Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.9%

Time: 9.1s

Alternatives: 13

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: 74.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99995)
   (/
    (/
     (+
      (- (* beta 2.0) (* (+ beta 2.0) (/ (+ beta 2.0) alpha)))
      (+ 2.0 (* beta (/ (- -2.0 beta) alpha))))
     alpha)
    2.0)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99995) {
		tmp = ((((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 2.0) - Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) / alpha))) + Float64(2.0 + Float64(beta * Float64(Float64(-2.0 - beta) / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 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], -0.99995], N[(N[(N[(N[(N[(beta * 2.0), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $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))) < -0.999950000000000006

   1. Initial program 8.3%

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

   3. Taylor expanded in alpha around inf 97.2%

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

   5. Simplified 97.2%

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

      1. +-commutative 97.2%

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

      2. pow2 97.2%

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

      3. *-un-lft-identity 97.2%

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

      4. times-frac 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99995)
     (/
      (/
       (+
        (- (* beta 2.0) (* (+ beta 2.0) (/ (+ beta 2.0) alpha)))
        (+ 2.0 (* beta (/ (- -2.0 beta) alpha))))
       alpha)
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = ((((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99995d0)) then
        tmp = ((((beta * 2.0d0) - ((beta + 2.0d0) * ((beta + 2.0d0) / alpha))) + (2.0d0 + (beta * (((-2.0d0) - beta) / alpha)))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99995) {
		tmp = ((((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99995:
		tmp = ((((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99995)
		tmp = Float64(Float64(Float64(Float64(Float64(beta * 2.0) - Float64(Float64(beta + 2.0) * Float64(Float64(beta + 2.0) / alpha))) + Float64(2.0 + Float64(beta * Float64(Float64(-2.0 - beta) / alpha)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99995)
		tmp = ((((beta * 2.0) - ((beta + 2.0) * ((beta + 2.0) / alpha))) + (2.0 + (beta * ((-2.0 - beta) / alpha)))) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99995], N[(N[(N[(N[(N[(beta * 2.0), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(beta * N[(N[(-2.0 - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999950000000000006

   1. Initial program 8.3%

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

   3. Taylor expanded in alpha around inf 97.2%

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

   5. Simplified 97.2%

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

      1. +-commutative 97.2%

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

      2. pow2 97.2%

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

      3. *-un-lft-identity 97.2%

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

      4. times-frac 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -0.999950000000000006 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99995:\\ \;\;\;\;\frac{\frac{\left(\beta \cdot 2 - \left(\beta + 2\right) \cdot \frac{\beta + 2}{\alpha}\right) + \left(2 + \beta \cdot \frac{-2 - \beta}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999999)
   (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ alpha (+ beta 2.0)))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= (-0.99999999d0)) then
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) / (alpha + (beta + 2.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 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)) <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + 2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999999)
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) / (alpha + (beta + 2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99999999], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + 2.0), $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))) < -0.99999998999999995

   1. Initial program 7.6%

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

   3. Taylor expanded in alpha around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-neg-frac2 98.7%

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

      3. associate--r+ 98.7%

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

      4. sub-neg 98.7%

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

      5. +-commutative 98.7%

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

      6. mul-1-neg 98.7%

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

      7. unsub-neg 98.7%

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

      8. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   if -0.99999998999999995 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-+l+ 99.7%

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

      2. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999999)
     (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999999d0)) then
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999999) {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99999999:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999999)
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999999)
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999999], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99999998999999995

