Octave 3.8, jcobi/1

Percentage Accurate: 73.9% → 99.8%

Time: 8.4s

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: 73.9% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \beta\\ t_1 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot t_0 - \frac{t_0}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_1} + \left(1 - \frac{\alpha}{t_1}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) beta)) (t_1 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
     (/ (- (* (/ (+ beta 2.0) (pow alpha 2.0)) t_0) (/ t_0 alpha)) 2.0)
     (/ (+ (/ beta t_1) (- 1.0 (/ alpha t_1))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - beta;
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((((beta + 2.0) / pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 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) :: t_1
    real(8) :: tmp
    t_0 = ((-2.0d0) - beta) - beta
    t_1 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999998d0)) then
        tmp = ((((beta + 2.0d0) / (alpha ** 2.0d0)) * t_0) - (t_0 / alpha)) / 2.0d0
    else
        tmp = ((beta / t_1) + (1.0d0 - (alpha / t_1))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - beta;
	double t_1 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((((beta + 2.0) / Math.pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-2.0 - beta) - beta
	t_1 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998:
		tmp = ((((beta + 2.0) / math.pow(alpha, 2.0)) * t_0) - (t_0 / alpha)) / 2.0
	else:
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(-2.0 - beta) - beta)
	t_1 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + 2.0) / (alpha ^ 2.0)) * t_0) - Float64(t_0 / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + Float64(1.0 - Float64(alpha / t_1))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-2.0 - beta) - beta;
	t_1 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998)
		tmp = ((((beta + 2.0) / (alpha ^ 2.0)) * t_0) - (t_0 / alpha)) / 2.0;
	else
		tmp = ((beta / t_1) + (1.0 - (alpha / t_1))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$1), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999998000000000054

   1. Initial program 6.7%

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

   2. Taylor expanded in alpha around -inf 91.8%

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

      1. associate-*r/ 91.8%

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

      2. distribute-lft-in 91.8%

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

      3. neg-mul-1 91.8%

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

      4. distribute-lft-neg-in 91.8%

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

      5. mul-1-neg 91.8%

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

      6. unpow2 91.8%

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

      7. associate-*r* 91.8%

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

      8. distribute-rgt-in 91.8%

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

      9. mul-1-neg 91.8%

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

      10. sub-neg 91.8%

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

      11. mul-1-neg 91.8%

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

   4. Simplified 99.7%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{\beta + 2}{{\alpha}^{2}} \cdot \left(\left(-2 - \beta\right) - \beta\right) - \frac{\left(-2 - \beta\right) - \beta}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
     (/ (+ beta 1.0) alpha)
     (/ (+ (/ 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)) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} 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)) <= (-0.999998d0)) then
        tmp = (beta + 1.0d0) / alpha
    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)) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} 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)) <= -0.999998:
		tmp = (beta + 1.0) / alpha
	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)) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	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)) <= -0.999998)
		tmp = (beta + 1.0) / alpha;
	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], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $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)) < -0.999998000000000054

   1. Initial program 6.7%

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

   2. Taylor expanded in alpha around inf 99.5%

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

      1. associate-*r/ 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 99.5%

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

      1. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. associate-+l- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
   (/ (+ beta 1.0) alpha)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 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 - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999998d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998)
		tmp = (beta + 1.0) / alpha;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999998000000000054

   1. Initial program 6.7%

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

   2. Taylor expanded in alpha around inf 99.5%

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

      1. associate-*r/ 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 99.5%

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

      1. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

   3. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999998) (/ (+ beta 1.0) alpha) (/ (+ 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 <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} 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 <= (-0.999998d0)) then
        tmp = (beta + 1.0d0) / alpha
    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 <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} 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 <= -0.999998:
		tmp = (beta + 1.0) / alpha
	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 <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	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 <= -0.999998)
		tmp = (beta + 1.0) / alpha;
	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, -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $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)) < -0.999998000000000054

   1. Initial program 6.7%

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

   2. Taylor expanded in alpha around inf 99.5%

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

      1. associate-*r/ 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in beta around 0 99.5%

