Octave 3.8, jcobi/1

Percentage Accurate: 75.0% → 99.7%

Time: 9.4s

Alternatives: 9

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9999998) (/ (+ beta 1.0) 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 <= -0.9999998) {
		tmp = (beta + 1.0) / 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 <= (-0.9999998d0)) then
        tmp = (beta + 1.0d0) / 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 <= -0.9999998) {
		tmp = (beta + 1.0) / 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 <= -0.9999998:
		tmp = (beta + 1.0) / 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 <= -0.9999998)
		tmp = Float64(Float64(beta + 1.0) / 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 <= -0.9999998)
		tmp = (beta + 1.0) / 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, -0.9999998], N[(N[(beta + 1.0), $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))) < -0.999999799999999994

   1. Initial program 6.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right)\right), \color{blue}{\alpha}\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot \beta\right)\right), \alpha\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\frac{1}{2} \cdot 2\right) \cdot \beta\right), \alpha\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + 1 \cdot \beta\right), \alpha\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \alpha\right) \]

      8. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \beta\right), \alpha\right) \]

   5. Simplified 99.5%

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

   if -0.999999799999999994 < (/.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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.0033:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.0033)
   (/ 1.0 (+ alpha 2.0))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.0033) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (1.0 + (beta / (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 <= 0.0033d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.0033) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 0.0033:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.0033)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 0.0033)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 0.0033], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.0033

   1. Initial program 64.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 64.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.7%

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

   if 0.0033 < beta

   1. Initial program 82.7%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 80.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 80.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.0033:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 0.0033:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 0.0033) (/ 1.0 (+ alpha 2.0)) (/ (+ beta 1.0) (+ beta 2.0))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.0033) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (beta + 1.0) / (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 <= 0.0033d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = (beta + 1.0d0) / (beta + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 0.0033) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = (beta + 1.0) / (beta + 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 0.0033:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = (beta + 1.0) / (beta + 2.0)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 0.0033)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(Float64(beta + 1.0) / Float64(beta + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 0.0033)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = (beta + 1.0) / (beta + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 0.0033], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 0.0033

   1. Initial program 64.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 64.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 98.7%

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

   if 0.0033 < beta

   1. Initial program 82.7%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 82.7%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 48.9%

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

   8. Taylor expanded in alpha around 0

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

      1. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 + \beta\right), \color{blue}{\left(2 + \beta\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\beta + 1\right), \left(\color{blue}{2} + \beta\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \left(\color{blue}{2} + \beta\right)\right) \]

      10. +-lowering-+.f64 80.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\beta, 1\right), \mathsf{+.f64}\left(2, \color{blue}{\beta}\right)\right) \]

   10. Simplified 80.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 0.0033:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\beta + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 320000000:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 320000000.0)
   (/ 1.0 (+ alpha 2.0))
   (+ 1.0 (/ (- -1.0 alpha) beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 320000000.0) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + ((-1.0 - alpha) / 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 <= 320000000.0d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 + (((-1.0d0) - alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 320000000.0) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 320000000.0:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = 1.0 + ((-1.0 - alpha) / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 320000000.0)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 - alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 320000000.0)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = 1.0 + ((-1.0 - alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 320000000.0], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 - alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e8

   1. Initial program 64.8%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 64.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 95.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 95.3%

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

   if 3.2e8 < beta

   1. Initial program 83.5%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 83.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 45.3%

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

   8. Taylor expanded in beta around inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(\frac{\alpha \cdot \left(1 + \frac{1}{\alpha}\right)}{\beta}\right)\right) \]

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\alpha \cdot \left(1 + \frac{1}{\alpha}\right)}{\beta}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha \cdot \left(1 + \frac{1}{\alpha}\right)\right), \color{blue}{\beta}\right)\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha \cdot 1 + \alpha \cdot \frac{1}{\alpha}\right), \beta\right)\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + \alpha \cdot \frac{1}{\alpha}\right), \beta\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\alpha + 1\right), \beta\right)\right) \]

      8. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, 1\right), \beta\right)\right) \]