   1. Initial program 7.6%

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

   3. Taylor expanded in alpha around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-neg-frac2 98.7%

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

      3. associate--r+ 98.7%

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

      4. sub-neg 98.7%

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

      5. +-commutative 98.7%

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

      6. mul-1-neg 98.7%

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

      7. unsub-neg 98.7%

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

      8. metadata-eval 98.7%

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

   5. Simplified 98.7%

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

   if -0.99999998999999995 < (/.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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999999:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -2.6 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha -1.9e-61)
     t_0
     (if (<= alpha -2.6e-121)
       1.0
       (if (<= alpha 1.95) t_0 (/ (* (+ beta 1.0) (/ 2.0 alpha)) 2.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -1.9e-61) {
		tmp = t_0;
	} else if (alpha <= -2.6e-121) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= (-1.9d-61)) then
        tmp = t_0
    else if (alpha <= (-2.6d-121)) then
        tmp = 1.0d0
    else if (alpha <= 1.95d0) then
        tmp = t_0
    else
        tmp = ((beta + 1.0d0) * (2.0d0 / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= -1.9e-61) {
		tmp = t_0;
	} else if (alpha <= -2.6e-121) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= -1.9e-61:
		tmp = t_0
	elif alpha <= -2.6e-121:
		tmp = 1.0
	elif alpha <= 1.95:
		tmp = t_0
	else:
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= -1.9e-61)
		tmp = t_0;
	elseif (alpha <= -2.6e-121)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(2.0 / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= -1.9e-61)
		tmp = t_0;
	elseif (alpha <= -2.6e-121)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, -1.9e-61], t$95$0, If[LessEqual[alpha, -2.6e-121], 1.0, If[LessEqual[alpha, 1.95], t$95$0, N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.8999999999999999e-61 or -2.59999999999999986e-121 < alpha < 1.94999999999999996

   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 74.1%

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

      1. +-commutative 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in alpha around 0 73.5%

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

   if -1.8999999999999999e-61 < alpha < -2.59999999999999986e-121

   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 inf 82.5%

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

   if 1.94999999999999996 < alpha

   1. Initial program 26.2%

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

   3. Taylor expanded in alpha around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-neg-frac2 80.4%

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

      3. associate--r+ 80.4%

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

      4. sub-neg 80.4%

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

      5. +-commutative 80.4%

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

      6. mul-1-neg 80.4%

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

      7. unsub-neg 80.4%

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

      8. metadata-eval 80.4%

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

   5. Simplified 80.4%

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

   6. Taylor expanded in beta around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l* 80.4%

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

      4. metadata-eval 80.4%

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

      5. associate-*r/ 80.4%

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

      6. distribute-lft1-in 80.4%

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

      7. associate-*r/ 80.4%

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

      8. metadata-eval 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq -2.6 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 35000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 35000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (- beta -2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 35000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 35000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta - (-2.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 35000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 35000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 35000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta - -2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 35000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta - -2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 35000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 35000

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.8%

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

   if 35000 < alpha

   1. Initial program 26.2%

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

   3. Taylor expanded in alpha around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-neg-frac2 80.4%

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

      3. associate--r+ 80.4%

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

      4. sub-neg 80.4%

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

      5. +-commutative 80.4%

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

      6. mul-1-neg 80.4%

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

      7. unsub-neg 80.4%

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

      8. metadata-eval 80.4%

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

   5. Simplified 80.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 35000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 25000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 25000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 25000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (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 <= 25000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (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 <= 25000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 25000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 25000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 25000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 25000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 25000

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.8%

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

   if 25000 < alpha

   1. Initial program 26.2%

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

   3. Taylor expanded in alpha around inf 80.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 25000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 17000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 17000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (* (+ beta 1.0) (/ 2.0 alpha)) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 17000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 1.0) * (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 <= 17000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + 1.0d0) * (2.0d0 / alpha)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 17000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 17000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 17000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + 1.0) * Float64(2.0 / alpha)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 17000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + 1.0) * (2.0 / alpha)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 17000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + 1.0), $MachinePrecision] * N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 17000

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.8%

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

   if 17000 < alpha

   1. Initial program 26.2%

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

   3. Taylor expanded in alpha around -inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. distribute-neg-frac2 80.4%

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

      3. associate--r+ 80.4%

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

      4. sub-neg 80.4%

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

      5. +-commutative 80.4%

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

      6. mul-1-neg 80.4%

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

      7. unsub-neg 80.4%

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

      8. metadata-eval 80.4%

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

   5. Simplified 80.4%

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

   6. Taylor expanded in beta around 0 80.4%

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

      1. associate-*r/ 80.4%

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

      2. *-commutative 80.4%

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

      3. associate-/l* 80.4%

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

      4. metadata-eval 80.4%

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

      5. associate-*r/ 80.4%

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

      6. distribute-lft1-in 80.4%

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

      7. associate-*r/ 80.4%

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

      8. metadata-eval 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 17000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + 1\right) \cdot \frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% 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 66.1%

