Octave 3.8, jcobi/1

Percentage Accurate: 74.6% → 99.7%

Time: 10.2s

Alternatives: 9

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: 74.6% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\beta \cdot \frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999)
     (/
      (+ (/ 2.0 alpha) (- (* beta (/ 2.0 alpha)) (/ 4.0 (* alpha 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 <= -0.999) {
		tmp = ((2.0 / alpha) + ((beta * (2.0 / alpha)) - (4.0 / (alpha * 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 <= (-0.999d0)) then
        tmp = ((2.0d0 / alpha) + ((beta * (2.0d0 / alpha)) - (4.0d0 / (alpha * 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 <= -0.999) {
		tmp = ((2.0 / alpha) + ((beta * (2.0 / alpha)) - (4.0 / (alpha * 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 <= -0.999:
		tmp = ((2.0 / alpha) + ((beta * (2.0 / alpha)) - (4.0 / (alpha * 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 <= -0.999)
		tmp = Float64(Float64(Float64(2.0 / alpha) + Float64(Float64(beta * Float64(2.0 / alpha)) - Float64(4.0 / Float64(alpha * 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 <= -0.999)
		tmp = ((2.0 / alpha) + ((beta * (2.0 / alpha)) - (4.0 / (alpha * 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, -0.999], N[(N[(N[(2.0 / alpha), $MachinePrecision] + N[(N[(beta * N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] - N[(4.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.998999999999999999

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around -inf 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in beta around 0 99.8%

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + \beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2} \]
   8. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}}{2} \]

      2. associate-*r/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot 1}{\alpha}} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{\frac{\color{blue}{2}}{\alpha} + \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      4. associate-*r/ 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\color{blue}{\frac{2 \cdot 1}{\alpha}} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{\color{blue}{2}}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      6. associate-*r/ 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \color{blue}{\frac{6 \cdot 1}{{\alpha}^{2}}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      7. metadata-eval 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{\color{blue}{6}}{{\alpha}^{2}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      8. unpow2 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\color{blue}{\alpha \cdot \alpha}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}\right)}{2} \]

      9. associate-*r/ 99.9%

         \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \color{blue}{\frac{4 \cdot 1}{{\alpha}^{2}}}\right)}{2} \]

      10. metadata-eval 99.9%

          \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{\color{blue}{4}}{{\alpha}^{2}}\right)}{2} \]

      11. unpow2 99.9%

          \[\leadsto \frac{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\color{blue}{\alpha \cdot \alpha}}\right)}{2} \]

   9. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} + \left(\beta \cdot \left(\frac{2}{\alpha} - \frac{6}{\alpha \cdot \alpha}\right) - \frac{4}{\alpha \cdot \alpha}\right)}}{2} \]

   10. Taylor expanded in alpha around inf 100.0%

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

       1. associate-*r/ 100.0%

          \[\leadsto \frac{\frac{2}{\alpha} + \left(\color{blue}{\frac{2 \cdot \beta}{\alpha}} - \frac{4}{\alpha \cdot \alpha}\right)}{2} \]

       2. associate-*l/ 99.9%

          \[\leadsto \frac{\frac{2}{\alpha} + \left(\color{blue}{\frac{2}{\alpha} \cdot \beta} - \frac{4}{\alpha \cdot \alpha}\right)}{2} \]

       3. *-commutative 99.9%

          \[\leadsto \frac{\frac{2}{\alpha} + \left(\color{blue}{\beta \cdot \frac{2}{\alpha}} - \frac{4}{\alpha \cdot \alpha}\right)}{2} \]

   12. Simplified 99.9%

       \[\leadsto \frac{\frac{2}{\alpha} + \left(\color{blue}{\beta \cdot \frac{2}{\alpha}} - \frac{4}{\alpha \cdot \alpha}\right)}{2} \]

   if -0.998999999999999999 < (/.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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\beta \cdot \frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999)
     (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 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 <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / 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 <= (-0.999d0)) then
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / 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 <= -0.999) {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / 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 <= -0.999:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / 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 <= -0.999)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / 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 <= -0.999)
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / 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, -0.999], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $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)) < -0.998999999999999999

   1. Initial program 6.5%

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

      1. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in alpha around inf 99.3%

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

      1. *-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in beta around 0 99.4%

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

   if -0.998999999999999999 < (/.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2200000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ (* 2.0 (/ beta alpha)) (* 2.0 (/ 1.0 alpha))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2200000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / 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 <= 2200000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 * (beta / alpha)) + (2.0d0 * (1.0d0 / alpha))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2200000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2200000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2200000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(2.0 * Float64(1.0 / alpha))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 2200000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 * (beta / alpha)) + (2.0 * (1.0 / alpha))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2200000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 98.3%

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

   if 2.2e9 < alpha

   1. Initial program 21.6%

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

      1. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in alpha around inf 84.5%

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

      1. *-commutative 84.5%

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

   6. Simplified 84.5%

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

   7. Taylor expanded in beta around 0 84.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + 2 \cdot \frac{1}{\alpha}}{2}\\ \end{array} \]

Alternative 4: 87.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7500000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7500000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7500000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} 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 <= 7500000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    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 <= 7500000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7500000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7500000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	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 <= 7500000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7500000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.5e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 98.3%