   10. Simplified 82.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 320000000:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1 - \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 320000000:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 320000000.0) (/ 1.0 (+ alpha 2.0)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 320000000.0) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 320000000.0d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 320000000.0) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 320000000.0:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 320000000.0)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 320000000.0)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 320000000.0], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2e8

   1. Initial program 64.8%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right) \cdot -1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right), -1\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}\right)\right), -1\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)\right), -1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right), 1\right)\right)\right), -1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\beta - \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\alpha + \beta\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\left(\beta + \alpha\right) + 2\right)\right), 1\right)\right)\right), -1\right)\right) \]

      12. associate-+l+ N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \left(\beta + \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \left(\alpha + 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

      14. +-lowering-+.f64 64.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\beta, \alpha\right), \mathsf{+.f64}\left(\beta, \mathsf{+.f64}\left(\alpha, 2\right)\right)\right), 1\right)\right)\right), -1\right)\right) \]

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\alpha \cdot \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)}\right), -1\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}} + 2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right), -1\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\alpha, \mathsf{+.f64}\left(\left(-2 \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha \cdot {\left(2 + 2 \cdot \beta\right)}^{2}}\right), \left(2 \cdot \frac{1}{2 + 2 \cdot \beta}\right)\right)\right)\right), -1\right)\right) \]

   7. Simplified 96.6%

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

   8. Taylor expanded in beta around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\alpha \cdot \left(1 + 2 \cdot \frac{1}{\alpha}\right)\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\alpha \cdot \left(2 \cdot \frac{1}{\alpha} + \color{blue}{1}\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(2 \cdot \frac{1}{\alpha}\right) \cdot \alpha + \color{blue}{1 \cdot \alpha}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot \left(\frac{1}{\alpha} \cdot \alpha\right) + \color{blue}{1} \cdot \alpha\right)\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 \cdot 1 + 1 \cdot \alpha\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \color{blue}{1} \cdot \alpha\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(2 + \alpha\right)\right) \]

      8. +-lowering-+.f64 95.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\alpha}\right)\right) \]

   10. Simplified 95.3%

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

   if 3.2e8 < beta

   1. Initial program 83.5%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 82.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 82.0%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\beta}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\beta}\right)\right) \]

      5. /-lowering-/.f64 82.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\beta}\right)\right) \]

   8. Simplified 82.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 320000000:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.7:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 1.7) (+ 0.5 (* beta 0.25)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 1.7d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 1.7) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 1.7:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 1.7)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 1.7)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 1.7], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.69999999999999996

   1. Initial program 65.1%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 63.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 63.3%

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

   6. Taylor expanded in beta around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\beta \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 61.7%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 61.7%

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

   if 1.69999999999999996 < beta

   1. Initial program 82.2%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 79.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 79.9%

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

   6. Taylor expanded in beta around inf

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

      1. sub-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)} \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\beta}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\beta}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{-1}{\beta}\right)\right) \]

      5. /-lowering-/.f64 79.2%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \color{blue}{\beta}\right)\right) \]

   8. Simplified 79.2%

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

Alternative 7: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} 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 + (beta * 0.25d0)
    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 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	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 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 65.3%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 63.1%

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

   6. Taylor expanded in beta around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot \beta\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\beta \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\beta, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 61.4%

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

   if 2 < beta

   1. Initial program 82.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 79.7%

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

Alternative 8: 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 65.3%

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

   3. Taylor expanded in alpha around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

      2. +-lowering-+.f64 63.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

   5. Simplified 63.1%

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

   6. Taylor expanded in beta around 0

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

   8. Simplified 61.0%

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

   if 2 < beta

   1. Initial program 82.0%

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

   3. Taylor expanded in beta around inf

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

   5. Simplified 79.7%

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

Alternative 9: 49.8% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{\beta}{2 + \beta}\right)}, 1\right), 2\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \left(2 + \beta\right)\right), 1\right), 2\right) \]

   2. +-lowering-+.f64 69.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\beta, \mathsf{+.f64}\left(2, \beta\right)\right), 1\right), 2\right) \]

5. Simplified 69.6%

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

6. Taylor expanded in beta around 0

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

8. Simplified 44.2%

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

Reproduce

herbie shell --seed 2024192 
(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))