Octave 3.8, jcobi/1

Percentage Accurate: 75.3% → 99.7%

Time: 12.4s

Alternatives: 9

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: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.95)
     (/
      (fma
       (/ (- (- -2.0 beta) beta) alpha)
       (/ (+ beta 2.0) alpha)
       (/ (+ beta (+ beta 2.0)) alpha))
      2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.95) {
		tmp = fma((((-2.0 - beta) - beta) / alpha), ((beta + 2.0) / alpha), ((beta + (beta + 2.0)) / alpha)) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(fma(Float64(Float64(Float64(-2.0 - beta) - beta) / alpha), Float64(Float64(beta + 2.0) / alpha), Float64(Float64(beta + Float64(beta + 2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return 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, -0.95], N[(N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] - beta), $MachinePrecision] / alpha), $MachinePrecision] * N[(N[(beta + 2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.94999999999999996

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in alpha around -inf 95.8%

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

   6. Simplified 99.7%

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

   if -0.94999999999999996 < (/.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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(-2 - \beta\right) - \beta}{\alpha}, \frac{\beta + 2}{\alpha}, \frac{\beta + \left(\beta + 2\right)}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t_0 \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.99999998)
     (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)
     (/ (+ t_0 1.0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999998d0)) then
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.99999998) {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.99999998:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.99999998)
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.99999998)
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999998], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.999999980000000011

   1. Initial program 6.0%

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

      1. +-commutative 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in alpha around -inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. mul-1-neg 99.9%

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

      7. remove-double-neg 99.9%

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

      8. neg-mul-1 99.9%

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

      9. mul-1-neg 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   if -0.999999980000000011 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99999998:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]

Alternative 3: 68.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)) (t_1 (/ (/ 2.0 alpha) 2.0)))
   (if (<= beta -1.12e-94)
     t_0
     (if (<= beta -4e-115)
       t_1
       (if (<= beta 1.05e-299)
         t_0
         (if (<= beta 1.05e-231) t_1 (if (<= beta 2.0) t_0 1.0)))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.12e-94) {
		tmp = t_0;
	} else if (beta <= -4e-115) {
		tmp = t_1;
	} else if (beta <= 1.05e-299) {
		tmp = t_0;
	} else if (beta <= 1.05e-231) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    t_1 = (2.0d0 / alpha) / 2.0d0
    if (beta <= (-1.12d-94)) then
        tmp = t_0
    else if (beta <= (-4d-115)) then
        tmp = t_1
    else if (beta <= 1.05d-299) then
        tmp = t_0
    else if (beta <= 1.05d-231) then
        tmp = t_1
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.12e-94) {
		tmp = t_0;
	} else if (beta <= -4e-115) {
		tmp = t_1;
	} else if (beta <= 1.05e-299) {
		tmp = t_0;
	} else if (beta <= 1.05e-231) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	t_1 = (2.0 / alpha) / 2.0
	tmp = 0
	if beta <= -1.12e-94:
		tmp = t_0
	elif beta <= -4e-115:
		tmp = t_1
	elif beta <= 1.05e-299:
		tmp = t_0
	elif beta <= 1.05e-231:
		tmp = t_1
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	t_1 = Float64(Float64(2.0 / alpha) / 2.0)
	tmp = 0.0
	if (beta <= -1.12e-94)
		tmp = t_0;
	elseif (beta <= -4e-115)
		tmp = t_1;
	elseif (beta <= 1.05e-299)
		tmp = t_0;
	elseif (beta <= 1.05e-231)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	t_1 = (2.0 / alpha) / 2.0;
	tmp = 0.0;
	if (beta <= -1.12e-94)
		tmp = t_0;
	elseif (beta <= -4e-115)
		tmp = t_1;
	elseif (beta <= 1.05e-299)
		tmp = t_0;
	elseif (beta <= 1.05e-231)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -1.12e-94], t$95$0, If[LessEqual[beta, -4e-115], t$95$1, If[LessEqual[beta, 1.05e-299], t$95$0, If[LessEqual[beta, 1.05e-231], t$95$1, If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.12e-94 or -4.0000000000000002e-115 < beta < 1.05000000000000005e-299 or 1.04999999999999995e-231 < beta < 2

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in alpha around 0 66.3%

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

   5. Taylor expanded in beta around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   6. Step-by-step derivation

      1. *-commutative 64.7%

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

   7. Simplified 64.7%

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

   if -1.12e-94 < beta < -4.0000000000000002e-115 or 1.05000000000000005e-299 < beta < 1.04999999999999995e-231

   1. Initial program 26.0%

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

      1. +-commutative 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in beta around 0 26.0%

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

      1. +-commutative 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in alpha around inf 79.8%

