Octave 3.8, jcobi/1

Percentage Accurate: 74.1% → 99.7%

Time: 18.9s

Alternatives: 13

Speedup: 1.1×

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: 74.1% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_0}^{2}, t\_0, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- beta alpha) (+ beta (+ alpha 2.0))))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/
      (/
       (-
        (+ 2.0 (* beta (- (- 2.0 (/ beta alpha)) (/ 2.0 alpha))))
        (/ 4.0 alpha))
       alpha)
      2.0)
     (/ (fma (pow t_0 2.0) t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = cbrt(((beta - alpha) / (beta + (alpha + 2.0))));
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = fma(pow(t_0, 2.0), t_0, 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = cbrt(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * Float64(Float64(2.0 - Float64(beta / alpha)) - Float64(2.0 / alpha)))) - Float64(4.0 / alpha)) / alpha) / 2.0);
	else
		tmp = Float64(fma((t_0 ^ 2.0), t_0, 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[Power[N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(N[(N[(2.0 + N[(beta * N[(N[(2.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 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) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

   6. Taylor expanded in beta around 0 92.7%

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

   7. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}} + 1}{2} \]

      2. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}}{2} \]

      3. pow2 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}\right)}^{2}}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}{2} \]

      4. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}\right)}^{2}, \sqrt[3]{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}}, 1\right)}{2} \]

      5. associate-+l+ 99.9%

         \[\leadsto \frac{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}\right)}^{2}, \sqrt[3]{\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}}, 1\right)}{2} \]

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}\right)}^{2}, \sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\alpha}{t\_0}\\ t_2 := 1 + \frac{\beta}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{t\_2}^{3} - {t\_1}^{3}}{{t\_2}^{2} + \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0)))
        (t_1 (/ alpha t_0))
        (t_2 (+ 1.0 (/ beta t_0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/
      (/
       (-
        (+ 2.0 (* beta (- (- 2.0 (/ beta alpha)) (/ 2.0 alpha))))
        (/ 4.0 alpha))
       alpha)
      2.0)
     (/
      (/
       (- (pow t_2 3.0) (pow t_1 3.0))
       (+ (pow t_2 2.0) (+ (* t_1 t_1) (* t_2 t_1))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = alpha / t_0;
	double t_2 = 1.0 + (beta / t_0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = ((pow(t_2, 3.0) - pow(t_1, 3.0)) / (pow(t_2, 2.0) + ((t_1 * t_1) + (t_2 * 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) :: t_2
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = alpha / t_0
    t_2 = 1.0d0 + (beta / t_0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999d0)) then
        tmp = (((2.0d0 + (beta * ((2.0d0 - (beta / alpha)) - (2.0d0 / alpha)))) - (4.0d0 / alpha)) / alpha) / 2.0d0
    else
        tmp = (((t_2 ** 3.0d0) - (t_1 ** 3.0d0)) / ((t_2 ** 2.0d0) + ((t_1 * t_1) + (t_2 * t_1)))) / 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 t_1 = alpha / t_0;
	double t_2 = 1.0 + (beta / t_0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = ((Math.pow(t_2, 3.0) - Math.pow(t_1, 3.0)) / (Math.pow(t_2, 2.0) + ((t_1 * t_1) + (t_2 * t_1)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = alpha / t_0
	t_2 = 1.0 + (beta / t_0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999:
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0
	else:
		tmp = ((math.pow(t_2, 3.0) - math.pow(t_1, 3.0)) / (math.pow(t_2, 2.0) + ((t_1 * t_1) + (t_2 * t_1)))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(alpha / t_0)
	t_2 = Float64(1.0 + Float64(beta / t_0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * Float64(Float64(2.0 - Float64(beta / alpha)) - Float64(2.0 / alpha)))) - Float64(4.0 / alpha)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64((t_2 ^ 3.0) - (t_1 ^ 3.0)) / Float64((t_2 ^ 2.0) + Float64(Float64(t_1 * t_1) + Float64(t_2 * t_1)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = alpha / t_0;
	t_2 = 1.0 + (beta / t_0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999)
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	else
		tmp = (((t_2 ^ 3.0) - (t_1 ^ 3.0)) / ((t_2 ^ 2.0) + ((t_1 * t_1) + (t_2 * t_1)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(N[(N[(2.0 + N[(beta * N[(N[(2.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

