Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.8%

Time: 11.7s

Alternatives: 14

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t_0}^{2}, t_0, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (cbrt (/ (- beta alpha) (+ beta (+ alpha 2.0))))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999)
     (/
      (+
       (* (/ (/ (+ beta 2.0) alpha) alpha) (- (- -2.0 beta) beta))
       (/ (- 2.0 (* beta -2.0)) 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)) <= -0.9999) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((2.0 - (beta * -2.0)) / 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)) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + 2.0) / alpha) / alpha) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(2.0 - Float64(beta * -2.0)) / 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], -0.9999], N[(N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $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) 2)) < -0.99990000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in alpha around -inf 94.9%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in beta around 0 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. unpow2 99.2%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. fma-def 99.9%

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

      3. pow2 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

   3. 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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}\right)}^{2}, \sqrt[3]{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}}, 1\right)}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t_0} + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999)
     (/
      (+
       (* (/ (/ (+ beta 2.0) alpha) alpha) (- (- -2.0 beta) beta))
       (/ (- 2.0 (* beta -2.0)) alpha))
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((2.0 - (beta * -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999d0)) then
        tmp = (((((beta + 2.0d0) / alpha) / alpha) * (((-2.0d0) - beta) - beta)) + ((2.0d0 - (beta * (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((2.0 - (beta * -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999:
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((2.0 - (beta * -2.0)) / alpha)) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + 2.0) / alpha) / alpha) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(2.0 - Float64(beta * -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999)
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((2.0 - (beta * -2.0)) / alpha)) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99990000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in alpha around -inf 94.9%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in beta around 0 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. unpow2 99.2%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t_1\right) - \frac{\alpha}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (- (+ 1.0 t_1) (/ alpha t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + t_1) - (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) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = ((1.0d0 + t_1) - (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 t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = ((1.0 + t_1) - (alpha / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = ((1.0 + t_1) - (alpha / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + t_1) - Float64(alpha / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = ((1.0 + t_1) - (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]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + t$95$1), $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) 2)) < -0.999999900000000053

   1. Initial program 6.8%

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

      1. div-sub 6.8%

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

      2. associate-+l- 10.4%

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

      3. +-commutative 10.4%

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

      4. associate-+l+ 10.4%

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

      5. +-commutative 10.4%

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

      6. associate-+l+ 10.4%

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

   3. Applied egg-rr 10.4%

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

   4. Taylor expanded in alpha around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. neg-mul-1 99.4%

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

      5. sub-neg 99.4%

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

   6. Simplified 99.4%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. div-sub 99.5%

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

      3. associate-+r- 99.5%

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

      4. +-commutative 99.5%

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

      5. associate-+l+ 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-+l+ 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}\right) - \frac{\alpha}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\beta}{t_0}\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{t_1 + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(1 - \frac{\alpha}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))) (t_1 (/ beta t_0)))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9999999)
     (/ (+ t_1 (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_1 (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    t_1 = beta / t_0
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.9999999d0)) then
        tmp = (t_1 + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_1 + (1.0d0 - (alpha / t_0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double t_1 = beta / t_0;
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999) {
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	t_1 = beta / t_0
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999:
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	t_1 = Float64(beta / t_0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9999999)
		tmp = Float64(Float64(t_1 + Float64(Float64(beta - -2.0) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(1.0 - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = beta + (alpha + 2.0);
	t_1 = beta / t_0;
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9999999)
		tmp = (t_1 + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_1 + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999999], N[(N[(t$95$1 + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999900000000053

   1. Initial program 6.8%

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

      1. div-sub 6.8%

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

      2. associate-+l- 10.4%

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

      3. +-commutative 10.4%

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

      4. associate-+l+ 10.4%

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

      5. +-commutative 10.4%

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

      6. associate-+l+ 10.4%

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

   3. Applied egg-rr 10.4%

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

   4. Taylor expanded in alpha around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. neg-mul-1 99.4%

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

      5. sub-neg 99.4%

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

   6. Simplified 99.4%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. associate-+l- 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l+ 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999999)
     (/ (+ (/ beta (+ beta (+ alpha 2.0))) (/ (- beta -2.0) alpha)) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999999) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.9999999d0)) then
        tmp = ((beta / (beta + (alpha + 2.0d0))) + ((beta - (-2.0d0)) / alpha)) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.9999999) {
		tmp = ((beta / (beta + (alpha + 2.0))) + ((beta - -2.0) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.9999999:
		tmp = ((beta / (beta + (alpha + 2.0))) + ((beta - -2.0) / alpha)) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.9999999)
		tmp = Float64(Float64(Float64(beta / Float64(beta + Float64(alpha + 2.0))) + Float64(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 <= -0.9999999)
		tmp = ((beta / (beta + (alpha + 2.0))) + ((beta - -2.0) / alpha)) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.9999999], N[(N[(N[(beta / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta - -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999900000000053