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

      1. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   if -0.999998000000000054 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 5: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 - \frac{1}{\beta + 2} \cdot \left(\alpha - \beta\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5.1e+17)
   (/ (- 1.0 (* (/ 1.0 (+ beta 2.0)) (- alpha beta))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.1e+17) {
		tmp = (1.0 - ((1.0 / (beta + 2.0)) * (alpha - beta))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5.1d+17) then
        tmp = (1.0d0 - ((1.0d0 / (beta + 2.0d0)) * (alpha - beta))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5.1e+17) {
		tmp = (1.0 - ((1.0 / (beta + 2.0)) * (alpha - beta))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5.1e+17:
		tmp = (1.0 - ((1.0 / (beta + 2.0)) * (alpha - beta))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5.1e+17)
		tmp = Float64(Float64(1.0 - Float64(Float64(1.0 / Float64(beta + 2.0)) * Float64(alpha - beta))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5.1e+17)
		tmp = (1.0 - ((1.0 / (beta + 2.0)) * (alpha - beta))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5.1e+17], N[(N[(1.0 - N[(N[(1.0 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha - beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.1e17

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in alpha around 0 98.0%

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

   if 5.1e17 < alpha

   1. Initial program 22.0%

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

   2. Taylor expanded in alpha around inf 84.3%

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

      1. associate-*r/ 84.3%

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

   4. Simplified 84.3%

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

   5. Taylor expanded in beta around 0 84.3%

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

      1. +-commutative 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 - \frac{1}{\beta + 2} \cdot \left(\alpha - \beta\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 6: 69.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 9.8 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha 4.2e-95)
     t_0
     (if (<= alpha 9.8e-64) 1.0 (if (<= alpha 1.1) t_0 (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= 4.2e-95) {
		tmp = t_0;
	} else if (alpha <= 9.8e-64) {
		tmp = 1.0;
	} else if (alpha <= 1.1) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	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 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= 4.2d-95) then
        tmp = t_0
    else if (alpha <= 9.8d-64) then
        tmp = 1.0d0
    else if (alpha <= 1.1d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= 4.2e-95) {
		tmp = t_0;
	} else if (alpha <= 9.8e-64) {
		tmp = 1.0;
	} else if (alpha <= 1.1) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= 4.2e-95:
		tmp = t_0
	elif alpha <= 9.8e-64:
		tmp = 1.0
	elif alpha <= 1.1:
		tmp = t_0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= 4.2e-95)
		tmp = t_0;
	elseif (alpha <= 9.8e-64)
		tmp = 1.0;
	elseif (alpha <= 1.1)
		tmp = t_0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= 4.2e-95)
		tmp = t_0;
	elseif (alpha <= 9.8e-64)
		tmp = 1.0;
	elseif (alpha <= 1.1)
		tmp = t_0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 4.2e-95], t$95$0, If[LessEqual[alpha, 9.8e-64], 1.0, If[LessEqual[alpha, 1.1], t$95$0, N[(1.0 / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 4.2e-95 or 9.8000000000000003e-64 < alpha < 1.1000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 74.8%

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

      1. +-commutative 74.8%

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

   4. Simplified 74.8%

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

   5. Taylor expanded in alpha around 0 73.3%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   6. Step-by-step derivation

      1. *-commutative 73.3%

         \[\leadsto 0.5 + \color{blue}{\alpha \cdot -0.25} \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{0.5 + \alpha \cdot -0.25} \]

   if 4.2e-95 < alpha < 9.8000000000000003e-64

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. div-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 100.0%

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

   9. Taylor expanded in beta around inf 75.0%

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

   if 1.1000000000000001 < alpha

   1. Initial program 24.0%

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

   2. Taylor expanded in beta around 0 5.7%

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

      1. +-commutative 5.7%

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

   4. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 68.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 9.8 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.1:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 7: 74.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.4 \cdot 10^{-63}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha 3.6e-96)
     t_0
     (if (<= alpha 3.4e-63)
       1.0
       (if (<= alpha 2.0) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= 3.6e-96) {
		tmp = t_0;
	} else if (alpha <= 3.4e-63) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	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 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= 3.6d-96) then
        tmp = t_0
    else if (alpha <= 3.4d-63) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= 3.6e-96) {
		tmp = t_0;
	} else if (alpha <= 3.4e-63) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= 3.6e-96:
		tmp = t_0
	elif alpha <= 3.4e-63:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= 3.6e-96)
		tmp = t_0;
	elseif (alpha <= 3.4e-63)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= 3.6e-96)
		tmp = t_0;
	elseif (alpha <= 3.4e-63)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 3.6e-96], t$95$0, If[LessEqual[alpha, 3.4e-63], 1.0, If[LessEqual[alpha, 2.0], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 3.60000000000000008e-96 or 3.39999999999999998e-63 < alpha < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 74.8%

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

      1. +-commutative 74.8%

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

   4. Simplified 74.8%

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

   5. Taylor expanded in alpha around 0 73.3%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   6. Step-by-step derivation