   6. Taylor expanded in beta around 0 92.7%

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

   7. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. sub-neg 99.9%

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

      2. flip3-+ 99.9%

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

      3. pow2 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. sqr-pow 99.3%

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

      2. fma-define 99.3%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      3. cube-neg 99.3%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, \color{blue}{-{\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      4. add-sqr-sqrt 56.4%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, -{\color{blue}{\left(\sqrt{\frac{\alpha}{\beta + \left(\alpha + 2\right)}} \cdot \sqrt{\frac{\alpha}{\beta + \left(\alpha + 2\right)}}\right)}}^{3}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      5. sqrt-prod 99.3%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, -{\color{blue}{\left(\sqrt{\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}}\right)}}^{3}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      6. sqr-neg 99.3%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, -{\left(\sqrt{\color{blue}{\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}}\right)}^{3}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      7. sqrt-prod 53.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, -{\color{blue}{\left(\sqrt{-\frac{\alpha}{\beta + \left(\alpha + 2\right)}} \cdot \sqrt{-\frac{\alpha}{\beta + \left(\alpha + 2\right)}}\right)}}^{3}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      8. add-sqr-sqrt 98.9%

         \[\leadsto \frac{\frac{\mathsf{fma}\left({\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, {\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{\left(\frac{3}{2}\right)}, -{\color{blue}{\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}}^{3}\right)}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right) - \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(-\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)\right)}}{2} \]

      9. fma-neg 98.9%

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

   10. Applied egg-rr 99.9%

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

       1. +-commutative 99.9%

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

       2. +-commutative 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{3} - {\left(\frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}^{3}}{{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right)}^{2} + \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)} + \left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) \cdot \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{-1}{t\_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, 1 + \frac{\beta}{t\_0}, \alpha \cdot t\_1\right) + \mathsf{fma}\left(t\_1, \alpha, \alpha \cdot \frac{1}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ -1.0 t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/
      (/
       (-
        (+ 2.0 (* beta (- (- 2.0 (/ beta alpha)) (/ 2.0 alpha))))
        (/ 4.0 alpha))
       alpha)
      2.0)
     (/
      (+
       (fma 1.0 (+ 1.0 (/ beta t_0)) (* alpha t_1))
       (fma t_1 alpha (* alpha (/ 1.0 t_0))))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = -1.0 / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = (fma(1.0, (1.0 + (beta / t_0)), (alpha * t_1)) + fma(t_1, alpha, (alpha * (1.0 / t_0)))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(-1.0 / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * Float64(Float64(2.0 - Float64(beta / alpha)) - Float64(2.0 / alpha)))) - Float64(4.0 / alpha)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(fma(1.0, Float64(1.0 + Float64(beta / t_0)), Float64(alpha * t_1)) + fma(t_1, alpha, Float64(alpha * Float64(1.0 / t_0)))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999], N[(N[(N[(N[(2.0 + N[(beta * N[(N[(2.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 * N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(alpha * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * alpha + N[(alpha * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

   6. Taylor expanded in beta around 0 92.7%

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

   7. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. div-inv 99.9%

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

      3. prod-diff 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\left(2 + \beta \cdot \left(\left(2 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)\right) - \frac{4}{\alpha}}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/
      (/
       (-
        (+ 2.0 (* beta (- (- 2.0 (/ beta alpha)) (/ 2.0 alpha))))
        (/ 4.0 alpha))
       alpha)
      2.0)
     (/ (- (+ 1.0 (/ beta t_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)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= (-0.999d0)) then
        tmp = (((2.0d0 + (beta * ((2.0d0 - (beta / alpha)) - (2.0d0 / alpha)))) - (4.0d0 / alpha)) / alpha) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (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)) <= -0.999) {
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= -0.999:
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0
	else:
		tmp = ((1.0 + (beta / t_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)) <= -0.999)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(beta * Float64(Float64(2.0 - Float64(beta / alpha)) - Float64(2.0 / alpha)))) - Float64(4.0 / alpha)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_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)) <= -0.999)
		tmp = (((2.0 + (beta * ((2.0 - (beta / alpha)) - (2.0 / alpha)))) - (4.0 / alpha)) / alpha) / 2.0;
	else
		tmp = ((1.0 + (beta / t_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], -0.999], N[(N[(N[(N[(2.0 + N[(beta * N[(N[(2.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 92.9%

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

   6. Taylor expanded in beta around 0 92.7%

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

   7. Taylor expanded in beta around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{t\_0}\right) - \frac{\alpha}{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (- (+ 1.0 (/ beta t_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)) <= -0.999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= (-0.999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((1.0d0 + (beta / t_0)) - (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)) <= -0.999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((1.0 + (beta / t_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)) <= -0.999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = ((1.0 + (beta / t_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)) <= -0.999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(beta / t_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)) <= -0.999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = ((1.0 + (beta / t_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], -0.999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. neg-mul-1 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. neg-mul-1 98.8%

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

      8. remove-double-neg 98.8%

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

   7. Simplified 98.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. div-sub 99.9%

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

      3. associate-+r- 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.999d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.999) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.999:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998999999999999999

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in alpha around -inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. neg-mul-1 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. neg-mul-1 98.8%