   1. Initial program 6.8%

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

      1. div-sub 6.8%

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

      2. associate-+l- 10.4%

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

      3. +-commutative 10.4%

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

      4. associate-+l+ 10.4%

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

      5. +-commutative 10.4%

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

      6. associate-+l+ 10.4%

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

   3. Applied egg-rr 10.4%

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

   4. Taylor expanded in alpha around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. metadata-eval 99.4%

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

      4. neg-mul-1 99.4%

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

      5. sub-neg 99.4%

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

   6. Simplified 99.4%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \frac{\beta - -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 6: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999999)
     (/ (/ (+ 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 <= -0.9999999) {
		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 <= (-0.9999999d0)) 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 <= -0.9999999) {
		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 <= -0.9999999:
		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 <= -0.9999999)
		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 <= -0.9999999)
		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, -0.9999999], 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) 2)) < -0.999999900000000053

   1. Initial program 6.8%

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

   2. Taylor expanded in alpha around -inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. sub-neg 99.3%

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

      3. mul-1-neg 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. neg-mul-1 99.3%

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

      6. mul-1-neg 99.3%

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

      7. remove-double-neg 99.3%

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

      8. neg-mul-1 99.3%

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

      9. mul-1-neg 99.3%

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

      10. remove-double-neg 99.3%

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

   4. Simplified 99.3%

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

   if -0.999999900000000053 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 7: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 8.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.89:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha -4.6e-67)
     1.0
     (if (<= alpha 8.8e-138)
       t_0
       (if (<= alpha 2.8e-119)
         1.0
         (if (<= alpha 0.89)
           t_0
           (if (or (<= alpha 6.2e+275) (not (<= alpha 3.3e+287)))
             (/ (/ 2.0 alpha) 2.0)
             (/ (/ (* beta 2.0) alpha) 2.0))))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 8.8e-138) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 0.89) {
		tmp = t_0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = ((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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= (-4.6d-67)) then
        tmp = 1.0d0
    else if (alpha <= 8.8d-138) then
        tmp = t_0
    else if (alpha <= 2.8d-119) then
        tmp = 1.0d0
    else if (alpha <= 0.89d0) then
        tmp = t_0
    else if ((alpha <= 6.2d+275) .or. (.not. (alpha <= 3.3d+287))) then
        tmp = (2.0d0 / alpha) / 2.0d0
    else
        tmp = ((beta * 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 8.8e-138) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 0.89) {
		tmp = t_0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = (2.0 / alpha) / 2.0;
	} else {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= -4.6e-67:
		tmp = 1.0
	elif alpha <= 8.8e-138:
		tmp = t_0
	elif alpha <= 2.8e-119:
		tmp = 1.0
	elif alpha <= 0.89:
		tmp = t_0
	elif (alpha <= 6.2e+275) or not (alpha <= 3.3e+287):
		tmp = (2.0 / alpha) / 2.0
	else:
		tmp = ((beta * 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 8.8e-138)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 0.89)
		tmp = t_0;
	elseif ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287))
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 8.8e-138)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 0.89)
		tmp = t_0;
	elseif ((alpha <= 6.2e+275) || ~((alpha <= 3.3e+287)))
		tmp = (2.0 / alpha) / 2.0;
	else
		tmp = ((beta * 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -4.6e-67], 1.0, If[LessEqual[alpha, 8.8e-138], t$95$0, If[LessEqual[alpha, 2.8e-119], 1.0, If[LessEqual[alpha, 0.89], t$95$0, If[Or[LessEqual[alpha, 6.2e+275], N[Not[LessEqual[alpha, 3.3e+287]], $MachinePrecision]], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < -4.6000000000000001e-67 or 8.7999999999999995e-138 < alpha < 2.8e-119

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around inf 77.3%

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

   if -4.6000000000000001e-67 < alpha < 8.7999999999999995e-138 or 2.8e-119 < alpha < 0.890000000000000013

