Octave 3.8, jcobi/1

Percentage Accurate: 74.2% → 99.9%

Time: 7.5s

Alternatives: 11

Speedup: 1.2×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99998)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99997999999999998

   1. Initial program 7.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around -inf 91.7%

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

   6. Simplified 99.9%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-sub 99.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]

      2. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]

      3. associate-+l+ 99.9%

         \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)}{2} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99998)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99998)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99997999999999998

   1. Initial program 7.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-sub 99.9%

         \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]

      2. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]

      3. associate-+l+ 99.9%

         \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)}{2} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99998)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99998)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99998], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99997999999999998

   1. Initial program 7.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 7.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 99.2%

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

   if -0.99997999999999998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 4: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+217} \lor \neg \left(\alpha \leq 1.08 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.8e+15)
   1.0
   (if (or (<= alpha 1.5e+217) (not (<= alpha 1.08e+260)))
     (/ (/ 2.0 alpha) 2.0)
     (/ (* 2.0 (/ beta alpha)) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.8e+15) {
		tmp = 1.0;
	} else if ((alpha <= 1.5e+217) || !(alpha <= 1.08e+260)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.8d+15) then
        tmp = 1.0d0
    else if ((alpha <= 1.5d+217) .or. (.not. (alpha <= 1.08d+260))) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.8e+15) {
		tmp = 1.0;
	} else if ((alpha <= 1.5e+217) || !(alpha <= 1.08e+260)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.8e+15:
		tmp = 1.0
	elif (alpha <= 1.5e+217) or not (alpha <= 1.08e+260):
		tmp = (2.0 / alpha) / 2.0
	else:
		tmp = (2.0 * (beta / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.8e+15)
		tmp = 1.0;
	elseif ((alpha <= 1.5e+217) || !(alpha <= 1.08e+260))
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.8e+15)
		tmp = 1.0;
	elseif ((alpha <= 1.5e+217) || ~((alpha <= 1.08e+260)))
		tmp = (2.0 / alpha) / 2.0;
	else
		tmp = (2.0 * (beta / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.8e+15], 1.0, If[Or[LessEqual[alpha, 1.5e+217], N[Not[LessEqual[alpha, 1.08e+260]], $MachinePrecision]], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 1.8e15

   1. Initial program 99.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around inf 49.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 1.8e15 < alpha < 1.49999999999999988e217 or 1.07999999999999995e260 < alpha

   1. Initial program 21.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 21.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 84.8%

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

   5. Taylor expanded in beta around 0 64.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 1.49999999999999988e217 < alpha < 1.07999999999999995e260

   1. Initial program 14.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 14.8%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 14.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 92.5%

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

   5. Taylor expanded in beta around inf 82.2%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+217} \lor \neg \left(\alpha \leq 1.08 \cdot 10^{+260}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.45 \cdot 10^{+217} \lor \neg \left(\alpha \leq 5.5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2e+15)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 1.45e+217) (not (<= alpha 5.5e+259)))
     (/ (/ 2.0 alpha) 2.0)
     (/ (* 2.0 (/ beta alpha)) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.45e+217) || !(alpha <= 5.5e+259)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2d+15) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 1.45d+217) .or. (.not. (alpha <= 5.5d+259))) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else
        tmp = (2.0d0 * (beta / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 1.45e+217) || !(alpha <= 5.5e+259)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = (2.0 * (beta / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2e+15:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 1.45e+217) or not (alpha <= 5.5e+259):
		tmp = (2.0 / alpha) / 2.0
	else:
		tmp = (2.0 * (beta / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2e+15)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 1.45e+217) || !(alpha <= 5.5e+259))
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(beta / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2e+15)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 1.45e+217) || ~((alpha <= 5.5e+259)))
		tmp = (2.0 / alpha) / 2.0;
	else
		tmp = (2.0 * (beta / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2e+15], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 1.45e+217], N[Not[LessEqual[alpha, 5.5e+259]], $MachinePrecision]], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 2e15

   1. Initial program 99.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around 0 97.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   if 2e15 < alpha < 1.44999999999999992e217 or 5.50000000000000029e259 < alpha

   1. Initial program 21.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 21.7%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 84.8%

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

   5. Taylor expanded in beta around 0 64.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 1.44999999999999992e217 < alpha < 5.50000000000000029e259

   1. Initial program 14.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 14.8%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 14.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 92.5%

