Octave 3.8, jcobi/1

Percentage Accurate: 74.5% → 99.6%

Time: 7.7s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.5% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.998:\\ \;\;\;\;0.5 \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + 0.5 \cdot \frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} + -1\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.998)
     (+
      (*
       0.5
       (* beta (- (* 2.0 (/ 1.0 alpha)) (* 6.0 (/ 1.0 (pow alpha 2.0))))))
      (* 0.5 (/ (- 2.0 (* 4.0 (/ 1.0 alpha))) alpha)))
     (/ (exp (log (- (/ beta t_0) (+ (/ alpha t_0) -1.0)))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998) {
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	} else {
		tmp = exp(log(((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.998d0)) then
        tmp = (0.5d0 * (beta * ((2.0d0 * (1.0d0 / alpha)) - (6.0d0 * (1.0d0 / (alpha ** 2.0d0)))))) + (0.5d0 * ((2.0d0 - (4.0d0 * (1.0d0 / alpha))) / alpha))
    else
        tmp = exp(log(((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998) {
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / Math.pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	} else {
		tmp = Math.exp(Math.log(((beta / t_0) - ((alpha / t_0) + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998:
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / math.pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha))
	else:
		tmp = math.exp(math.log(((beta / t_0) - ((alpha / t_0) + -1.0)))) / 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(alpha + beta) + 2.0)) <= -0.998)
		tmp = Float64(Float64(0.5 * Float64(beta * Float64(Float64(2.0 * Float64(1.0 / alpha)) - Float64(6.0 * Float64(1.0 / (alpha ^ 2.0)))))) + Float64(0.5 * Float64(Float64(2.0 - Float64(4.0 * Float64(1.0 / alpha))) / alpha)));
	else
		tmp = Float64(exp(log(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) + -1.0)))) / 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) / ((alpha + beta) + 2.0)) <= -0.998)
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / (alpha ^ 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	else
		tmp = exp(log(((beta / t_0) - ((alpha / t_0) + -1.0)))) / 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(0.5 * N[(beta * N[(N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] - N[(6.0 * N[(1.0 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(2.0 - N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[Log[N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in alpha around inf 8.0%

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

   6. Taylor expanded in alpha around inf 97.1%

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

   7. Taylor expanded in beta around 0 99.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + 0.5 \cdot \frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}} \]

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. rem-log-exp 99.9%

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

   10. Applied egg-rr 99.9%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.998:\\ \;\;\;\;0.5 \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + 0.5 \cdot \frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.998)
     (+
      (*
       0.5
       (* beta (- (* 2.0 (/ 1.0 alpha)) (* 6.0 (/ 1.0 (pow alpha 2.0))))))
      (* 0.5 (/ (- 2.0 (* 4.0 (/ 1.0 alpha))) alpha)))
     (/ (- (/ beta t_0) (- (/ alpha t_0) 1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998) {
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.998d0)) then
        tmp = (0.5d0 * (beta * ((2.0d0 * (1.0d0 / alpha)) - (6.0d0 * (1.0d0 / (alpha ** 2.0d0)))))) + (0.5d0 * ((2.0d0 - (4.0d0 * (1.0d0 / alpha))) / alpha))
    else
        tmp = ((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998) {
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / Math.pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.998:
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / math.pow(alpha, 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha))
	else:
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 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(alpha + beta) + 2.0)) <= -0.998)
		tmp = Float64(Float64(0.5 * Float64(beta * Float64(Float64(2.0 * Float64(1.0 / alpha)) - Float64(6.0 * Float64(1.0 / (alpha ^ 2.0)))))) + Float64(0.5 * Float64(Float64(2.0 - Float64(4.0 * Float64(1.0 / alpha))) / alpha)));
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) - 1.0)) / 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) / ((alpha + beta) + 2.0)) <= -0.998)
		tmp = (0.5 * (beta * ((2.0 * (1.0 / alpha)) - (6.0 * (1.0 / (alpha ^ 2.0)))))) + (0.5 * ((2.0 - (4.0 * (1.0 / alpha))) / alpha));
	else
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.998], N[(N[(0.5 * N[(beta * N[(N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision] - N[(6.0 * N[(1.0 / N[Power[alpha, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(2.0 - N[(4.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.998