      1. *-commutative 73.3%

         \[\leadsto 0.5 + \color{blue}{\alpha \cdot -0.25} \]

   7. Simplified 73.3%

      \[\leadsto \color{blue}{0.5 + \alpha \cdot -0.25} \]

   if 3.60000000000000008e-96 < alpha < 3.39999999999999998e-63

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. div-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 100.0%

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

   9. Taylor expanded in beta around inf 75.0%

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

   if 2 < alpha

   1. Initial program 24.0%

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

   2. Taylor expanded in alpha around inf 82.5%

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

      1. associate-*r/ 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in beta around 0 82.6%

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

      1. +-commutative 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{-96}:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{elif}\;\alpha \leq 3.4 \cdot 10^{-63}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 8: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.6e+24)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.6e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.6d+24) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.6e+24) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.6e+24:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.6e+24)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.6e+24)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3.6e+24], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.59999999999999983e24

   1. Initial program 99.1%

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

   2. Taylor expanded in alpha around 0 96.7%

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

   if 3.59999999999999983e24 < alpha

   1. Initial program 21.3%

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

   2. Taylor expanded in alpha around inf 85.0%

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

      1. associate-*r/ 85.1%

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

   4. Simplified 85.1%

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

   5. Taylor expanded in beta around 0 85.1%

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

      1. +-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 9: 68.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 3.6e-95)
   0.5
   (if (<= alpha 5.5e-64) 1.0 (if (<= alpha 2.0) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.6e-95) {
		tmp = 0.5;
	} else if (alpha <= 5.5e-64) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 3.6d-95) then
        tmp = 0.5d0
    else if (alpha <= 5.5d-64) then
        tmp = 1.0d0
    else if (alpha <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 3.6e-95) {
		tmp = 0.5;
	} else if (alpha <= 5.5e-64) {
		tmp = 1.0;
	} else if (alpha <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 3.6e-95:
		tmp = 0.5
	elif alpha <= 5.5e-64:
		tmp = 1.0
	elif alpha <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 3.6e-95)
		tmp = 0.5;
	elseif (alpha <= 5.5e-64)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 3.6e-95)
		tmp = 0.5;
	elseif (alpha <= 5.5e-64)
		tmp = 1.0;
	elseif (alpha <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 3.6e-95], 0.5, If[LessEqual[alpha, 5.5e-64], 1.0, If[LessEqual[alpha, 2.0], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 3.6e-95 or 5.4999999999999999e-64 < alpha < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 74.8%

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

      1. +-commutative 74.8%

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

   4. Simplified 74.8%

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

   5. Taylor expanded in alpha around 0 72.3%

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

   if 3.6e-95 < alpha < 5.4999999999999999e-64

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. div-inv 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 100.0%

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

   9. Taylor expanded in beta around inf 75.0%

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

   if 2 < alpha

   1. Initial program 24.0%

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

   2. Taylor expanded in beta around 0 5.7%

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

      1. +-commutative 5.7%

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

   4. Simplified 5.7%

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

   5. Taylor expanded in alpha around inf 68.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.6 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 5.5 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 10: 70.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.05) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.05) {
		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 <= 2.05d0) 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 <= 2.05) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.05:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.05)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.05], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0499999999999998

   1. Initial program 66.7%

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

   2. Taylor expanded in beta around 0 66.3%

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

      1. +-commutative 66.3%

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

   4. Simplified 66.3%

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

   5. Taylor expanded in alpha around 0 63.6%

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

   if 2.0499999999999998 < beta

   1. Initial program 86.5%

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

      1. clear-num 86.6%

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

      2. associate-/r/ 86.6%

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

      3. +-commutative 86.6%

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

      4. associate-+l+ 86.6%

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

   3. Applied egg-rr 86.6%

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

      1. clear-num 86.5%

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

      2. inv-pow 86.5%

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

      3. +-commutative 86.5%

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

      4. *-commutative 86.5%

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

      5. div-inv 86.5%

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

      6. +-commutative 86.5%

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

      7. associate-+l+ 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. unpow-1 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in alpha around 0 83.8%

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

   9. Taylor expanded in beta around inf 80.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.05:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 49.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

double code(double alpha, double beta) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.5;
}

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in beta around 0 48.8%

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

   1. +-commutative 48.8%

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

4. Simplified 48.8%

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

5. Taylor expanded in alpha around 0 48.1%

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

6. Final simplification 48.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023311 
(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))