Octave 3.8, jcobi/1

Percentage Accurate: 74.5% → 99.6%

Time: 13.1s

Alternatives: 8

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: 74.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999995)
   (/ (+ 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)) <= -0.999999995) {
		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)) <= -0.999999995) {
		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)) <= -0.999999995:
		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)) <= -0.999999995)
		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], -0.999999995], 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) #s(literal 2 binary64))) < -0.99999999500000003

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in alpha around inf 99.9%

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

   6. Taylor expanded in alpha around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -0.99999999500000003 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-exp-log 99.8%

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

      2. +-commutative 99.8%

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

      3. log1p-define 99.8%

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

      4. associate-+l+ 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.999999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.999999995) (/ (+ 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 <= -0.999999995) {
		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 <= (-0.999999995d0)) 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 <= -0.999999995) {
		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 <= -0.999999995:
		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 <= -0.999999995)
		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 <= -0.999999995)
		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, -0.999999995], 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))) < -0.99999999500000003

   1. Initial program 5.8%

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

      1. +-commutative 5.8%

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

   3. Simplified 5.8%

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

   5. Taylor expanded in alpha around inf 99.9%

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

   6. Taylor expanded in alpha around 0 99.9%

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

      1. associate-*r/ 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

   if -0.99999999500000003 < (/.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999995:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \alpha \cdot -0.25\\ \mathbf{if}\;\alpha \leq -5.2 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\alpha \leq -1.55 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* alpha -0.25))))
   (if (<= alpha -5.2e-97)
     t_0
     (if (<= alpha -1.55e-206) 1.0 (if (<= alpha 0.92) t_0 (/ 1.0 alpha))))))

C

double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5.2e-97) {
		tmp = t_0;
	} else if (alpha <= -1.55e-206) {
		tmp = 1.0;
	} else if (alpha <= 0.92) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (alpha * (-0.25d0))
    if (alpha <= (-5.2d-97)) then
        tmp = t_0
    else if (alpha <= (-1.55d-206)) then
        tmp = 1.0d0
    else if (alpha <= 0.92d0) then
        tmp = t_0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 0.5 + (alpha * -0.25);
	double tmp;
	if (alpha <= -5.2e-97) {
		tmp = t_0;
	} else if (alpha <= -1.55e-206) {
		tmp = 1.0;
	} else if (alpha <= 0.92) {
		tmp = t_0;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 0.5 + (alpha * -0.25)
	tmp = 0
	if alpha <= -5.2e-97:
		tmp = t_0
	elif alpha <= -1.55e-206:
		tmp = 1.0
	elif alpha <= 0.92:
		tmp = t_0
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(0.5 + Float64(alpha * -0.25))
	tmp = 0.0
	if (alpha <= -5.2e-97)
		tmp = t_0;
	elseif (alpha <= -1.55e-206)
		tmp = 1.0;
	elseif (alpha <= 0.92)
		tmp = t_0;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 0.5 + (alpha * -0.25);
	tmp = 0.0;
	if (alpha <= -5.2e-97)
		tmp = t_0;
	elseif (alpha <= -1.55e-206)
		tmp = 1.0;
	elseif (alpha <= 0.92)
		tmp = t_0;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(0.5 + N[(alpha * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, -5.2e-97], t$95$0, If[LessEqual[alpha, -1.55e-206], 1.0, If[LessEqual[alpha, 0.92], t$95$0, N[(1.0 / alpha), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -5.20000000000000014e-97 or -1.5500000000000001e-206 < alpha < 0.92000000000000004

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in beta around 0 75.7%

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

      1. +-commutative 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in alpha around 0 75.0%

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

      1. *-commutative 75.0%

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

   10. Simplified 75.0%

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

   if -5.20000000000000014e-97 < alpha < -1.5500000000000001e-206

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

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

   6. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-inv 100.0%

         \[\leadsto e^{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)} \cdot \frac{1}{2}\right)}} \]