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in alpha around inf 8.0%

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

   6. Taylor expanded in alpha around inf 97.1%

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

   7. Taylor expanded in beta around 0 99.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 6 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + 0.5 \cdot \frac{2 - 4 \cdot \frac{1}{\alpha}}{\alpha}} \]

   if -0.998 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

   6. 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. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.999999996:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.999999996)
     (+ (/ 1.0 alpha) (/ beta alpha))
     (/ (- (/ beta t_0) (- (/ alpha t_0) 1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = ((beta / t_0) - ((alpha / 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 + 2.0d0)
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.999999996d0)) then
        tmp = (1.0d0 / alpha) + (beta / alpha)
    else
        tmp = ((beta / t_0) - ((alpha / 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 + 2.0);
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.999999996:
		tmp = (1.0 / alpha) + (beta / alpha)
	else:
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 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(alpha + beta) + 2.0)) <= -0.999999996)
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) - 1.0)) / 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) / ((alpha + beta) + 2.0)) <= -0.999999996)
		tmp = (1.0 / alpha) + (beta / alpha);
	else
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999999996], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999996000000002

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around inf 99.4%

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

   6. Taylor expanded in beta around 0 99.4%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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%

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

   3. Simplified 99.5%

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

      1. div-sub 99.5%

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

      2. associate-+l- 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+l+ 99.5%

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

   6. Applied egg-rr 99.5%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999996:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta \cdot \left(1 + \frac{\alpha}{\beta}\right) + 2} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999996)
   (+ (/ 1.0 alpha) (/ beta alpha))
   (/ (+ (/ (- beta alpha) (+ (* beta (+ 1.0 (/ alpha beta))) 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 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) :: tmp
    if (((beta - alpha) / ((alpha + beta) + 2.0d0)) <= (-0.9999996d0)) then
        tmp = (1.0d0 / alpha) + (beta / alpha)
    else
        tmp = (((beta - alpha) / ((beta * (1.0d0 + (alpha / beta))) + 2.0d0)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999996:
		tmp = (1.0 / alpha) + (beta / alpha)
	else:
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999996)
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(beta * Float64(1.0 + Float64(alpha / beta))) + 2.0)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999996)
		tmp = (1.0 / alpha) + (beta / alpha);
	else
		tmp = (((beta - alpha) / ((beta * (1.0 + (alpha / beta))) + 2.0)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999996], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta * N[(1.0 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999959999999999

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in alpha around inf 98.9%

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

   6. Taylor expanded in beta around 0 98.9%

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

   if -0.99999959999999999 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in beta around inf 99.7%

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

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999999996:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.999999996)
     (+ (/ 1.0 alpha) (/ beta alpha))
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.999999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((alpha + beta) + 2.0d0)
    if (t_0 <= (-0.999999996d0)) then
        tmp = (1.0d0 / alpha) + (beta / alpha)
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.999999996) {
		tmp = (1.0 / alpha) + (beta / alpha);
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((alpha + beta) + 2.0)
	tmp = 0
	if t_0 <= -0.999999996:
		tmp = (1.0 / alpha) + (beta / alpha)
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.999999996)
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.999999996)
		tmp = (1.0 / alpha) + (beta / alpha);
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.999999996], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999996000000002

   1. Initial program 6.4%

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

      1. +-commutative 6.4%

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

   3. Simplified 6.4%

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

   5. Taylor expanded in alpha around inf 99.4%

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

   6. Taylor expanded in beta around 0 99.4%

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

   if -0.999999996000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.5%

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

Alternative 6: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.65:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.65)
   (/ (+ (/ (- beta alpha) (+ beta 2.0)) 1.0) 2.0)
   (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.65) {
		tmp = (((beta - alpha) / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.65d0) then
        tmp = (((beta - alpha) / (beta + 2.0d0)) + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.65) {
		tmp = (((beta - alpha) / (beta + 2.0)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.65:
		tmp = (((beta - alpha) / (beta + 2.0)) + 1.0) / 2.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.65)
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / Float64(beta + 2.0)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.65)
		tmp = (((beta - alpha) / (beta + 2.0)) + 1.0) / 2.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.65], N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6499999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around inf 98.8%