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

   if 2 < beta

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in beta around inf 87.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.12 \cdot 10^{-94}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.85 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -2.2 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (* beta 0.5)) 2.0)) (t_1 (/ (/ 2.0 alpha) 2.0)))
   (if (<= beta -1.85e-96)
     t_0
     (if (<= beta -2.2e-114)
       t_1
       (if (<= beta 3.1e-299)
         t_0
         (if (<= beta 1e-231)
           t_1
           (if (<= beta 2.0) t_0 (/ (- 2.0 (/ 2.0 beta)) 2.0))))))))

C

double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.85e-96) {
		tmp = t_0;
	} else if (beta <= -2.2e-114) {
		tmp = t_1;
	} else if (beta <= 3.1e-299) {
		tmp = t_0;
	} else if (beta <= 1e-231) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + (beta * 0.5d0)) / 2.0d0
    t_1 = (2.0d0 / alpha) / 2.0d0
    if (beta <= (-1.85d-96)) then
        tmp = t_0
    else if (beta <= (-2.2d-114)) then
        tmp = t_1
    else if (beta <= 3.1d-299) then
        tmp = t_0
    else if (beta <= 1d-231) then
        tmp = t_1
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = (2.0d0 - (2.0d0 / beta)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (1.0 + (beta * 0.5)) / 2.0;
	double t_1 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.85e-96) {
		tmp = t_0;
	} else if (beta <= -2.2e-114) {
		tmp = t_1;
	} else if (beta <= 3.1e-299) {
		tmp = t_0;
	} else if (beta <= 1e-231) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (1.0 + (beta * 0.5)) / 2.0
	t_1 = (2.0 / alpha) / 2.0
	tmp = 0
	if beta <= -1.85e-96:
		tmp = t_0
	elif beta <= -2.2e-114:
		tmp = t_1
	elif beta <= 3.1e-299:
		tmp = t_0
	elif beta <= 1e-231:
		tmp = t_1
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = (2.0 - (2.0 / beta)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(1.0 + Float64(beta * 0.5)) / 2.0)
	t_1 = Float64(Float64(2.0 / alpha) / 2.0)
	tmp = 0.0
	if (beta <= -1.85e-96)
		tmp = t_0;
	elseif (beta <= -2.2e-114)
		tmp = t_1;
	elseif (beta <= 3.1e-299)
		tmp = t_0;
	elseif (beta <= 1e-231)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 - Float64(2.0 / beta)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (1.0 + (beta * 0.5)) / 2.0;
	t_1 = (2.0 / alpha) / 2.0;
	tmp = 0.0;
	if (beta <= -1.85e-96)
		tmp = t_0;
	elseif (beta <= -2.2e-114)
		tmp = t_1;
	elseif (beta <= 3.1e-299)
		tmp = t_0;
	elseif (beta <= 1e-231)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = (2.0 - (2.0 / beta)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(1.0 + N[(beta * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -1.85e-96], t$95$0, If[LessEqual[beta, -2.2e-114], t$95$1, If[LessEqual[beta, 3.1e-299], t$95$0, If[LessEqual[beta, 1e-231], t$95$1, If[LessEqual[beta, 2.0], t$95$0, N[(N[(2.0 - N[(2.0 / beta), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.84999999999999993e-96 or -2.20000000000000011e-114 < beta < 3.1e-299 or 9.9999999999999999e-232 < beta < 2

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in alpha around 0 66.3%

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

   5. Taylor expanded in beta around 0 64.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \beta} + 1}{2} \]
   6. Step-by-step derivation

      1. *-commutative 64.7%

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

   7. Simplified 64.7%

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

   if -1.84999999999999993e-96 < beta < -2.20000000000000011e-114 or 3.1e-299 < beta < 9.9999999999999999e-232

   1. Initial program 26.0%

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

      1. +-commutative 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in beta around 0 26.0%

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

      1. +-commutative 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in alpha around inf 79.8%

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

   if 2 < beta

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in alpha around 0 88.2%

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

   5. Taylor expanded in beta around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. metadata-eval 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.85 \cdot 10^{-96}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -2.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 10^{-231}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 5: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -1.25 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-299}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 alpha) 2.0)))
   (if (<= beta -1.25e-95)
     0.5
     (if (<= beta -1.4e-115)
       t_0
       (if (<= beta 2.3e-299)
         0.5
         (if (<= beta 1.75e-231) t_0 (if (<= beta 2.0) 0.5 1.0)))))))