      3. metadata-eval 100.0%

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

      4. log-prod 100.0%

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

      5. add-log-exp 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \left(\frac{\color{blue}{1}}{2}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \color{blue}{0.5}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log 0.5}} \]

   9. Taylor expanded in beta around inf 68.1%

      \[\leadsto \color{blue}{e^{\log 0.5 + \log 2}} \]
   10. Step-by-step derivation

       1. exp-sum 68.1%

          \[\leadsto \color{blue}{e^{\log 0.5} \cdot e^{\log 2}} \]

       2. rem-exp-log 68.1%

          \[\leadsto \color{blue}{0.5} \cdot e^{\log 2} \]

       3. rem-exp-log 68.1%

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

       4. metadata-eval 68.1%

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

   11. Simplified 68.1%

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

   if 0.92000000000000004 < alpha

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

   3. Simplified 22.2%

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

   5. Taylor expanded in beta around 0 7.8%

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

      1. +-commutative 7.8%

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

   7. Simplified 7.8%

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

   8. Taylor expanded in alpha around inf 69.6%

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

Alternative 4: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -1.3 \cdot 10^{-96}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -5 \cdot 10^{-206}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 1.7:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -1.3e-96)
   0.5
   (if (<= alpha -5e-206) 1.0 (if (<= alpha 1.7) 0.5 (/ 1.0 alpha)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.3e-96) {
		tmp = 0.5;
	} else if (alpha <= -5e-206) {
		tmp = 1.0;
	} else if (alpha <= 1.7) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= (-1.3d-96)) then
        tmp = 0.5d0
    else if (alpha <= (-5d-206)) then
        tmp = 1.0d0
    else if (alpha <= 1.7d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -1.3e-96) {
		tmp = 0.5;
	} else if (alpha <= -5e-206) {
		tmp = 1.0;
	} else if (alpha <= 1.7) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -1.3e-96:
		tmp = 0.5
	elif alpha <= -5e-206:
		tmp = 1.0
	elif alpha <= 1.7:
		tmp = 0.5
	else:
		tmp = 1.0 / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -1.3e-96)
		tmp = 0.5;
	elseif (alpha <= -5e-206)
		tmp = 1.0;
	elseif (alpha <= 1.7)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= -1.3e-96)
		tmp = 0.5;
	elseif (alpha <= -5e-206)
		tmp = 1.0;
	elseif (alpha <= 1.7)
		tmp = 0.5;
	else
		tmp = 1.0 / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -1.3e-96], 0.5, If[LessEqual[alpha, -5e-206], 1.0, If[LessEqual[alpha, 1.7], 0.5, N[(1.0 / alpha), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -1.3000000000000001e-96 or -5e-206 < alpha < 1.69999999999999996

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 98.3%

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

   6. Taylor expanded in beta around 0 74.0%

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

   if -1.3000000000000001e-96 < alpha < -5e-206

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

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

   6. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-inv 100.0%

         \[\leadsto e^{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)} \cdot \frac{1}{2}\right)}} \]

      3. metadata-eval 100.0%

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

      4. log-prod 100.0%

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

      5. add-log-exp 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \left(\frac{\color{blue}{1}}{2}\right)} \]

      7. metadata-eval 100.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \color{blue}{0.5}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log 0.5}} \]

   9. Taylor expanded in beta around inf 68.1%

      \[\leadsto \color{blue}{e^{\log 0.5 + \log 2}} \]
   10. Step-by-step derivation

       1. exp-sum 68.1%

          \[\leadsto \color{blue}{e^{\log 0.5} \cdot e^{\log 2}} \]

       2. rem-exp-log 68.1%

          \[\leadsto \color{blue}{0.5} \cdot e^{\log 2} \]