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

   if 1.6499999999999999 < alpha

   1. Initial program 25.3%

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

      1. +-commutative 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in alpha around inf 81.6%

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

   6. Taylor expanded in beta around 0 81.6%

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

Alternative 7: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{-58}:\\ \;\;\;\;0.5 + 0.25 \cdot \beta\\ \mathbf{elif}\;\beta \leq 0.38:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.7e-58)
   (+ 0.5 (* 0.25 beta))
   (if (<= beta 0.38) (/ 1.0 alpha) (- 1.0 (/ 1.0 beta)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.7e-58) {
		tmp = 0.5 + (0.25 * beta);
	} else if (beta <= 0.38) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 5.7d-58) then
        tmp = 0.5d0 + (0.25d0 * beta)
    else if (beta <= 0.38d0) then
        tmp = 1.0d0 / alpha
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.7e-58) {
		tmp = 0.5 + (0.25 * beta);
	} else if (beta <= 0.38) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 5.7e-58:
		tmp = 0.5 + (0.25 * beta)
	elif beta <= 0.38:
		tmp = 1.0 / alpha
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.7e-58)
		tmp = Float64(0.5 + Float64(0.25 * beta));
	elseif (beta <= 0.38)
		tmp = Float64(1.0 / alpha);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 5.7e-58)
		tmp = 0.5 + (0.25 * beta);
	elseif (beta <= 0.38)
		tmp = 1.0 / alpha;
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 5.7e-58], N[(0.5 + N[(0.25 * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 0.38], N[(1.0 / alpha), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 5.70000000000000032e-58

   1. Initial program 70.2%

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

      1. +-commutative 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in beta around inf 66.0%

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

   6. Taylor expanded in beta around inf 65.8%

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

   7. Taylor expanded in beta around 0 65.6%

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

   if 5.70000000000000032e-58 < beta < 0.38

   1. Initial program 33.4%

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

      1. +-commutative 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in alpha around inf 33.5%

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

   6. Taylor expanded in beta around 0 32.9%

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

   7. Taylor expanded in alpha around inf 61.9%

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

   if 0.38 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in beta around inf 82.3%

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

   6. Taylor expanded in beta around inf 83.1%

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

   7. Taylor expanded in beta around inf 81.9%

      \[\leadsto \color{blue}{1 - \frac{1}{\beta}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 120000000:\\ \;\;\;\;\frac{\frac{\beta}{2 + \beta} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 120000000.0)
   (/ (+ (/ beta (+ 2.0 beta)) 1.0) 2.0)
   (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 120000000.0) {
		tmp = ((beta / (2.0 + beta)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 120000000.0d0) then
        tmp = ((beta / (2.0d0 + beta)) + 1.0d0) / 2.0d0
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 120000000.0) {
		tmp = ((beta / (2.0 + beta)) + 1.0) / 2.0;
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 120000000.0:
		tmp = ((beta / (2.0 + beta)) + 1.0) / 2.0
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 120000000.0)
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + beta)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 120000000.0)
		tmp = ((beta / (2.0 + beta)) + 1.0) / 2.0;
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 120000000.0], N[(N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.2e8

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in alpha around 0 95.7%

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

   if 1.2e8 < alpha

   1. Initial program 22.4%

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

      1. +-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in alpha around inf 84.0%

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

   6. Taylor expanded in beta around 0 84.0%

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

Alternative 9: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{-58}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 0.095:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.7e-58) 0.5 (if (<= beta 0.095) (/ 1.0 alpha) 1.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e-58) {
		tmp = 0.5;
	} else if (beta <= 0.095) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.7d-58) then
        tmp = 0.5d0
    else if (beta <= 0.095d0) then
        tmp = 1.0d0 / alpha
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.7e-58) {
		tmp = 0.5;
	} else if (beta <= 0.095) {
		tmp = 1.0 / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.7e-58:
		tmp = 0.5
	elif beta <= 0.095:
		tmp = 1.0 / alpha
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.7e-58)
		tmp = 0.5;
	elseif (beta <= 0.095)
		tmp = Float64(1.0 / alpha);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.7e-58)
		tmp = 0.5;
	elseif (beta <= 0.095)
		tmp = 1.0 / alpha;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.7e-58], 0.5, If[LessEqual[beta, 0.095], N[(1.0 / alpha), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.6999999999999999e-58