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

      8. remove-double-neg 98.8%

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

   7. Simplified 98.8%

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

   if -0.998999999999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

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

Alternative 7: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\beta \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -2.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= beta -2.8e-67)
     t_0
     (if (<= beta -2.7e-108)
       (/ (/ 2.0 alpha) 2.0)
       (if (<= beta 2.0) t_0 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -2.8e-67) {
		tmp = t_0;
	} else if (beta <= -2.7e-108) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (beta <= (-2.8d-67)) then
        tmp = t_0
    else if (beta <= (-2.7d-108)) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (beta <= -2.8e-67) {
		tmp = t_0;
	} else if (beta <= -2.7e-108) {
		tmp = (2.0 / alpha) / 2.0;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if beta <= -2.8e-67:
		tmp = t_0
	elif beta <= -2.7e-108:
		tmp = (2.0 / alpha) / 2.0
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -2.8e-67)
		tmp = t_0;
	elseif (beta <= -2.7e-108)
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -2.8e-67)
		tmp = t_0;
	elseif (beta <= -2.7e-108)
		tmp = (2.0 / alpha) / 2.0;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -2.8e-67], t$95$0, If[LessEqual[beta, -2.7e-108], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -2.8000000000000001e-67 or -2.70000000000000005e-108 < beta < 2

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in alpha around 0 67.2%

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

   6. Taylor expanded in beta around 0 67.2%

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

      1. *-commutative 67.2%

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

   8. Simplified 67.2%

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

   if -2.8000000000000001e-67 < beta < -2.70000000000000005e-108

   1. Initial program 32.1%

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

      1. +-commutative 32.1%

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

   3. Simplified 32.1%

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

   5. Taylor expanded in alpha around -inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. sub-neg 72.3%

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

      3. distribute-lft-in 72.3%

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

      4. neg-mul-1 72.3%

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

      5. mul-1-neg 72.3%

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

      6. remove-double-neg 72.3%

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

      7. neg-mul-1 72.3%

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

      8. remove-double-neg 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in beta around 0 72.3%

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

   if 2 < beta

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in beta around inf 76.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -2.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 145000:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 145000.0)
   (/ (/ 2.0 (+ alpha 2.0)) 2.0)
   (if (<= beta 1.32e+26)
     1.0
     (if (<= beta 2.7e+61) (/ (/ (* beta 2.0) alpha) 2.0) 1.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 145000.0) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else if (beta <= 1.32e+26) {
		tmp = 1.0;
	} else if (beta <= 2.7e+61) {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 145000.0d0) then
        tmp = (2.0d0 / (alpha + 2.0d0)) / 2.0d0
    else if (beta <= 1.32d+26) then
        tmp = 1.0d0
    else if (beta <= 2.7d+61) then
        tmp = ((beta * 2.0d0) / alpha) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 145000.0) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else if (beta <= 1.32e+26) {
		tmp = 1.0;
	} else if (beta <= 2.7e+61) {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 145000.0:
		tmp = (2.0 / (alpha + 2.0)) / 2.0
	elif beta <= 1.32e+26:
		tmp = 1.0
	elif beta <= 2.7e+61:
		tmp = ((beta * 2.0) / alpha) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 145000.0)
		tmp = Float64(Float64(2.0 / Float64(alpha + 2.0)) / 2.0);
	elseif (beta <= 1.32e+26)
		tmp = 1.0;
	elseif (beta <= 2.7e+61)
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 145000.0)
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	elseif (beta <= 1.32e+26)
		tmp = 1.0;
	elseif (beta <= 2.7e+61)
		tmp = ((beta * 2.0) / alpha) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 145000.0], N[(N[(2.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 1.32e+26], 1.0, If[LessEqual[beta, 2.7e+61], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if beta < 145000

   1. Initial program 65.5%

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

      1. +-commutative 65.5%

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

   3. Simplified 65.5%

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

      1. +-commutative 65.5%

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

      2. div-sub 65.5%

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

      3. associate-+r- 65.5%

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

      4. associate-+l+ 65.5%

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

      5. associate-+l+ 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. flip3-+ 65.5%

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

      2. clear-num 65.5%

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

      3. frac-sub 65.5%

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

   8. Applied egg-rr 65.5%

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

      1. associate-/r* 65.5%

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

   10. Simplified 65.5%

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

   11. Taylor expanded in beta around 0 98.4%

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

       1. +-commutative 98.4%

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

   13. Simplified 98.4%

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

   if 145000 < beta < 1.32e26 or 2.7000000000000002e61 < beta

   1. Initial program 85.6%

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

      1. +-commutative 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in beta around inf 85.4%

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

   if 1.32e26 < beta < 2.7000000000000002e61

   1. Initial program 35.4%

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

      1. +-commutative 35.4%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in alpha around -inf 70.1%

