Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.9%

Time: 10.3s

Alternatives: 11

Speedup: 0.9×

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: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
   (/
    (+
     (* (/ (/ (+ beta 2.0) alpha) alpha) (- (- -2.0 beta) beta))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999996:
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + 2.0) / alpha) / alpha) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999996], 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[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 7.1%

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

   3. Taylor expanded in alpha around -inf 93.0%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. unpow2 99.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   if -0.999995999999999996 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ 99.8%

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

      2. add-exp-log 99.8%

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

      3. flip3-+ 99.9%

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

      4. +-commutative 99.9%

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

      5. log1p-udef 99.9%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}}}{2} \]

      6. +-commutative 99.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\beta + \alpha\right)} + 2}\right)}}{2} \]

      7. associate-+l+ 99.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\beta + \left(\alpha + 2\right)}}\right)}}{2} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}}{2} \]
   5. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \color{blue}{\left(2 + \alpha\right)}}\right)}}{2} \]

      2. +-commutative 99.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(2 + \alpha\right) + \beta}}\right)}}{2} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + 2\right)} + \beta}\right)}}{2} \]

   6. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{\alpha}{t_0} + \beta \cdot \frac{-1}{t_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999996)
     (/
      (+
       (* (/ (/ (+ beta 2.0) alpha) alpha) (- (- -2.0 beta) beta))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/ (- 1.0 (+ (/ alpha t_0) (* beta (/ -1.0 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)) <= -0.999996) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 - ((alpha / t_0) + (beta * (-1.0 / 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)) <= (-0.999996d0)) then
        tmp = (((((beta + 2.0d0) / alpha) / alpha) * (((-2.0d0) - beta) - beta)) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha / t_0) + (beta * ((-1.0d0) / 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)) <= -0.999996) {
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = (1.0 - ((alpha / t_0) + (beta * (-1.0 / 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)) <= -0.999996:
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = (1.0 - ((alpha / t_0) + (beta * (-1.0 / 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)) <= -0.999996)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta + 2.0) / alpha) / alpha) * Float64(Float64(-2.0 - beta) - beta)) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha / t_0) + Float64(beta * Float64(-1.0 / 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)) <= -0.999996)
		tmp = (((((beta + 2.0) / alpha) / alpha) * ((-2.0 - beta) - beta)) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = (1.0 - ((alpha / t_0) + (beta * (-1.0 / 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], -0.999996], 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[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha / t$95$0), $MachinePrecision] + N[(beta * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999995999999999996

   1. Initial program 7.1%

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

   3. Taylor expanded in alpha around -inf 93.0%

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

   5. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. unpow2 99.7%

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

      3. times-frac 99.7%

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

      4. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   if -0.999995999999999996 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-lft-in 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. distribute-rgt-neg-out 99.8%

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

      2. neg-sub0 99.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \color{blue}{\left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \alpha\right)}\right) + 1}{2} \]

      3. add-sqr-sqrt 58.4%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(\sqrt{\alpha} \cdot \sqrt{\alpha}\right)}\right)\right) + 1}{2} \]

      4. sqrt-prod 95.9%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\sqrt{\alpha \cdot \alpha}}\right)\right) + 1}{2} \]

      5. sqr-neg 95.9%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \sqrt{\color{blue}{\left(-\alpha\right) \cdot \left(-\alpha\right)}}\right)\right) + 1}{2} \]

      6. sqrt-unprod 40.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(\sqrt{-\alpha} \cdot \sqrt{-\alpha}\right)}\right)\right) + 1}{2} \]

      7. add-sqr-sqrt 96.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(-\alpha\right)}\right)\right) + 1}{2} \]

      8. associate-*l/ 96.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \color{blue}{\frac{1 \cdot \left(-\alpha\right)}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