   1. Initial program 70.2%

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

      1. +-commutative 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in beta around inf 66.0%

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

   6. Taylor expanded in beta around inf 65.8%

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

   7. Taylor expanded in beta around 0 65.3%

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

   if 2.6999999999999999e-58 < beta < 0.095000000000000001

   1. Initial program 33.4%

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

      1. +-commutative 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in alpha around inf 33.5%

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

   6. Taylor expanded in beta around 0 32.9%

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

   7. Taylor expanded in alpha around inf 61.9%

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

   if 0.095000000000000001 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in beta around inf 81.0%

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

   6. Taylor expanded in beta around inf 81.8%

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

Alternative 10: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.65:\\ \;\;\;\;0.5 + -0.25 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha} + \frac{\beta}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.65) (+ 0.5 (* -0.25 alpha)) (+ (/ 1.0 alpha) (/ beta alpha))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.65) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.65d0) then
        tmp = 0.5d0 + ((-0.25d0) * alpha)
    else
        tmp = (1.0d0 / alpha) + (beta / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.65) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = (1.0 / alpha) + (beta / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.65:
		tmp = 0.5 + (-0.25 * alpha)
	else:
		tmp = (1.0 / alpha) + (beta / alpha)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.65)
		tmp = Float64(0.5 + Float64(-0.25 * alpha));
	else
		tmp = Float64(Float64(1.0 / alpha) + Float64(beta / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.65)
		tmp = 0.5 + (-0.25 * alpha);
	else
		tmp = (1.0 / alpha) + (beta / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.65], N[(0.5 + N[(-0.25 * alpha), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.6499999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around inf 85.5%

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

   6. Taylor expanded in beta around 0 71.9%

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

   7. Taylor expanded in alpha around 0 70.8%

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

   if 1.6499999999999999 < alpha

   1. Initial program 25.3%

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

      1. +-commutative 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in alpha around inf 81.6%

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

   6. Taylor expanded in beta around 0 81.6%

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

Alternative 11: 69.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 0.95:\\ \;\;\;\;0.5 + -0.25 \cdot \alpha\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 0.95) (+ 0.5 (* -0.25 alpha)) (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 0.95d0) then
        tmp = 0.5d0 + ((-0.25d0) * alpha)
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 0.95) {
		tmp = 0.5 + (-0.25 * alpha);
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 0.95:
		tmp = 0.5 + (-0.25 * alpha)
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 0.95)
		tmp = Float64(0.5 + Float64(-0.25 * alpha));
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 0.95)
		tmp = 0.5 + (-0.25 * alpha);
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 0.95], N[(0.5 + N[(-0.25 * alpha), $MachinePrecision]), $MachinePrecision], N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 0.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around inf 85.5%

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

   6. Taylor expanded in beta around 0 71.9%

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

   7. Taylor expanded in alpha around 0 70.8%

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

   if 0.94999999999999996 < alpha

   1. Initial program 25.3%

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

      1. +-commutative 25.3%

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

   3. Simplified 25.3%

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

   5. Taylor expanded in alpha around inf 25.3%

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

   6. Taylor expanded in beta around 0 8.7%

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

   7. Taylor expanded in alpha around inf 65.9%

      \[\leadsto \color{blue}{\frac{1}{\alpha}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.1) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.1d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.1) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.1:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.1)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.1], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2.10000000000000009

   1. Initial program 66.2%

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

      1. +-commutative 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in beta around inf 61.7%

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

   6. Taylor expanded in beta around inf 61.9%

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

   7. Taylor expanded in beta around 0 60.9%

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

   if 2.10000000000000009 < beta

   1. Initial program 86.2%

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

      1. +-commutative 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in beta around inf 81.7%

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

   6. Taylor expanded in beta around inf 82.5%

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

Alternative 13: 49.8% 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 72.6%

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

   1. +-commutative 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in beta around inf 68.3%

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

6. Taylor expanded in beta around inf 68.6%

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

7. Taylor expanded in beta around 0 46.8%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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))