Octave 3.8, jcobi/1

Percentage Accurate: 75.5% → 99.4%

Time: 5.5s

Alternatives: 8

Speedup: 1.2×

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: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -1.0)
   (/ (+ beta 1.0) alpha)
   (/ (exp (log1p (/ (- beta alpha) (+ beta (+ alpha 2.0))))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = exp(log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = Math.exp(Math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -1.0:
		tmp = (beta + 1.0) / alpha
	else:
		tmp = math.exp(math.log1p(((beta - alpha) / (beta + (alpha + 2.0))))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = Float64(exp(log1p(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[Exp[N[Log[1 + N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -1

   1. Initial program 5.6%

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

      1. +-commutative 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in alpha around inf 100.0%

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

   5. Taylor expanded in alpha around 0 100.0%

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

      1. metadata-eval 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. times-frac 100.0%

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

      5. *-lft-identity 100.0%

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

      6. fma-udef 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-lft1-in 100.0%

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

      9. *-commutative 100.0%

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

      10. times-frac 100.0%

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

      11. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. +-commutative 100.0%

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

      3. log1p-udef 100.0%

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

      4. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -1.0) (/ (+ 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 <= -1.0) {
		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 <= (-1.0d0)) 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 <= -1.0) {
		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 <= -1.0:
		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 <= -1.0)
		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 <= -1.0)
		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, -1.0], 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) 2)) < -1

   1. Initial program 5.6%

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

      1. +-commutative 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in alpha around inf 100.0%

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

   5. Taylor expanded in alpha around 0 100.0%

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

      1. metadata-eval 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-udef 100.0%

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

      4. times-frac 100.0%

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

      5. *-lft-identity 100.0%

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

      6. fma-udef 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-lft1-in 100.0%

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

      9. *-commutative 100.0%

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

      10. times-frac 100.0%

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

      11. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{if}\;\alpha \leq 5 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (* alpha 0.5)) 2.0)))
   (if (<= alpha 5e-287)
     t_0
     (if (<= alpha 1e-271)
       1.0
       (if (<= alpha 1.95) t_0 (/ (+ beta 1.0) alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 5e-287) {
		tmp = t_0;
	} else if (alpha <= 1e-271) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    if (alpha <= 5d-287) then
        tmp = t_0
    else if (alpha <= 1d-271) then
        tmp = 1.0d0
    else if (alpha <= 1.95d0) then
        tmp = t_0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	double tmp;
	if (alpha <= 5e-287) {
		tmp = t_0;
	} else if (alpha <= 1e-271) {
		tmp = 1.0;
	} else if (alpha <= 1.95) {
		tmp = t_0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 - (alpha * 0.5)) / 2.0
	tmp = 0
	if alpha <= 5e-287:
		tmp = t_0
	elif alpha <= 1e-271:
		tmp = 1.0
	elif alpha <= 1.95:
		tmp = t_0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0)
	tmp = 0.0
	if (alpha <= 5e-287)
		tmp = t_0;
	elseif (alpha <= 1e-271)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 - (alpha * 0.5)) / 2.0;
	tmp = 0.0;
	if (alpha <= 5e-287)
		tmp = t_0;
	elseif (alpha <= 1e-271)
		tmp = 1.0;
	elseif (alpha <= 1.95)
		tmp = t_0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[alpha, 5e-287], t$95$0, If[LessEqual[alpha, 1e-271], 1.0, If[LessEqual[alpha, 1.95], t$95$0, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 5.00000000000000025e-287 or 9.99999999999999963e-272 < alpha < 1.94999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in beta around 0 73.3%

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

      1. +-commutative 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in alpha around 0 72.3%

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

      1. *-commutative 72.3%

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

   9. Simplified 72.3%

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

   if 5.00000000000000025e-287 < alpha < 9.99999999999999963e-272

   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. Taylor expanded in beta around inf 88.4%

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

   if 1.94999999999999996 < alpha

   1. Initial program 21.9%

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

      1. +-commutative 21.9%

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

   3. Simplified 21.9%

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

   4. Taylor expanded in alpha around inf 83.8%

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

   5. Taylor expanded in alpha around 0 83.8%

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

      1. metadata-eval 83.8%

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

      2. +-commutative 83.8%

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

      3. fma-udef 83.8%

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

      4. times-frac 83.8%

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

      5. *-lft-identity 83.8%

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

      6. fma-udef 83.8%

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

      7. *-commutative 83.8%

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

      8. distribute-lft1-in 83.8%

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

      9. *-commutative 83.8%

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

      10. times-frac 83.8%

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

      11. metadata-eval 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\alpha \leq 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.95:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 4: 73.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{-287}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8e-287)
   0.5
   (if (<= alpha 6.2e-272)
     1.0
     (if (<= alpha 80000000000.0) 0.5 (/ (+ beta 1.0) alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8e-287) {
		tmp = 0.5;
	} else if (alpha <= 6.2e-272) {
		tmp = 1.0;
	} else if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 8d-287) then
        tmp = 0.5d0
    else if (alpha <= 6.2d-272) then
        tmp = 1.0d0
    else if (alpha <= 80000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8e-287) {
		tmp = 0.5;
	} else if (alpha <= 6.2e-272) {
		tmp = 1.0;
	} else if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 8e-287:
		tmp = 0.5
	elif alpha <= 6.2e-272:
		tmp = 1.0
	elif alpha <= 80000000000.0:
		tmp = 0.5
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 8e-287)
		tmp = 0.5;
	elseif (alpha <= 6.2e-272)
		tmp = 1.0;
	elseif (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 8e-287)
		tmp = 0.5;
	elseif (alpha <= 6.2e-272)
		tmp = 1.0;
	elseif (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 8e-287], 0.5, If[LessEqual[alpha, 6.2e-272], 1.0, If[LessEqual[alpha, 80000000000.0], 0.5, N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 8.00000000000000017e-287 or 6.20000000000000059e-272 < alpha < 8e10