C

double code(double alpha, double beta) {
	double t_0 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.25e-95) {
		tmp = 0.5;
	} else if (beta <= -1.4e-115) {
		tmp = t_0;
	} else if (beta <= 2.3e-299) {
		tmp = 0.5;
	} else if (beta <= 1.75e-231) {
		tmp = t_0;
	} else 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) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / alpha) / 2.0d0
    if (beta <= (-1.25d-95)) then
        tmp = 0.5d0
    else if (beta <= (-1.4d-115)) then
        tmp = t_0
    else if (beta <= 2.3d-299) then
        tmp = 0.5d0
    else if (beta <= 1.75d-231) then
        tmp = t_0
    else 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 t_0 = (2.0 / alpha) / 2.0;
	double tmp;
	if (beta <= -1.25e-95) {
		tmp = 0.5;
	} else if (beta <= -1.4e-115) {
		tmp = t_0;
	} else if (beta <= 2.3e-299) {
		tmp = 0.5;
	} else if (beta <= 1.75e-231) {
		tmp = t_0;
	} else if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (2.0 / alpha) / 2.0
	tmp = 0
	if beta <= -1.25e-95:
		tmp = 0.5
	elif beta <= -1.4e-115:
		tmp = t_0
	elif beta <= 2.3e-299:
		tmp = 0.5
	elif beta <= 1.75e-231:
		tmp = t_0
	elif beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(2.0 / alpha) / 2.0)
	tmp = 0.0
	if (beta <= -1.25e-95)
		tmp = 0.5;
	elseif (beta <= -1.4e-115)
		tmp = t_0;
	elseif (beta <= 2.3e-299)
		tmp = 0.5;
	elseif (beta <= 1.75e-231)
		tmp = t_0;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (2.0 / alpha) / 2.0;
	tmp = 0.0;
	if (beta <= -1.25e-95)
		tmp = 0.5;
	elseif (beta <= -1.4e-115)
		tmp = t_0;
	elseif (beta <= 2.3e-299)
		tmp = 0.5;
	elseif (beta <= 1.75e-231)
		tmp = t_0;
	elseif (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -1.25e-95], 0.5, If[LessEqual[beta, -1.4e-115], t$95$0, If[LessEqual[beta, 2.3e-299], 0.5, If[LessEqual[beta, 1.75e-231], t$95$0, If[LessEqual[beta, 2.0], 0.5, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -1.2499999999999999e-95 or -1.39999999999999994e-115 < beta < 2.3000000000000001e-299 or 1.7500000000000001e-231 < beta < 2

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in beta around 0 66.7%

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

      1. +-commutative 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in alpha around 0 64.1%

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

   if -1.2499999999999999e-95 < beta < -1.39999999999999994e-115 or 2.3000000000000001e-299 < beta < 1.7500000000000001e-231

   1. Initial program 26.0%

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

      1. +-commutative 26.0%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in beta around 0 26.0%

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

      1. +-commutative 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in alpha around inf 79.8%

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

   if 2 < beta

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in beta around inf 87.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -1.25 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq -1.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-299}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{-231}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 87.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 31.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 31.5)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 31.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 31.5d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (2.0d0 / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 31.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 31.5:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 31.5)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 31.5)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 31.5], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 31.5

   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 99.3%

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

   if 31.5 < alpha

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in beta around 0 7.8%

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

      1. +-commutative 7.8%

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

   6. Simplified 7.8%

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

   7. Taylor expanded in alpha around inf 71.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 31.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 31.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 31.5)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ beta (+ beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 31.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 31.5d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((beta + (beta + 2.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 31.5) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 31.5:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 31.5)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta + Float64(beta + 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 31.5)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((beta + (beta + 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 31.5], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 31.5

   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 99.3%

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

   if 31.5 < alpha

   1. Initial program 23.3%

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

      1. +-commutative 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in alpha around -inf 83.2%

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

      1. associate-*r/ 83.2%

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

      2. sub-neg 83.2%

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

      3. mul-1-neg 83.2%

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

      4. distribute-lft-in 83.2%

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

      5. neg-mul-1 83.2%

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

      6. mul-1-neg 83.2%

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

      7. remove-double-neg 83.2%

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

      8. neg-mul-1 83.2%

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

      9. mul-1-neg 83.2%

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

      10. remove-double-neg 83.2%

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

      11. +-commutative 83.2%

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

   6. Simplified 83.2%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 31.5:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 71.7% accurate, 4.2× 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 62.7%

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

      1. +-commutative 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in beta around 0 60.8%

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

      1. +-commutative 60.8%

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

   6. Simplified 60.8%

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

   7. Taylor expanded in alpha around 0 58.3%

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

   if 2 < beta

   1. Initial program 88.4%

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

      1. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in beta around inf 87.4%

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

4. Final simplification 68.7%

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

Alternative 9: 49.3% 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 71.8%

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

   1. +-commutative 71.8%

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

3. Simplified 71.8%

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

4. Taylor expanded in beta around 0 44.3%

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

   1. +-commutative 44.3%

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

6. Simplified 44.3%

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

7. Taylor expanded in alpha around 0 43.7%

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

8. Final simplification 43.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023196 
(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))