Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

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: 74.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(1 - \left(\left(1 + {x}^{4} \cdot 0.25\right) + -1\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.42)
   (*
    (- 1.0 (+ (+ 1.0 (* (pow x 4.0) 0.25)) -1.0))
    (/ 1.0 (+ 1.0 (* 0.5 (* x x)))))
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = (1.0 - ((1.0 + (pow(x, 4.0) * 0.25)) + -1.0)) * (1.0 / (1.0 + (0.5 * (x * x))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = (1.0d0 - ((1.0d0 + ((x ** 4.0d0) * 0.25d0)) + (-1.0d0))) * (1.0d0 / (1.0d0 + (0.5d0 * (x * x))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = (1.0 - ((1.0 + (Math.pow(x, 4.0) * 0.25)) + -1.0)) * (1.0 / (1.0 + (0.5 * (x * x))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = (1.0 - ((1.0 + (math.pow(x, 4.0) * 0.25)) + -1.0)) * (1.0 / (1.0 + (0.5 * (x * x))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(Float64(1.0 - Float64(Float64(1.0 + Float64((x ^ 4.0) * 0.25)) + -1.0)) * Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = (1.0 - ((1.0 + ((x ^ 4.0) * 0.25)) + -1.0)) * (1.0 / (1.0 + (0.5 * (x * x))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], N[(N[(1.0 - N[(N[(1.0 + N[(N[Power[x, 4.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 63.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 63.7%

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

   4. Simplified 63.7%

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

      1. flip-+ 63.4%

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

      2. div-inv 63.4%

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

      3. metadata-eval 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

      6. swap-sqr 63.4%

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

      7. pow2 63.4%

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

      8. pow2 63.4%

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

      9. pow-prod-up 63.4%

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

      10. metadata-eval 63.4%

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

      11. metadata-eval 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot \color{blue}{0.25}\right) \cdot \frac{1}{1 - -0.5 \cdot \left(x \cdot x\right)} \]

      12. cancel-sign-sub-inv 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{\color{blue}{1 + \left(--0.5\right) \cdot \left(x \cdot x\right)}} \]

      13. metadata-eval 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{1 + \color{blue}{0.5} \cdot \left(x \cdot x\right)} \]

   6. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 63.4%

         \[\leadsto \left(1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{4} \cdot 0.25\right)\right)}\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      2. expm1-udef 63.4%

         \[\leadsto \left(1 - \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{4} \cdot 0.25\right)} - 1\right)}\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      3. log1p-udef 63.4%

         \[\leadsto \left(1 - \left(e^{\color{blue}{\log \left(1 + {x}^{4} \cdot 0.25\right)}} - 1\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      4. add-exp-log 63.4%

         \[\leadsto \left(1 - \left(\color{blue}{\left(1 + {x}^{4} \cdot 0.25\right)} - 1\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   8. Applied egg-rr 63.4%

      \[\leadsto \left(1 - \color{blue}{\left(\left(1 + {x}^{4} \cdot 0.25\right) - 1\right)}\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(1 - \left(\left(1 + {x}^{4} \cdot 0.25\right) + -1\right)\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\right) + -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.42) (+ (+ (+ 2.0 (fma -0.5 (* x x) 1.0)) -1.0) -1.0) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = ((2.0 + fma(-0.5, (x * x), 1.0)) + -1.0) + -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(Float64(Float64(2.0 + fma(-0.5, Float64(x * x), 1.0)) + -1.0) + -1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], N[(N[(N[(2.0 + N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 63.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 63.7%

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

   4. Simplified 63.7%

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

      1. expm1-log1p-u 63.0%

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

      2. expm1-udef 63.0%

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

      3. log1p-udef 63.0%

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

      4. add-exp-log 63.7%

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

      5. +-commutative 63.7%

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

      6. fma-def 63.7%

         \[\leadsto \left(1 + \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)}\right) - 1 \]

   6. Applied egg-rr 63.7%

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

      1. expm1-log1p-u 60.8%

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

      2. expm1-udef 60.8%

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

      3. log1p-udef 61.5%

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

      4. +-commutative 61.5%

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

      5. add-exp-log 63.7%

         \[\leadsto \left(\color{blue}{\left(\left(1 + \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\right) + 1\right)} - 1\right) - 1 \]