      9. *-un-lft-identity 96.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{-\alpha}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      10. add-sqr-sqrt 40.8%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{-\alpha} \cdot \sqrt{-\alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      11. sqrt-unprod 96.0%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{\left(-\alpha\right) \cdot \left(-\alpha\right)}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      12. sqr-neg 96.0%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\sqrt{\color{blue}{\alpha \cdot \alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      13. sqrt-prod 58.4%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{\alpha} \cdot \sqrt{\alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      14. add-sqr-sqrt 99.9%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\alpha}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \color{blue}{\left(0 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}\right) + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 99.9%

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

      2. distribute-neg-frac 99.9%

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

      3. +-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999996:\\ \;\;\;\;\frac{\frac{\frac{\beta + 2}{\alpha}}{\alpha} \cdot \left(\left(-2 - \beta\right) - \beta\right) + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{\alpha}{\beta + \left(\alpha + 2\right)} + \beta \cdot \frac{-1}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (alpha + 2.0d0)
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.999998d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha / t_0) + (beta * ((-1.0d0) / t_0)))) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.2%

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

   3. Taylor expanded in alpha around -inf 99.3%

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

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

   5. Simplified 99.3%

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

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

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

      2. distribute-lft-in 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-rgt-neg-out 99.7%

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

      2. neg-sub0 99.7%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \color{blue}{\left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \alpha\right)}\right) + 1}{2} \]

      3. add-sqr-sqrt 58.5%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(\sqrt{\alpha} \cdot \sqrt{\alpha}\right)}\right)\right) + 1}{2} \]

      4. sqrt-prod 95.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\sqrt{\alpha \cdot \alpha}}\right)\right) + 1}{2} \]

      5. sqr-neg 95.8%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \sqrt{\color{blue}{\left(-\alpha\right) \cdot \left(-\alpha\right)}}\right)\right) + 1}{2} \]

      6. sqrt-unprod 40.6%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(\sqrt{-\alpha} \cdot \sqrt{-\alpha}\right)}\right)\right) + 1}{2} \]

      7. add-sqr-sqrt 96.4%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{1}{\beta + \left(\alpha + 2\right)} \cdot \color{blue}{\left(-\alpha\right)}\right)\right) + 1}{2} \]

      8. associate-*l/ 96.4%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \color{blue}{\frac{1 \cdot \left(-\alpha\right)}{\beta + \left(\alpha + 2\right)}}\right)\right) + 1}{2} \]

      9. *-un-lft-identity 96.4%

         \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{-\alpha}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      10. add-sqr-sqrt 40.6%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{-\alpha} \cdot \sqrt{-\alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      11. sqrt-unprod 95.8%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{\left(-\alpha\right) \cdot \left(-\alpha\right)}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      12. sqr-neg 95.8%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\sqrt{\color{blue}{\alpha \cdot \alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      13. sqrt-prod 58.5%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\sqrt{\alpha} \cdot \sqrt{\alpha}}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

      14. add-sqr-sqrt 99.7%

          \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \left(0 - \frac{\color{blue}{\alpha}}{\beta + \left(\alpha + 2\right)}\right)\right) + 1}{2} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\left(\frac{1}{\beta + \left(\alpha + 2\right)} \cdot \beta + \color{blue}{\left(0 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}\right) + 1}{2} \]
   9. Step-by-step derivation

      1. neg-sub0 99.7%

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

      2. distribute-neg-frac 99.7%

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

      3. +-commutative 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / 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) / t_0) <= (-0.999998d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / 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) / t_0) <= -0.999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.999998)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / 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) / t_0) <= -0.999998)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.999998], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / 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)) < -0.999998000000000054

   1. Initial program 6.2%

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

   3. Taylor expanded in alpha around -inf 99.3%

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

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

   5. Simplified 99.3%

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

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

   1. Initial program 99.7%

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

4. Final simplification 99.6%

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

Alternative 5: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1350000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1350000000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1350000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1350000000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1350000000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1350000000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1350000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1350000000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1350000000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.35e9