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

   3. Taylor expanded in alpha around 0 63.7%

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

   4. Taylor expanded in beta around 0 63.7%

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

   if 2 < beta

   1. Initial program 88.6%

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

   3. Taylor expanded in alpha around 0 87.9%

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

   4. Taylor expanded in beta around inf 87.1%

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

      1. sub-neg 87.1%

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

      2. associate-*r/ 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-neg-frac 87.1%

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

      5. metadata-eval 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 71.6%

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

Alternative 10: 71.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5;
	} 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 = 0.5d0
    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 = 0.5;
	} else {
		tmp = (2.0 + (-2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 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 66.1%

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

   3. Taylor expanded in alpha around 0 63.7%

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

   4. Taylor expanded in beta around 0 63.6%

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

   if 2 < beta

   1. Initial program 88.6%

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

   3. Taylor expanded in alpha around 0 87.9%

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

   4. Taylor expanded in beta around inf 87.1%

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

      1. sub-neg 87.1%

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

      2. associate-*r/ 87.1%

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

      3. metadata-eval 87.1%

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

      4. distribute-neg-frac 87.1%

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

      5. metadata-eval 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 71.6%

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

Alternative 11: 69.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.95) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 1.95d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.95) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.95:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.95)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.95], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.94999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in alpha around 0 98.8%

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

   4. Taylor expanded in beta around 0 69.7%

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

   if 1.94999999999999996 < alpha

   1. Initial program 26.2%

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

   3. Taylor expanded in alpha around inf 78.8%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in beta around 0 70.5%

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

      1. sub-neg 70.5%

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

      2. associate-*r/ 70.5%

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

      3. metadata-eval 70.5%

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

      4. distribute-neg-frac 70.5%

         \[\leadsto \frac{\frac{2 + \color{blue}{\frac{-4}{\alpha}}}{\alpha}}{2} \]

      5. metadata-eval 70.5%

         \[\leadsto \frac{\frac{2 + \frac{\color{blue}{-4}}{\alpha}}{\alpha}}{2} \]

   8. Simplified 70.5%

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

      1. add-cube-cbrt 69.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{2 + \frac{-4}{\alpha}}{\alpha}}{2}} \cdot \sqrt[3]{\frac{\frac{2 + \frac{-4}{\alpha}}{\alpha}}{2}}\right) \cdot \sqrt[3]{\frac{\frac{2 + \frac{-4}{\alpha}}{\alpha}}{2}}} \]

      2. pow3 69.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{2 + \frac{-4}{\alpha}}{\alpha}}{2}}\right)}^{3}} \]

      3. associate-/l/ 69.1%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{2 + \frac{-4}{\alpha}}{2 \cdot \alpha}}}\right)}^{3} \]

   10. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{2 + \frac{-4}{\alpha}}{2 \cdot \alpha}}\right)}^{3}} \]

   11. Taylor expanded in alpha around inf 68.1%

       \[\leadsto \color{blue}{2 \cdot \frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{\alpha}} \]
   12. Step-by-step derivation

       1. rem-cube-cbrt 69.3%

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

       2. associate-*r/ 69.3%

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

       3. metadata-eval 69.3%

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

   13. Simplified 69.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.95:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.5% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		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 <= 2.0d0) 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 <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

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

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 66.1%

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

   3. Taylor expanded in alpha around 0 63.7%

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

   4. Taylor expanded in beta around 0 63.6%

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

   if 2 < beta

   1. Initial program 88.6%

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

   3. Taylor expanded in beta around inf 86.4%

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

4. Final simplification 71.4%

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

Alternative 13: 49.5% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 73.8%

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

3. Taylor expanded in alpha around 0 71.9%

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

4. Taylor expanded in beta around 0 47.8%

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

5. Final simplification 47.8%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))