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

   5. Taylor expanded in beta around inf 82.2%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 1.45 \cdot 10^{+217} \lor \neg \left(\alpha \leq 5.5 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -3.1e-276)
     t_0
     (if (<= beta 4.3e-296)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = t_0;
	} else if (beta <= 4.3e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= (-3.1d-276)) then
        tmp = t_0
    else if (beta <= 4.3d-296) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = t_0;
	} else if (beta <= 4.3e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -3.1e-276:
		tmp = t_0
	elif beta <= 4.3e-296:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -3.1e-276)
		tmp = t_0;
	elseif (beta <= 4.3e-296)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -3.1e-276)
		tmp = t_0;
	elseif (beta <= 4.3e-296)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -3.1e-276], t$95$0, If[LessEqual[beta, 4.3e-296], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -3.09999999999999989e-276 or 4.29999999999999978e-296 < beta < 2

   1. Initial program 77.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 77.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around 0 74.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   5. Taylor expanded in beta around 0 74.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   6. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   7. Simplified 74.0%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -3.09999999999999989e-276 < beta < 4.29999999999999978e-296

   1. Initial program 25.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 81.2%

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

   5. Taylor expanded in beta around 0 81.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 2 < beta

   1. Initial program 83.3%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 83.3%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around inf 80.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 0.00106:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -3.1e-276)
   (/ (+ 1.0 (* beta 0.5)) 2.0)
   (if (<= beta 4.2e-296)
     (/ (/ 2.0 alpha) 2.0)
     (if (<= beta 0.00106) (/ (- 1.0 (* alpha 0.5)) 2.0) 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 4.2e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 0.00106) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-3.1d-276)) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else if (beta <= 4.2d-296) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 0.00106d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 4.2e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 0.00106) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -3.1e-276:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	elif beta <= 4.2e-296:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 0.00106:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -3.1e-276)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	elseif (beta <= 4.2e-296)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 0.00106)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -3.1e-276)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	elseif (beta <= 4.2e-296)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 0.00106)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -3.1e-276], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 4.2e-296], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 0.00106], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 4 regimes

2. if beta < -3.09999999999999989e-276

   1. Initial program 78.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around 0 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   5. Taylor expanded in beta around 0 75.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   6. Step-by-step derivation

      1. *-commutative 75.1%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   7. Simplified 75.1%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -3.09999999999999989e-276 < beta < 4.1999999999999999e-296

   1. Initial program 25.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 81.2%

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

   5. Taylor expanded in beta around 0 81.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 4.1999999999999999e-296 < beta < 0.00105999999999999996

   1. Initial program 78.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around 0 78.0%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   6. Simplified 78.0%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   7. Taylor expanded in alpha around 0 73.8%

      \[\leadsto \frac{1 - \color{blue}{0.5 \cdot \alpha}}{2} \]
   8. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   9. Simplified 73.8%

      \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   if 0.00105999999999999996 < beta

   1. Initial program 82.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around inf 79.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 0.00106:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 0.00106:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta -3.1e-276)
   (/ (+ 1.0 (* beta 0.5)) 2.0)
   (if (<= beta 4.2e-296)
     (/ (/ 2.0 alpha) 2.0)
     (if (<= beta 0.00106)
       (/ (- 1.0 (* alpha 0.5)) 2.0)
       (/ (- 2.0 (/ 2.0 beta)) 2.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 4.2e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 0.00106) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= (-3.1d-276)) then
        tmp = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    else if (beta <= 4.2d-296) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 0.00106d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= -3.1e-276) {
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	} else if (beta <= 4.2e-296) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 0.00106) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= -3.1e-276:
		tmp = (1.0 + (beta * 0.5)) / 2.0
	elif beta <= 4.2e-296:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 0.00106:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= -3.1e-276)
		tmp = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0);
	elseif (beta <= 4.2e-296)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 0.00106)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= -3.1e-276)
		tmp = (1.0 + (beta * 0.5)) / 2.0;
	elseif (beta <= 4.2e-296)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 0.00106)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, -3.1e-276], N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 4.2e-296], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 0.00106], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < -3.09999999999999989e-276

   1. Initial program 78.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around 0 76.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   5. Taylor expanded in beta around 0 75.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   6. Step-by-step derivation