      6. +-commutative 63.7%

         \[\leadsto \left(\left(\color{blue}{\left(\mathsf{fma}\left(-0.5, x \cdot x, 1\right) + 1\right)} + 1\right) - 1\right) - 1 \]

      7. associate-+l+ 63.7%

         \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(-0.5, x \cdot x, 1\right) + \left(1 + 1\right)\right)} - 1\right) - 1 \]

      8. metadata-eval 63.7%

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

   8. Applied egg-rr 63.7%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(\left(2 + \mathsf{fma}\left(-0.5, x \cdot x, 1\right)\right) + -1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 74.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(2 + \left(x \cdot x\right) \cdot -0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.42) (+ (+ 2.0 (* (* x x) -0.5)) -1.0) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 + ((x * x) * -0.5)) + -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = (2.0d0 + ((x * x) * (-0.5d0))) + (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = (2.0 + ((x * x) * -0.5)) + -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = (2.0 + ((x * x) * -0.5)) + -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * x) * -0.5)) + -1.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = (2.0 + ((x * x) * -0.5)) + -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 63.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 63.7%

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

   4. Simplified 63.7%

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

      1. flip-+ 63.4%

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

      2. div-inv 63.4%

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

      3. metadata-eval 63.4%

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

      4. *-commutative 63.4%

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

      5. *-commutative 63.4%

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

      6. swap-sqr 63.4%

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

      7. pow2 63.4%

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

      8. pow2 63.4%

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

      9. pow-prod-up 63.4%

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

      10. metadata-eval 63.4%

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

      11. metadata-eval 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot \color{blue}{0.25}\right) \cdot \frac{1}{1 - -0.5 \cdot \left(x \cdot x\right)} \]

      12. cancel-sign-sub-inv 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{\color{blue}{1 + \left(--0.5\right) \cdot \left(x \cdot x\right)}} \]

      13. metadata-eval 63.4%

          \[\leadsto \left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{1 + \color{blue}{0.5} \cdot \left(x \cdot x\right)} \]

   6. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\left(1 - {x}^{4} \cdot 0.25\right) \cdot \frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}} \]
   7. Step-by-step derivation

      1. un-div-inv 63.4%

         \[\leadsto \color{blue}{\frac{1 - {x}^{4} \cdot 0.25}{1 + 0.5 \cdot \left(x \cdot x\right)}} \]

      2. metadata-eval 63.4%

         \[\leadsto \frac{\color{blue}{1 \cdot 1} - {x}^{4} \cdot 0.25}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      3. *-commutative 63.4%

         \[\leadsto \frac{1 \cdot 1 - \color{blue}{0.25 \cdot {x}^{4}}}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      4. metadata-eval 63.4%

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

      5. sqr-pow 63.4%

         \[\leadsto \frac{1 \cdot 1 - \left(0.5 \cdot 0.5\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)}}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      6. metadata-eval 63.4%

         \[\leadsto \frac{1 \cdot 1 - \left(0.5 \cdot 0.5\right) \cdot \left({x}^{\color{blue}{2}} \cdot {x}^{\left(\frac{4}{2}\right)}\right)}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      7. pow2 63.4%

         \[\leadsto \frac{1 \cdot 1 - \left(0.5 \cdot 0.5\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)}{1 + 0.5 \cdot \left(x \cdot x\right)} \]

      8. metadata-eval 63.4%

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

      9. pow2 63.4%

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

      10. swap-sqr 63.4%

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

      11. flip-- 63.7%

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

      12. cancel-sign-sub-inv 63.7%

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

      13. metadata-eval 63.7%

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

      14. expm1-log1p-u 63.0%

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

      15. expm1-udef 63.0%

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

   8. Applied egg-rr 63.7%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\left(2 + \left(x \cdot x\right) \cdot -0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 74.3% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.42) (+ 1.0 (* (* x x) -0.5)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.42], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 63.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 63.7%

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

   4. Simplified 63.7%

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

   if 1.4199999999999999 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 34.5% accurate, 67.3× 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%

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

   3. Applied egg-rr 13.2%

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

   if 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 35.6%

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

Alternative 7: 74.4% accurate, 67.3× 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. Taylor expanded in x around 0 63.7%

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

   if 360 < x

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 73.0%

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

Alternative 8: 51.5% 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%

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

3. Applied egg-rr 53.5%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 53.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))