       3. rem-exp-log 68.1%

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

       4. metadata-eval 68.1%

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

   11. Simplified 68.1%

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

   if 1.69999999999999996 < alpha

   1. Initial program 22.2%

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

      1. +-commutative 22.2%

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

   3. Simplified 22.2%

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

   5. Taylor expanded in beta around 0 7.8%

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

      1. +-commutative 7.8%

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

   7. Simplified 7.8%

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

   8. Taylor expanded in alpha around inf 69.6%

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

Alternative 5: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.15e-27)
   (/ 1.0 (+ alpha 2.0))
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.15e-27) {
		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 <= 2.15d-27) 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 <= 2.15e-27) {
		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 <= 2.15e-27:
		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 <= 2.15e-27)
		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 <= 2.15e-27)
		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, 2.15e-27], 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 < 2.15000000000000001e-27

   1. Initial program 64.4%

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

      1. +-commutative 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in beta around 0 63.5%

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

      1. +-commutative 63.5%

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

   7. Simplified 63.5%

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

      1. clear-num 63.5%

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

      2. inv-pow 63.5%

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

   9. Applied egg-rr 63.5%

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

       1. unpow-1 63.5%

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

   11. Simplified 63.5%

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

   12. Taylor expanded in alpha around 0 99.1%

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

       1. +-commutative 99.1%

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

   14. Simplified 99.1%

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

   if 2.15000000000000001e-27 < beta

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in alpha around 0 82.9%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.15 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 22.5:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 22.5) (/ 1.0 (+ alpha 2.0)) (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 22.5) {
		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 <= 22.5d0) 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 <= 22.5) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 22.5:
		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 <= 22.5)
		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 <= 22.5)
		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, 22.5], 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 < 22.5

   1. Initial program 65.2%

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

      1. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in beta around 0 64.1%

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

      1. +-commutative 64.1%

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

   7. Simplified 64.1%

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

      1. clear-num 64.1%

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

      2. inv-pow 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. unpow-1 64.1%

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

   11. Simplified 64.1%

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

   12. Taylor expanded in alpha around 0 98.8%

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

       1. +-commutative 98.8%

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

   14. Simplified 98.8%

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

   if 22.5 < beta

   1. Initial program 83.2%

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

      1. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in alpha around 0 82.1%

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

   6. Taylor expanded in beta around inf 81.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 22.5:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% 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 65.2%

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

      1. +-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in alpha around 0 62.4%

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

   6. Taylor expanded in beta around 0 61.2%

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

   if 2 < beta

   1. Initial program 83.2%

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

      1. +-commutative 83.2%

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

   3. Simplified 83.2%

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

      1. add-exp-log 83.2%

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

      2. +-commutative 83.2%

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

      3. log1p-define 83.2%

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

      4. associate-+l+ 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. add-exp-log 83.2%

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

      2. div-inv 83.2%

         \[\leadsto e^{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right)} \cdot \frac{1}{2}\right)}} \]

      3. metadata-eval 83.2%

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

      4. log-prod 83.2%

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

      5. add-log-exp 83.2%

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

      6. metadata-eval 83.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \left(\frac{\color{blue}{1}}{2}\right)} \]

      7. metadata-eval 83.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log \color{blue}{0.5}} \]

   8. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}\right) + \log 0.5}} \]

   9. Taylor expanded in beta around inf 80.3%

      \[\leadsto \color{blue}{e^{\log 0.5 + \log 2}} \]
   10. Step-by-step derivation

       1. exp-sum 80.3%

          \[\leadsto \color{blue}{e^{\log 0.5} \cdot e^{\log 2}} \]

       2. rem-exp-log 80.3%

          \[\leadsto \color{blue}{0.5} \cdot e^{\log 2} \]

       3. rem-exp-log 80.3%

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

       4. metadata-eval 80.3%

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

   11. Simplified 80.3%

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

Alternative 8: 49.5% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 71.1%

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

   1. +-commutative 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in alpha around 0 68.8%

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

6. Taylor expanded in beta around 0 46.5%

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

Reproduce

herbie shell --seed 2024110 
(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))