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 81.2%

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

      1. +-commutative 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in alpha around 0 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if 0.890000000000000013 < alpha < 6.1999999999999998e275 or 3.3000000000000002e287 < alpha

   1. Initial program 19.5%

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

   2. Taylor expanded in beta around 0 7.7%

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

      1. +-commutative 7.7%

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

   4. Simplified 7.7%

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

   5. Taylor expanded in alpha around inf 70.7%

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

   if 6.1999999999999998e275 < alpha < 3.3000000000000002e287

   1. Initial program 4.5%

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

   2. Taylor expanded in alpha around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. mul-1-neg 100.0%

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

      10. remove-double-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 8.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.89:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -1 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha -1e-67)
     1.0
     (if (<= alpha 5e-143)
       t_0
       (if (<= alpha 2.8e-119)
         1.0
         (if (<= alpha 1.95)
           t_0
           (if (or (<= alpha 6.2e+275) (not (<= alpha 3.3e+287)))
             (/ (/ (+ beta 2.0) alpha) 2.0)
             (/ (/ (* beta 2.0) alpha) 2.0))))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -1e-67) {
		tmp = 1.0;
	} else if (alpha <= 5e-143) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = ((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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= (-1d-67)) then
        tmp = 1.0d0
    else if (alpha <= 5d-143) then
        tmp = t_0
    else if (alpha <= 2.8d-119) then
        tmp = 1.0d0
    else if (alpha <= 1.95d0) then
        tmp = t_0
    else if ((alpha <= 6.2d+275) .or. (.not. (alpha <= 3.3d+287))) then
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    else
        tmp = ((beta * 2.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -1e-67) {
		tmp = 1.0;
	} else if (alpha <= 5e-143) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= -1e-67:
		tmp = 1.0
	elif alpha <= 5e-143:
		tmp = t_0
	elif alpha <= 2.8e-119:
		tmp = 1.0
	elif alpha <= 1.95:
		tmp = t_0
	elif (alpha <= 6.2e+275) or not (alpha <= 3.3e+287):
		tmp = ((beta + 2.0) / alpha) / 2.0
	else:
		tmp = ((beta * 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -1e-67)
		tmp = 1.0;
	elseif (alpha <= 5e-143)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	elseif ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287))
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -1e-67)
		tmp = 1.0;
	elseif (alpha <= 5e-143)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	elseif ((alpha <= 6.2e+275) || ~((alpha <= 3.3e+287)))
		tmp = ((beta + 2.0) / alpha) / 2.0;
	else
		tmp = ((beta * 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -1e-67], 1.0, If[LessEqual[alpha, 5e-143], t$95$0, If[LessEqual[alpha, 2.8e-119], 1.0, If[LessEqual[alpha, 1.95], t$95$0, If[Or[LessEqual[alpha, 6.2e+275], N[Not[LessEqual[alpha, 3.3e+287]], $MachinePrecision]], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if alpha < -9.99999999999999943e-68 or 5.0000000000000002e-143 < alpha < 2.8e-119

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around inf 77.3%

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

   if -9.99999999999999943e-68 < alpha < 5.0000000000000002e-143 or 2.8e-119 < alpha < 1.94999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 81.2%

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

      1. +-commutative 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in alpha around 0 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if 1.94999999999999996 < alpha < 6.1999999999999998e275 or 3.3000000000000002e287 < alpha