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0 97.9%

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

   if 1.35e9 < alpha

   1. Initial program 25.8%

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

   3. Taylor expanded in beta around 0 5.6%

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

      1. +-commutative 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in alpha around inf 66.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1350000000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 6200000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 6200000.0)
   (/ (+ (/ beta (+ beta 2.0)) 1.0) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 6200000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 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 <= 6200000.0d0) then
        tmp = ((beta / (beta + 2.0d0)) + 1.0d0) / 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 <= 6200000.0) {
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 6200000.0:
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 6200000.0)
		tmp = Float64(Float64(Float64(beta / Float64(beta + 2.0)) + 1.0) / 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 <= 6200000.0)
		tmp = ((beta / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 6200000.0], N[(N[(N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 < 6.2e6

   1. Initial program 99.9%

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

   3. Taylor expanded in alpha around 0 97.9%

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

   if 6.2e6 < alpha

   1. Initial program 25.8%

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

   3. Taylor expanded in alpha around -inf 80.1%

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

      1. associate-*r/ 80.1%

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

      2. sub-neg 80.1%

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

      3. mul-1-neg 80.1%

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

      4. distribute-lft-in 80.1%

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

      5. neg-mul-1 80.1%

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

      6. mul-1-neg 80.1%

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

      7. remove-double-neg 80.1%

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

      8. neg-mul-1 80.1%

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

      9. mul-1-neg 80.1%

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

      10. remove-double-neg 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 6200000:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.85:\\ \;\;\;\;\frac{\alpha \cdot -0.5 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.85) (/ (+ (* alpha -0.5) 1.0) 2.0) (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.85) {
		tmp = ((alpha * -0.5) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.85d0) then
        tmp = ((alpha * (-0.5d0)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.85) {
		tmp = ((alpha * -0.5) + 1.0) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.85:
		tmp = ((alpha * -0.5) + 1.0) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.85)
		tmp = Float64(Float64(Float64(alpha * -0.5) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.85)
		tmp = ((alpha * -0.5) + 1.0) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.85], N[(N[(N[(alpha * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.849999999999999978

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0 73.4%

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

      1. +-commutative 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in alpha around 0 72.6%

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

      1. *-commutative 72.6%

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

   8. Simplified 72.6%

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

   if 0.849999999999999978 < alpha

   1. Initial program 27.2%

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

   3. Taylor expanded in beta around 0 6.5%

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

      1. +-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in alpha around inf 65.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.85:\\ \;\;\;\;\frac{\alpha \cdot -0.5 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 66.4%

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

   3. Taylor expanded in alpha around 0 64.2%

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

   4. Taylor expanded in beta around 0 63.8%

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

      1. *-commutative 63.8%

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

   6. Simplified 63.8%

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

   if 2 < beta

   1. Initial program 85.8%

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

   3. Taylor expanded in alpha around 0 82.6%

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

   4. Taylor expanded in beta around inf 82.6%

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

      1. associate-*r/ 82.6%

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

      2. metadata-eval 82.6%

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

   6. Simplified 82.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.92:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.92) 0.5 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.92) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.92d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.92) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.92:
		tmp = 0.5
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.92)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.92)
		tmp = 0.5;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.92], 0.5, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.92000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in beta around 0 73.4%

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

      1. +-commutative 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in alpha around 0 71.9%

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

   if 0.92000000000000004 < alpha

   1. Initial program 27.2%

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

   3. Taylor expanded in beta around 0 6.5%

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

      1. +-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in alpha around inf 65.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 0.92:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 17:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 17.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 17.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 17.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 17.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 17.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 17.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 17.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 17.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 17

   1. Initial program 66.0%

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

   3. Taylor expanded in beta around 0 64.9%

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

      1. +-commutative 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in alpha around 0 62.7%

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

   if 17 < beta

   1. Initial program 86.8%

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

   3. Taylor expanded in beta around inf 83.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 17:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in beta around 0 48.3%

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

   1. +-commutative 48.3%

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

5. Simplified 48.3%

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

6. Taylor expanded in alpha around 0 48.1%

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

7. Final simplification 48.1%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))