Octave 3.8, jcobi/1

Percentage Accurate: 74.3% → 99.9%

Time: 9.9s

Alternatives: 11

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
   (/
    (/
     (+
      2.0
      (-
       (* beta (+ 2.0 (- (* (/ beta alpha) -2.0) (/ 6.0 alpha))))
       (/ 4.0 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)) <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / 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)) <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(Float64(beta / alpha) * -2.0) - Float64(6.0 / alpha)))) - Float64(4.0 / 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], -0.99999], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(N[(beta / alpha), $MachinePrecision] * -2.0), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $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))) < -0.999990000000000046

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in beta around 0 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}}}{\alpha}}{2} \]
   9. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. div-inv 99.8%

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

      2. fma-define 99.9%

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

      3. associate-+l+ 99.9%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
   (/
    (/
     (+
      2.0
      (-
       (* beta (+ 2.0 (- (* (/ beta alpha) -2.0) (/ 6.0 alpha))))
       (/ 4.0 alpha)))
     alpha)
    2.0)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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)) <= (-0.99999d0)) then
        tmp = ((2.0d0 + ((beta * (2.0d0 + (((beta / alpha) * (-2.0d0)) - (6.0d0 / alpha)))) - (4.0d0 / alpha))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 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)) <= -0.99999) {
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * Float64(2.0 + Float64(Float64(Float64(beta / alpha) * -2.0) - Float64(6.0 / alpha)))) - Float64(4.0 / alpha))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 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)) <= -0.99999)
		tmp = ((2.0 + ((beta * (2.0 + (((beta / alpha) * -2.0) - (6.0 / alpha)))) - (4.0 / alpha))) / alpha) / 2.0;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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], -0.99999], N[(N[(N[(2.0 + N[(N[(beta * N[(2.0 + N[(N[(N[(beta / alpha), $MachinePrecision] * -2.0), $MachinePrecision] - N[(6.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999990000000000046

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in alpha around inf 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in beta around 0 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\left(2 + \beta \cdot \left(\left(2 + -2 \cdot \frac{\beta}{\alpha}\right) - 6 \cdot \frac{1}{\alpha}\right)\right) - 4 \cdot \frac{1}{\alpha}}}{\alpha}}{2} \]
   9. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. associate--l+ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot \left(2 + \left(\frac{\beta}{\alpha} \cdot -2 - \frac{6}{\alpha}\right)\right) - \frac{4}{\alpha}\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999)
   (/
    (*
     0.5
     (+
      (+ 2.0 (* beta 2.0))
      (* (+ beta 2.0) (/ (- (- -2.0 beta) beta) alpha))))
    alpha)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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)) <= (-0.99999d0)) then
        tmp = (0.5d0 * ((2.0d0 + (beta * 2.0d0)) + ((beta + 2.0d0) * ((((-2.0d0) - beta) - beta) / alpha)))) / alpha
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 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)) <= -0.99999) {
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999:
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999)
		tmp = Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(beta * 2.0)) + Float64(Float64(beta + 2.0) * Float64(Float64(Float64(-2.0 - beta) - beta) / alpha)))) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 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)) <= -0.99999)
		tmp = (0.5 * ((2.0 + (beta * 2.0)) + ((beta + 2.0) * (((-2.0 - beta) - beta) / alpha)))) / alpha;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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], -0.99999], N[(N[(0.5 * N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + 2.0), $MachinePrecision] * N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999990000000000046

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

   3. Simplified 7.9%

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

      1. clear-num 8.0%

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

      2. associate-/r/ 8.0%

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

      3. associate-+l+ 8.0%

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

   6. Applied egg-rr 8.0%

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

   7. Taylor expanded in alpha around inf 92.7%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot \beta\right) + 0.5 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   8. Step-by-step derivation

   9. Simplified 99.9%

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

   if -0.999990000000000046 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   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. Step-by-step derivation

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(2 + \beta \cdot 2\right) + \left(\beta + 2\right) \cdot \frac{\left(-2 - \beta\right) - \beta}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99999995)
   (/ (* (- (- -2.0 beta) beta) -0.5) alpha)
   (/ (+ 1.0 (* (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995) {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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)) <= (-0.99999995d0)) then
        tmp = ((((-2.0d0) - beta) - beta) * (-0.5d0)) / alpha
    else
        tmp = (1.0d0 + ((beta - alpha) * (1.0d0 / (beta + (alpha + 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)) <= -0.99999995) {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	} else {
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99999995:
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha
	else:
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99999995)
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * -0.5) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) * Float64(1.0 / Float64(beta + Float64(alpha + 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)) <= -0.99999995)
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	else
		tmp = (1.0 + ((beta - alpha) * (1.0 / (beta + (alpha + 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], -0.99999995], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999949999999971

