata-eval 78.7%

         \[\leadsto \sqrt{\frac{\color{blue}{4}}{{x}^{2} \cdot {x}^{2}}} \]

      5. pow-sqr 78.7%

         \[\leadsto \sqrt{\frac{4}{\color{blue}{{x}^{\left(2 \cdot 2\right)}}}} \]

      6. metadata-eval 78.7%

         \[\leadsto \sqrt{\frac{4}{{x}^{\color{blue}{4}}}} \]

   8. Applied egg-rr 78.7%

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

      1. pow1/2 78.7%

         \[\leadsto \color{blue}{{\left(\frac{4}{{x}^{4}}\right)}^{0.5}} \]

      2. div-inv 78.7%

         \[\leadsto {\color{blue}{\left(4 \cdot \frac{1}{{x}^{4}}\right)}}^{0.5} \]

      3. unpow-prod-down 78.7%

         \[\leadsto \color{blue}{{4}^{0.5} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{0.5}} \]

      4. pow-flip 78.7%

         \[\leadsto {4}^{0.5} \cdot {\color{blue}{\left({x}^{\left(-4\right)}\right)}}^{0.5} \]

      5. pow-pow 52.8%

         \[\leadsto {4}^{0.5} \cdot \color{blue}{{x}^{\left(\left(-4\right) \cdot 0.5\right)}} \]

      6. metadata-eval 52.8%

         \[\leadsto {4}^{0.5} \cdot {x}^{\left(\color{blue}{-4} \cdot 0.5\right)} \]

      7. metadata-eval 52.8%

         \[\leadsto {4}^{0.5} \cdot {x}^{\color{blue}{-2}} \]

      8. unpow1/2 52.8%

         \[\leadsto \color{blue}{\sqrt{4}} \cdot {x}^{-2} \]

      9. metadata-eval 52.8%

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

      10. *-commutative 52.8%

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

      11. sqr-pow 52.8%

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

      12. associate-*l* 52.8%

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

      13. metadata-eval 52.8%

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

      14. unpow-1 52.8%

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

      15. metadata-eval 52.8%

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

      16. unpow-1 52.8%

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

   10. Applied egg-rr 52.8%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(2 \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 50.9%

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

4. Final simplification 50.9%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))