   1. Initial program 19.5%

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

      1. div-sub 19.5%

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

      2. associate-+l- 22.0%

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

      3. +-commutative 22.0%

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

      4. associate-+l+ 22.0%

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

      5. +-commutative 22.0%

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

      6. associate-+l+ 22.0%

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

   3. Applied egg-rr 22.0%

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

   4. Taylor expanded in alpha around inf 87.3%

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

      1. associate-*r/ 87.3%

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

      2. distribute-lft-in 87.3%

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

      3. metadata-eval 87.3%

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

      4. neg-mul-1 87.3%

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

      5. sub-neg 87.3%

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

   6. Simplified 87.3%

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

   7. Taylor expanded in alpha around 0 72.8%

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

      1. +-commutative 72.8%

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

   9. Simplified 72.8%

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

   if 6.1999999999999998e275 < alpha < 3.3000000000000002e287

   1. Initial program 4.5%

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

   2. Taylor expanded in alpha around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. mul-1-neg 100.0%

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

      10. remove-double-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{-143}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.39:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)))
   (if (<= alpha -4.6e-67)
     1.0
     (if (<= alpha 1.02e-137)
       t_0
       (if (<= alpha 2.8e-119)
         1.0
         (if (<= alpha 0.39) t_0 (/ (/ 2.0 alpha) 2.0)))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 1.02e-137) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 0.39) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    if (alpha <= (-4.6d-67)) then
        tmp = 1.0d0
    else if (alpha <= 1.02d-137) then
        tmp = t_0
    else if (alpha <= 2.8d-119) then
        tmp = 1.0d0
    else if (alpha <= 0.39d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    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 (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 1.02e-137) {
		tmp = t_0;
	} else if (alpha <= 2.8e-119) {
		tmp = 1.0;
	} else if (alpha <= 0.39) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	tmp = 0
	if alpha <= -4.6e-67:
		tmp = 1.0
	elif alpha <= 1.02e-137:
		tmp = t_0
	elif alpha <= 2.8e-119:
		tmp = 1.0
	elif alpha <= 0.39:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 1.02e-137)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 0.39)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 1.02e-137)
		tmp = t_0;
	elseif (alpha <= 2.8e-119)
		tmp = 1.0;
	elseif (alpha <= 0.39)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	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[alpha, -4.6e-67], 1.0, If[LessEqual[alpha, 1.02e-137], t$95$0, If[LessEqual[alpha, 2.8e-119], 1.0, If[LessEqual[alpha, 0.39], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -4.6000000000000001e-67 or 1.02e-137 < alpha < 2.8e-119

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around inf 77.3%

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

   if -4.6000000000000001e-67 < alpha < 1.02e-137 or 2.8e-119 < alpha < 0.39000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 99.4%

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

   3. Taylor expanded in beta around 0 78.7%

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

   if 0.39000000000000001 < alpha

   1. Initial program 18.7%

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

   2. Taylor expanded in beta around 0 7.6%

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

      1. +-commutative 7.6%

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

   4. Simplified 7.6%

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

   5. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{-119}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.39:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.35:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha -4.6e-67)
     1.0
     (if (<= alpha 1.02e-137)
       t_0
       (if (<= alpha 2.7e-118)
         1.0
         (if (<= alpha 1.35) t_0 (/ (/ 2.0 alpha) 2.0)))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 1.02e-137) {
		tmp = t_0;
	} else if (alpha <= 2.7e-118) {
		tmp = 1.0;
	} else if (alpha <= 1.35) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= (-4.6d-67)) then
        tmp = 1.0d0
    else if (alpha <= 1.02d-137) then
        tmp = t_0
    else if (alpha <= 2.7d-118) then
        tmp = 1.0d0
    else if (alpha <= 1.35d0) then
        tmp = t_0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -4.6e-67) {
		tmp = 1.0;
	} else if (alpha <= 1.02e-137) {
		tmp = t_0;
	} else if (alpha <= 2.7e-118) {
		tmp = 1.0;
	} else if (alpha <= 1.35) {
		tmp = t_0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= -4.6e-67:
		tmp = 1.0
	elif alpha <= 1.02e-137:
		tmp = t_0
	elif alpha <= 2.7e-118:
		tmp = 1.0
	elif alpha <= 1.35:
		tmp = t_0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 1.02e-137)
		tmp = t_0;
	elseif (alpha <= 2.7e-118)
		tmp = 1.0;
	elseif (alpha <= 1.35)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -4.6e-67)
		tmp = 1.0;
	elseif (alpha <= 1.02e-137)
		tmp = t_0;
	elseif (alpha <= 2.7e-118)
		tmp = 1.0;
	elseif (alpha <= 1.35)
		tmp = t_0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -4.6e-67], 1.0, If[LessEqual[alpha, 1.02e-137], t$95$0, If[LessEqual[alpha, 2.7e-118], 1.0, If[LessEqual[alpha, 1.35], t$95$0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -4.6000000000000001e-67 or 1.02e-137 < alpha < 2.69999999999999994e-118

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around inf 77.3%

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

   if -4.6000000000000001e-67 < alpha < 1.02e-137 or 2.69999999999999994e-118 < alpha < 1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around 0 81.2%