   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. Step-by-step derivation

      1. clear-num 7.1%

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

      2. associate-/r/ 7.1%

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

      3. associate-+l+ 7.1%

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

   6. Applied egg-rr 7.1%

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

   7. Taylor expanded in alpha around -inf 98.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. associate--r+ 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-neg-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. sub-neg 98.9%

         \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(-2 - \beta\right)} - \beta\right)}{\alpha} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}} \]

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.7%

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

      3. associate-+l+ 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.99999995:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999995)
     (/ (* (- (- -2.0 beta) beta) -0.5) alpha)
     (/ (+ 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 <= -0.99999995) {
		tmp = (((-2.0 - beta) - beta) * -0.5) / 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) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999995d0)) then
        tmp = ((((-2.0d0) - beta) - beta) * (-0.5d0)) / 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) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	} 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 <= -0.99999995:
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha
	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 <= -0.99999995)
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * -0.5) / 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) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999995)
		tmp = (((-2.0 - beta) - beta) * -0.5) / 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[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999995], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $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))) < -0.999999949999999971

   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. Step-by-step derivation

      1. clear-num 7.1%

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

      2. associate-/r/ 7.1%

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

      3. associate-+l+ 7.1%

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

   6. Applied egg-rr 7.1%

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

   7. Taylor expanded in alpha around -inf 98.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.9%

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

      2. associate--r+ 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. +-commutative 98.9%

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

      7. distribute-neg-in 98.9%

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

      8. metadata-eval 98.9%

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

      9. sub-neg 98.9%

         \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(-2 - \beta\right)} - \beta\right)}{\alpha} \]

   9. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}} \]

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999995:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7900000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7900000000.0)
   (/ (+ 1.0 (/ (- beta alpha) (+ beta 2.0))) 2.0)
   (/ (* (- (- -2.0 beta) beta) -0.5) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7900000000.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / 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 <= 7900000000.0d0) then
        tmp = (1.0d0 + ((beta - alpha) / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((((-2.0d0) - beta) - beta) * (-0.5d0)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7900000000.0) {
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7900000000.0:
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0
	else:
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7900000000.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * -0.5) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7900000000.0)
		tmp = (1.0 + ((beta - alpha) / (beta + 2.0))) / 2.0;
	else
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7900000000.0], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 7.9e9

   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 97.6%

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

   if 7.9e9 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

      1. clear-num 21.8%

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

      2. associate-/r/ 21.8%

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

      3. associate-+l+ 21.8%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in alpha around -inf 84.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 84.5%

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

      2. associate--r+ 84.5%

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

      3. sub-neg 84.5%

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

      4. mul-1-neg 84.5%

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

      5. distribute-neg-in 84.5%

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

      6. +-commutative 84.5%

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

      7. distribute-neg-in 84.5%

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

      8. metadata-eval 84.5%

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

      9. sub-neg 84.5%

         \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(-2 - \beta\right)} - \beta\right)}{\alpha} \]

   9. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7900000000:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 17200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 17200000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (* (- (- -2.0 beta) beta) -0.5) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 17200000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / 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 <= 17200000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((((-2.0d0) - beta) - beta) * (-0.5d0)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 17200000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 17200000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 17200000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * -0.5) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 17200000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 17200000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.72e10

   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 97.0%

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

   if 1.72e10 < alpha

   1. Initial program 21.7%

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

      1. +-commutative 21.7%

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

   3. Simplified 21.7%

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

      1. clear-num 21.8%

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

      2. associate-/r/ 21.8%

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

      3. associate-+l+ 21.8%

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

   6. Applied egg-rr 21.8%

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

   7. Taylor expanded in alpha around -inf 84.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 84.5%

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

      2. associate--r+ 84.5%

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

      3. sub-neg 84.5%

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

      4. mul-1-neg 84.5%

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

      5. distribute-neg-in 84.5%

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

      6. +-commutative 84.5%

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

      7. distribute-neg-in 84.5%

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

      8. metadata-eval 84.5%

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

      9. sub-neg 84.5%

         \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(-2 - \beta\right)} - \beta\right)}{\alpha} \]