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. +-commutative 100.0%

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

      3. log1p-udef 100.0%

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

      4. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 97.9%

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

   9. Taylor expanded in beta around 0 70.9%

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

   if 8.00000000000000017e-287 < alpha < 6.20000000000000059e-272

   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. Taylor expanded in beta around inf 88.4%

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

   if 8e10 < alpha

   1. Initial program 21.1%

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

      1. +-commutative 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in alpha around inf 84.6%

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

   5. Taylor expanded in alpha around 0 84.6%

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

      1. metadata-eval 84.6%

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

      2. +-commutative 84.6%

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

      3. fma-udef 84.6%

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

      4. times-frac 84.6%

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

      5. *-lft-identity 84.6%

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

      6. fma-udef 84.6%

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

      7. *-commutative 84.6%

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

      8. distribute-lft1-in 84.6%

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

      9. *-commutative 84.6%

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

      10. times-frac 84.6%

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

      11. metadata-eval 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8 \cdot 10^{-287}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 5: 93.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 680000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 680000000000.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 680000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 680000000000.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 680000000000.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 680000000000.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 680000000000.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 680000000000.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 680000000000.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.8e11

   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. Taylor expanded in alpha around 0 98.0%

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

   if 6.8e11 < alpha

   1. Initial program 21.1%

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

      1. +-commutative 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in alpha around inf 84.6%

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

   5. Taylor expanded in alpha around 0 84.6%

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

      1. metadata-eval 84.6%

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

      2. +-commutative 84.6%

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

      3. fma-udef 84.6%

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

      4. times-frac 84.6%

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

      5. *-lft-identity 84.6%

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

      6. fma-udef 84.6%

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

      7. *-commutative 84.6%

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

      8. distribute-lft1-in 84.6%

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

      9. *-commutative 84.6%

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

      10. times-frac 84.6%

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

      11. metadata-eval 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 680000000000:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 6: 67.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 7.2 \cdot 10^{-287}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 7.2e-287)
   0.5
   (if (<= alpha 6.2e-272)
     1.0
     (if (<= alpha 80000000000.0) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.2e-287) {
		tmp = 0.5;
	} else if (alpha <= 6.2e-272) {
		tmp = 1.0;
	} else if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 7.2d-287) then
        tmp = 0.5d0
    else if (alpha <= 6.2d-272) then
        tmp = 1.0d0
    else if (alpha <= 80000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 7.2e-287) {
		tmp = 0.5;
	} else if (alpha <= 6.2e-272) {
		tmp = 1.0;
	} else if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 7.2e-287:
		tmp = 0.5
	elif alpha <= 6.2e-272:
		tmp = 1.0
	elif alpha <= 80000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 7.2e-287)
		tmp = 0.5;
	elseif (alpha <= 6.2e-272)
		tmp = 1.0;
	elseif (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 7.2e-287)
		tmp = 0.5;
	elseif (alpha <= 6.2e-272)
		tmp = 1.0;
	elseif (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 7.2e-287], 0.5, If[LessEqual[alpha, 6.2e-272], 1.0, If[LessEqual[alpha, 80000000000.0], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < 7.2000000000000003e-287 or 6.20000000000000059e-272 < alpha < 8e10

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. +-commutative 100.0%

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

      3. log1p-udef 100.0%

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

      4. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 97.9%

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

   9. Taylor expanded in beta around 0 70.9%

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

   if 7.2000000000000003e-287 < alpha < 6.20000000000000059e-272

   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. Taylor expanded in beta around inf 88.4%

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

   if 8e10 < alpha

   1. Initial program 21.1%

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

      1. +-commutative 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in alpha around inf 84.6%

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

   5. Taylor expanded in beta around 0 76.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 7.2 \cdot 10^{-287}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 7: 68.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 80000000000.0) 0.5 (/ 1.0 alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 80000000000.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 80000000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 80000000000.0:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 80000000000.0)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 80000000000.0], 0.5, N[(1.0 / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 8e10

   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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. +-commutative 100.0%

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

      3. log1p-udef 100.0%

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

      4. associate-+l+ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in alpha around 0 98.0%

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

   9. Taylor expanded in beta around 0 69.1%

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

   if 8e10 < alpha

   1. Initial program 21.1%

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

      1. +-commutative 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in alpha around inf 84.6%

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

   5. Taylor expanded in beta around 0 76.4%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 80000000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8: 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 70.1%

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

   1. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. add-exp-log 70.1%

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

   2. +-commutative 70.1%

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

   3. log1p-udef 70.1%

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

   4. associate-+l+ 70.1%

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

5. Applied egg-rr 70.1%

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

   1. +-commutative 70.1%

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

7. Simplified 70.1%

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

8. Taylor expanded in alpha around 0 68.6%

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

9. Taylor expanded in beta around 0 45.7%

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

10. Final simplification 45.7%

    \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023178 
(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))