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

      1. +-commutative 81.2%

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

   4. Simplified 81.2%

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

   5. Taylor expanded in alpha around 0 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

   if 1.3500000000000001 < alpha

   1. Initial program 18.7%

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

   2. Taylor expanded in beta around 0 7.6%

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

      1. +-commutative 7.6%

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

   4. Simplified 7.6%

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

   5. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -4.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 2.7 \cdot 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.35:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 11: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 10200.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (if (or (<= alpha 6.2e+275) (not (<= alpha 3.3e+287)))
     (/ (/ (+ beta 2.0) alpha) 2.0)
     (/ (/ (* beta 2.0) alpha) 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 10200.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = ((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 <= 10200.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else if ((alpha <= 6.2d+275) .or. (.not. (alpha <= 3.3d+287))) then
        tmp = ((beta + 2.0d0) / alpha) / 2.0d0
    else
        tmp = ((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 <= 10200.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else if ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287)) {
		tmp = ((beta + 2.0) / alpha) / 2.0;
	} else {
		tmp = ((beta * 2.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 10200.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	elif (alpha <= 6.2e+275) or not (alpha <= 3.3e+287):
		tmp = ((beta + 2.0) / alpha) / 2.0
	else:
		tmp = ((beta * 2.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 10200.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	elseif ((alpha <= 6.2e+275) || !(alpha <= 3.3e+287))
		tmp = Float64(Float64(Float64(beta + 2.0) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta * 2.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 10200.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	elseif ((alpha <= 6.2e+275) || ~((alpha <= 3.3e+287)))
		tmp = ((beta + 2.0) / alpha) / 2.0;
	else
		tmp = ((beta * 2.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 10200.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[alpha, 6.2e+275], N[Not[LessEqual[alpha, 3.3e+287]], $MachinePrecision]], N[(N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta * 2.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 10200

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 98.3%

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

   if 10200 < alpha < 6.1999999999999998e275 or 3.3000000000000002e287 < alpha

   1. Initial program 19.5%

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

      1. div-sub 19.5%

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

      2. associate-+l- 22.0%

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

      3. +-commutative 22.0%

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

      4. associate-+l+ 22.0%

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

      5. +-commutative 22.0%

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

      6. associate-+l+ 22.0%

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

   3. Applied egg-rr 22.0%

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

   4. Taylor expanded in alpha around inf 87.3%

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

      1. associate-*r/ 87.3%

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

      2. distribute-lft-in 87.3%

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

      3. metadata-eval 87.3%

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

      4. neg-mul-1 87.3%

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

      5. sub-neg 87.3%

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

   6. Simplified 87.3%

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

   7. Taylor expanded in alpha around 0 72.8%

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

      1. +-commutative 72.8%

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

   9. Simplified 72.8%

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

   if 6.1999999999999998e275 < alpha < 3.3000000000000002e287

   1. Initial program 4.5%

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

   2. Taylor expanded in alpha around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. mul-1-neg 100.0%

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

      7. remove-double-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. mul-1-neg 100.0%

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

      10. remove-double-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+275} \lor \neg \left(\alpha \leq 3.3 \cdot 10^{+287}\right):\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 12: 93.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 11200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 11200.0)
   (/ (+ 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 <= 11200.0) {
		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 <= 11200.0d0) 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 <= 11200.0) {
		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 <= 11200.0:
		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 <= 11200.0)
		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 <= 11200.0)
		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, 11200.0], 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 < 11200

   1. Initial program 100.0%

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

   2. Taylor expanded in alpha around 0 98.3%

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

   if 11200 < alpha

   1. Initial program 18.7%

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

   2. Taylor expanded in alpha around -inf 88.0%

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

      1. associate-*r/ 88.0%

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

      2. sub-neg 88.0%

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

      3. mul-1-neg 88.0%

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

      4. distribute-lft-in 88.0%

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

      5. neg-mul-1 88.0%

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

      6. mul-1-neg 88.0%

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

      7. remove-double-neg 88.0%

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

      8. neg-mul-1 88.0%

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

      9. mul-1-neg 88.0%

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

      10. remove-double-neg 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 11200:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 13: 51.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 9200.0) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 9200.0) {
		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 <= 9200.0d0) 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 <= 9200.0) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 9200.0:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 9200.0)
		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 <= 9200.0)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 9200.0], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9200

   1. Initial program 100.0%

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

   2. Taylor expanded in beta around inf 42.5%

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

   if 9200 < alpha

   1. Initial program 18.7%

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

   2. Taylor expanded in beta around 0 7.6%

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

      1. +-commutative 7.6%

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

   4. Simplified 7.6%

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

   5. Taylor expanded in alpha around inf 67.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 14: 37.0% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 69.8%

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

2. Taylor expanded in beta around inf 32.1%

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

3. Final simplification 32.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023318 
(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))