   9. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 17200000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.9)
   (+ 0.5 (* alpha -0.25))
   (/ (* (- (- -2.0 beta) beta) -0.5) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / 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 <= 1.9d0) then
        tmp = 0.5d0 + (alpha * (-0.25d0))
    else
        tmp = ((((-2.0d0) - beta) - beta) * (-0.5d0)) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.9) {
		tmp = 0.5 + (alpha * -0.25);
	} else {
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.9:
		tmp = 0.5 + (alpha * -0.25)
	else:
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.9)
		tmp = Float64(0.5 + Float64(alpha * -0.25));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) - beta) * -0.5) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 1.9)
		tmp = 0.5 + (alpha * -0.25);
	else
		tmp = (((-2.0 - beta) - beta) * -0.5) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.9], N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] * -0.5), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.8999999999999999

   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 0 73.8%

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

      1. +-commutative 73.8%

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

   7. Simplified 73.8%

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

   8. Taylor expanded in alpha around 0 73.0%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot \alpha} \]
   9. Step-by-step derivation

      1. *-commutative 73.0%

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

   10. Simplified 73.0%

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

   if 1.8999999999999999 < alpha

   1. Initial program 26.8%

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

      1. +-commutative 26.8%

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

   3. Simplified 26.8%

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

      1. clear-num 26.8%

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

      2. associate-/r/ 26.9%

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

      3. associate-+l+ 26.9%

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in alpha around -inf 79.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{-1 \cdot \beta - \left(2 + \beta\right)}{\alpha}} \]
   8. Step-by-step derivation

      1. associate-*r/ 79.9%

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

      2. associate--r+ 79.9%

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

      3. sub-neg 79.9%

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

      4. mul-1-neg 79.9%

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

      5. distribute-neg-in 79.9%

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

      6. +-commutative 79.9%

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

      7. distribute-neg-in 79.9%

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

      8. metadata-eval 79.9%

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

      9. sub-neg 79.9%

         \[\leadsto \frac{-0.5 \cdot \left(\color{blue}{\left(-2 - \beta\right)} - \beta\right)}{\alpha} \]

   9. Simplified 79.9%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(-2 - \beta\right) - \beta\right)}{\alpha}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.9:\\ \;\;\;\;0.5 + \alpha \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 - \beta\right) - \beta\right) \cdot -0.5}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 73.4%

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

      1. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in beta around 0 71.1%

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

      1. +-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in alpha around 0 67.6%

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

   if 2 < beta

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.7%

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

      2. associate-/r/ 79.7%

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

      3. associate-+l+ 79.7%

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

   6. Applied egg-rr 79.7%

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

   7. Taylor expanded in beta around inf 76.8%

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

      1. associate-*r/ 76.8%

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

      2. distribute-lft-in 76.8%

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

      3. metadata-eval 76.8%

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

      4. neg-mul-1 76.8%

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

      5. *-commutative 76.8%

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

      6. distribute-rgt-neg-in 76.8%

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

      7. metadata-eval 76.8%

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

   9. Simplified 76.8%

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

   10. Taylor expanded in alpha around 0 77.6%

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

       1. associate-*r/ 77.6%

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

       2. metadata-eval 77.6%

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

   12. Simplified 77.6%

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

   13. Taylor expanded in beta around 0 77.6%

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

4. Final simplification 70.9%

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

Alternative 10: 71.0% 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 73.4%

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

      1. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in beta around 0 71.1%

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

      1. +-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in alpha around 0 67.6%

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

   if 2 < beta

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

   3. Simplified 79.6%

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

      1. clear-num 79.7%

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

      2. associate-/r/ 79.7%

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

      3. associate-+l+ 79.7%

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

   6. Applied egg-rr 79.7%

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

   7. Taylor expanded in beta around inf 76.9%

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

Alternative 11: 49.2% 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 75.4%

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

   1. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in beta around 0 52.0%

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

   1. +-commutative 52.0%

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

7. Simplified 52.0%

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

8. Taylor expanded in alpha around 0 50.7%

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

Reproduce

herbie shell --seed 2024144 
(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))