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

   if 7.5e9 < alpha

   1. Initial program 21.6%

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

      1. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in beta around 0 5.9%

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

      1. +-commutative 5.9%

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

   6. Simplified 5.9%

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

   7. Taylor expanded in alpha around inf 73.3%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7500000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5000000000.0)
   (/ (+ 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 <= 5000000000.0) {
		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 <= 5000000000.0d0) 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 <= 5000000000.0) {
		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 <= 5000000000.0:
		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 <= 5000000000.0)
		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 <= 5000000000.0)
		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, 5000000000.0], 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 < 5e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in alpha around 0 98.3%

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

   if 5e9 < alpha

   1. Initial program 21.6%

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

      1. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in alpha around inf 84.5%

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

      1. *-commutative 84.5%

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

   6. Simplified 84.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 68.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.95) (/ (- 1.0 (* alpha 0.5)) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} 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 <= 0.95d0) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    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 <= 0.95) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.95:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.95)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	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 <= 0.95)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.95], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around 0 74.1%

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

      1. +-commutative 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in alpha around 0 73.7%

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

      1. *-commutative 73.7%

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

   9. Simplified 73.7%

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

   if 0.94999999999999996 < alpha

   1. Initial program 24.2%

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

      1. +-commutative 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in beta around 0 7.2%

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

      1. +-commutative 7.2%

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

   6. Simplified 7.2%

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

   7. Taylor expanded in alpha around inf 71.4%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 68.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 2200000000.0) 0.5 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 2200000000.0) {
		tmp = 0.5;
	} 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 <= 2200000000.0d0) then
        tmp = 0.5d0
    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 <= 2200000000.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 2200000000.0:
		tmp = 0.5
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 2200000000.0)
		tmp = 0.5;
	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 <= 2200000000.0)
		tmp = 0.5;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 2200000000.0], 0.5, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2e9

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around 0 73.5%

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

      1. +-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in alpha around 0 72.2%

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

   if 2.2e9 < alpha

   1. Initial program 21.6%

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

      1. +-commutative 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in beta around 0 5.9%

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

      1. +-commutative 5.9%

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

   6. Simplified 5.9%

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

   7. Taylor expanded in alpha around inf 73.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2200000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 71.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 48.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5;
	} 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 <= 48.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 48.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 48.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 48.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 48.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 48.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 48

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in beta around 0 66.0%

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

      1. +-commutative 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in alpha around 0 64.2%

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

   if 48 < beta

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. div-sub 89.7%

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

      3. sub-neg 89.7%

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

      4. +-commutative 89.7%

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

      5. associate-+r+ 91.1%

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

      6. sub-neg 91.1%

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

      7. associate-+l- 89.7%

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

      8. div-sub 89.7%

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

      9. div-sub 89.7%

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

      10. metadata-eval 89.7%

          \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      11. associate-/r* 89.8%

          \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2}} \]

      12. sub-neg 89.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

      13. +-commutative 89.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(-\beta\right) + \alpha}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

      14. neg-sub0 89.8%

          \[\leadsto 0.5 - \frac{\color{blue}{\left(0 - \beta\right)} + \alpha}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

      15. associate-+l- 89.8%

          \[\leadsto 0.5 - \frac{\color{blue}{0 - \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

      16. sub0-neg 89.8%

          \[\leadsto 0.5 - \frac{\color{blue}{-\left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

      17. neg-mul-1 89.8%

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

      18. times-frac 89.8%

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

      19. *-commutative 89.8%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{0.5 - \frac{\beta - \alpha}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]

   4. Taylor expanded in beta around inf 87.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 48:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 37.0% 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 73.6%

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

   1. +-commutative 73.6%

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

   2. div-sub 73.6%

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

   3. sub-neg 73.6%

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

   4. +-commutative 73.6%

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

   5. associate-+r+ 74.2%

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

   6. sub-neg 74.2%

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

   7. associate-+l- 73.6%

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

   8. div-sub 73.6%

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

   9. div-sub 73.6%

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

   10. metadata-eval 73.6%

       \[\leadsto \color{blue}{0.5} - \frac{\frac{\alpha - \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   11. associate-/r* 73.7%

       \[\leadsto 0.5 - \color{blue}{\frac{\alpha - \beta}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2}} \]

   12. sub-neg 73.7%

       \[\leadsto 0.5 - \frac{\color{blue}{\alpha + \left(-\beta\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

   13. +-commutative 73.7%

       \[\leadsto 0.5 - \frac{\color{blue}{\left(-\beta\right) + \alpha}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

   14. neg-sub0 73.7%

       \[\leadsto 0.5 - \frac{\color{blue}{\left(0 - \beta\right)} + \alpha}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

   15. associate-+l- 73.7%

       \[\leadsto 0.5 - \frac{\color{blue}{0 - \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

   16. sub0-neg 73.7%

       \[\leadsto 0.5 - \frac{\color{blue}{-\left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \]

   17. neg-mul-1 73.7%

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

   18. times-frac 73.6%

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

   19. *-commutative 73.6%

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

3. Simplified 73.7%

   \[\leadsto \color{blue}{0.5 - \frac{\beta - \alpha}{\mathsf{fma}\left(\beta + \alpha, -2, -4\right)}} \]

4. Taylor expanded in beta around inf 35.6%

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

5. Final simplification 35.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023185 
(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))