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

      1. associate-*r/ 70.1%

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

      2. sub-neg 70.1%

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

      3. distribute-lft-in 70.1%

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

      4. neg-mul-1 70.1%

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

      5. mul-1-neg 70.1%

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

      6. remove-double-neg 70.1%

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

      7. neg-mul-1 70.1%

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

      8. remove-double-neg 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in beta around inf 70.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 145000:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e-11)
   (/ (/ 2.0 (+ alpha 2.0)) 2.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e-11) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 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 <= 3d-11) then
        tmp = (2.0d0 / (alpha + 2.0d0)) / 2.0d0
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e-11) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 3e-11:
		tmp = (2.0 / (alpha + 2.0)) / 2.0
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e-11)
		tmp = Float64(Float64(2.0 / Float64(alpha + 2.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e-11)
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 3e-11], N[(N[(2.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3e-11

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

   3. Simplified 65.7%

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

      1. +-commutative 65.7%

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

      2. div-sub 65.7%

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

      3. associate-+r- 65.7%

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

      4. associate-+l+ 65.7%

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

      5. associate-+l+ 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. flip3-+ 65.7%

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

      2. clear-num 65.7%

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

      3. frac-sub 65.7%

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

   8. Applied egg-rr 65.7%

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

      1. associate-/r* 65.7%

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

   10. Simplified 65.7%

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

   11. Taylor expanded in beta around 0 99.1%

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

       1. +-commutative 99.1%

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

   13. Simplified 99.1%

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

   if 3e-11 < beta

   1. Initial program 77.7%

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

      1. +-commutative 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in alpha around 0 76.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.4e+37)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.4e+37) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (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 <= 7.4d+37) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (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 <= 7.4e+37) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7.4e+37:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7.4e+37)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7.4e+37)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7.4e+37], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.3999999999999999e37

   1. Initial program 98.5%

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

      1. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in alpha around 0 97.4%

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

   if 7.3999999999999999e37 < alpha

   1. Initial program 15.4%

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

      1. +-commutative 15.4%

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

   3. Simplified 15.4%

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

   5. Taylor expanded in alpha around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. sub-neg 90.6%

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

      3. distribute-lft-in 90.6%

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

      4. neg-mul-1 90.6%

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

      5. mul-1-neg 90.6%

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

      6. remove-double-neg 90.6%

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

      7. neg-mul-1 90.6%

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

      8. remove-double-neg 90.6%

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

   7. Simplified 90.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 145000:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 145000.0) (/ (/ 2.0 (+ alpha 2.0)) 2.0) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 145000.0) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 145000.0d0) then
        tmp = (2.0d0 / (alpha + 2.0d0)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 145000.0) {
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 145000.0:
		tmp = (2.0 / (alpha + 2.0)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 145000.0)
		tmp = Float64(Float64(2.0 / Float64(alpha + 2.0)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 145000.0)
		tmp = (2.0 / (alpha + 2.0)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 145000.0], N[(N[(2.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 145000

   1. Initial program 65.5%

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

      1. +-commutative 65.5%

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

   3. Simplified 65.5%

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

      1. +-commutative 65.5%

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

      2. div-sub 65.5%

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

      3. associate-+r- 65.5%

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

      4. associate-+l+ 65.5%

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

      5. associate-+l+ 65.5%

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

   6. Applied egg-rr 65.5%

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

      1. flip3-+ 65.5%

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

      2. clear-num 65.5%

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

      3. frac-sub 65.5%

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

   8. Applied egg-rr 65.5%

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

      1. associate-/r* 65.5%

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

   10. Simplified 65.5%

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

   11. Taylor expanded in beta around 0 98.4%

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

       1. +-commutative 98.4%

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

   13. Simplified 98.4%

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

   if 145000 < beta

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in beta around inf 77.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 145000:\\ \;\;\;\;\frac{\frac{2}{\alpha + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 5500000.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5500000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 5500000.0d0) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 5500000.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 5500000.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 5500000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 5500000.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 5500000.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.5e6

   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. Add Preprocessing

   5. Taylor expanded in beta around inf 44.6%

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

   if 5.5e6 < alpha

   1. Initial program 19.0%

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

      1. +-commutative 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in alpha around -inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. sub-neg 87.5%

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

      3. distribute-lft-in 87.5%

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

      4. neg-mul-1 87.5%

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

      5. mul-1-neg 87.5%

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

      6. remove-double-neg 87.5%

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

      7. neg-mul-1 87.5%

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

      8. remove-double-neg 87.5%

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

   7. Simplified 87.5%

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

   8. Taylor expanded in beta around 0 67.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.3% 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 69.6%

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

   1. +-commutative 69.6%

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

3. Simplified 69.6%

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

5. Taylor expanded in beta around inf 34.0%

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

6. Final simplification 34.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))