      1. *-commutative 75.1%

         \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   7. Simplified 75.1%

      \[\leadsto \frac{\color{blue}{\beta \cdot 0.5} + 1}{2} \]

   if -3.09999999999999989e-276 < beta < 4.1999999999999999e-296

   1. Initial program 25.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 81.2%

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

   5. Taylor expanded in beta around 0 81.2%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha}}}{2} \]

   if 4.1999999999999999e-296 < beta < 0.00105999999999999996

   1. Initial program 78.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around 0 78.0%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{2 + \alpha}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \frac{1 - \frac{\alpha}{\color{blue}{\alpha + 2}}}{2} \]

   6. Simplified 78.0%

      \[\leadsto \frac{\color{blue}{1 - \frac{\alpha}{\alpha + 2}}}{2} \]

   7. Taylor expanded in alpha around 0 73.8%

      \[\leadsto \frac{1 - \color{blue}{0.5 \cdot \alpha}}{2} \]
   8. Step-by-step derivation

      1. *-commutative 73.8%

         \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   9. Simplified 73.8%

      \[\leadsto \frac{1 - \color{blue}{\alpha \cdot 0.5}}{2} \]

   if 0.00105999999999999996 < beta

   1. Initial program 82.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]
   4. Step-by-step derivation

      1. div-sub 82.5%

         \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)} + 1}{2} \]

      2. associate-+l- 85.0%

         \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\beta + \alpha\right) + 2} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}}{2} \]

      3. associate-+l+ 85.0%

         \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + \left(\alpha + 2\right)}} - \left(\frac{\alpha}{\left(\beta + \alpha\right) + 2} - 1\right)}{2} \]

      4. associate-+l+ 85.0%

         \[\leadsto \frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}} - 1\right)}{2} \]

   5. Applied egg-rr 85.0%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + \left(\alpha + 2\right)} - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} - 1\right)}}{2} \]

   6. Taylor expanded in beta around -inf 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. neg-mul-1 79.0%

         \[\leadsto \frac{2 - \frac{\left(2 + \alpha\right) - \color{blue}{\left(-\alpha\right)}}{\beta}}{2} \]

      5. associate--l+ 79.0%

         \[\leadsto \frac{2 - \frac{\color{blue}{2 + \left(\alpha - \left(-\alpha\right)\right)}}{\beta}}{2} \]

      6. sub-neg 79.0%

         \[\leadsto \frac{2 - \frac{2 + \color{blue}{\left(\alpha + \left(-\left(-\alpha\right)\right)\right)}}{\beta}}{2} \]

      7. remove-double-neg 79.0%

         \[\leadsto \frac{2 - \frac{2 + \left(\alpha + \color{blue}{\alpha}\right)}{\beta}}{2} \]

   8. Simplified 79.0%

      \[\leadsto \frac{\color{blue}{2 - \frac{2 + \left(\alpha + \alpha\right)}{\beta}}}{2} \]

   9. Taylor expanded in alpha around 0 79.8%

      \[\leadsto \frac{2 - \color{blue}{\frac{2}{\beta}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 0.00106:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 9: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.75e+15)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.75e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.75d+15) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.75e+15) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.75e+15:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.75e+15)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.75e+15)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3.75e+15], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.75e15

   1. Initial program 99.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around 0 97.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   if 3.75e15 < alpha

   1. Initial program 20.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 20.8%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 85.8%

      \[\leadsto \frac{\color{blue}{\frac{2 + 2 \cdot \beta}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 51.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2.55e+15) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.55e+15) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 2.55d+15) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2.55e+15) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2.55e+15:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2.55e+15)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2.55e+15)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2.55e+15], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.55e15

   1. Initial program 99.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in beta around inf 49.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 2.55e15 < alpha

   1. Initial program 20.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. +-commutative 20.8%

         \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

   3. Simplified 20.8%

      \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

   4. Taylor expanded in alpha around inf 85.8%

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

   5. Taylor expanded in beta around 0 58.6%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.55 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 36.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 1.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 77.8%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. Step-by-step derivation

   1. +-commutative 77.8%

      \[\leadsto \frac{\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2} + 1}{2} \]

3. Simplified 77.8%

   \[\leadsto \color{blue}{\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}} \]

4. Taylor expanded in beta around inf 40.5%

   \[\leadsto \frac{\color{blue}{2}}{2} \]

5. Final simplification 40.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))