Octave 3.8, jcobi/1

Percentage Accurate: 76.0% → 99.9%

Time: 8.6s

Alternatives: 11

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: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{-2 - \left(\beta + \beta\right)}{\alpha} \cdot \left(\beta + 2\right) + \left(\left(\beta + \beta\right) - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999)
     (/
      (/
       (+
        (* (/ (- -2.0 (+ beta beta)) alpha) (+ beta 2.0))
        (- (+ beta beta) -2.0))
       alpha)
      2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = (((((-2.0 - (beta + beta)) / alpha) * (beta + 2.0)) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_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) + 2.0d0
    if (((beta - alpha) / t_0) <= (-0.9999d0)) then
        tmp = ((((((-2.0d0) - (beta + beta)) / alpha) * (beta + 2.0d0)) + ((beta + beta) - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = (((((-2.0 - (beta + beta)) / alpha) * (beta + 2.0)) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.9999:
		tmp = (((((-2.0 - (beta + beta)) / alpha) * (beta + 2.0)) + ((beta + beta) - -2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 - Float64(beta + beta)) / alpha) * Float64(beta + 2.0)) + Float64(Float64(beta + beta) - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.9999)
		tmp = (((((-2.0 - (beta + beta)) / alpha) * (beta + 2.0)) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.9999], N[(N[(N[(N[(N[(N[(-2.0 - N[(beta + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99990000000000001

   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 99.9%

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

   6. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-div 100.0%

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

      3. associate--l- 100.0%

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

      4. +-commutative 100.0%

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

      5. associate--l- 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{-2 - \left(\beta + \beta\right)}{\alpha} \cdot \left(\beta + 2\right) + \left(\left(\beta + \beta\right) - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{-2 \cdot \left(\beta \cdot \beta\right)}{\alpha} + \left(\left(\beta + \beta\right) - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999)
     (/
      (/ (+ (/ (* -2.0 (* beta beta)) alpha) (- (+ beta beta) -2.0)) alpha)
      2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = ((((-2.0 * (beta * beta)) / alpha) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_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) + 2.0d0
    if (((beta - alpha) / t_0) <= (-0.9999d0)) then
        tmp = (((((-2.0d0) * (beta * beta)) / alpha) + ((beta + beta) - (-2.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = ((((-2.0 * (beta * beta)) / alpha) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.9999:
		tmp = ((((-2.0 * (beta * beta)) / alpha) + ((beta + beta) - -2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(beta * beta)) / alpha) + Float64(Float64(beta + beta) - -2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.9999)
		tmp = ((((-2.0 * (beta * beta)) / alpha) + ((beta + beta) - -2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.9999], N[(N[(N[(N[(N[(-2.0 * N[(beta * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99990000000000001

   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 99.9%

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

   6. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. sub-div 100.0%

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

      3. associate--l- 100.0%

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

      4. +-commutative 100.0%

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

      5. associate--l- 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in beta around inf 99.3%

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

       1. associate-*r/ 99.3%

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

       2. unpow2 99.3%

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

   11. Simplified 99.3%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{\frac{-2 \cdot \left(\beta \cdot \beta\right)}{\alpha} + \left(\left(\beta + \beta\right) - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\frac{\beta - \alpha}{t_0} \leq -0.9999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{t_0}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= (/ (- beta alpha) t_0) -0.9999)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (- 1.0 (/ (- alpha beta) t_0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_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) + 2.0d0
    if (((beta - alpha) / t_0) <= (-0.9999d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 - ((alpha - beta) / t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (((beta - alpha) / t_0) <= -0.9999) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if ((beta - alpha) / t_0) <= -0.9999:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / t_0) <= -0.9999)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(alpha - beta) / t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (((beta - alpha) / t_0) <= -0.9999)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 - ((alpha - beta) / t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision], -0.9999], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(N[(alpha - beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99990000000000001

   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 98.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{\left(\beta + \alpha\right) + 2}}{2}\\ \end{array} \]

Alternative 4: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq -5.5 \cdot 10^{-303}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{-174}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -3.4e-174)
     t_0
     (if (<= beta -1.5e-207)
       t_1
       (if (<= beta -5.5e-303)
         0.5
         (if (<= beta 2.05e-293)
           t_1
           (if (<= beta 7e-174)
             0.5
             (if (<= beta 2.7e-127) 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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -1.5e-207) {
		tmp = t_1;
	} else if (beta <= -5.5e-303) {
		tmp = 0.5;
	} else if (beta <= 2.05e-293) {
		tmp = t_1;
	} else if (beta <= 7e-174) {
		tmp = 0.5;
	} else if (beta <= 2.7e-127) {
		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 <= (-3.4d-174)) then
        tmp = t_0
    else if (beta <= (-1.5d-207)) then
        tmp = t_1
    else if (beta <= (-5.5d-303)) then
        tmp = 0.5d0
    else if (beta <= 2.05d-293) then
        tmp = t_1
    else if (beta <= 7d-174) then
        tmp = 0.5d0
    else if (beta <= 2.7d-127) 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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -1.5e-207) {
		tmp = t_1;
	} else if (beta <= -5.5e-303) {
		tmp = 0.5;
	} else if (beta <= 2.05e-293) {
		tmp = t_1;
	} else if (beta <= 7e-174) {
		tmp = 0.5;
	} else if (beta <= 2.7e-127) {
		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 <= -3.4e-174:
		tmp = t_0
	elif beta <= -1.5e-207:
		tmp = t_1
	elif beta <= -5.5e-303:
		tmp = 0.5
	elif beta <= 2.05e-293:
		tmp = t_1
	elif beta <= 7e-174:
		tmp = 0.5
	elif beta <= 2.7e-127:
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -1.5e-207)
		tmp = t_1;
	elseif (beta <= -5.5e-303)
		tmp = 0.5;
	elseif (beta <= 2.05e-293)
		tmp = t_1;
	elseif (beta <= 7e-174)
		tmp = 0.5;
	elseif (beta <= 2.7e-127)
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -1.5e-207)
		tmp = t_1;
	elseif (beta <= -5.5e-303)
		tmp = 0.5;
	elseif (beta <= 2.05e-293)
		tmp = t_1;
	elseif (beta <= 7e-174)
		tmp = 0.5;
	elseif (beta <= 2.7e-127)
		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, -3.4e-174], t$95$0, If[LessEqual[beta, -1.5e-207], t$95$1, If[LessEqual[beta, -5.5e-303], 0.5, If[LessEqual[beta, 2.05e-293], t$95$1, If[LessEqual[beta, 7e-174], 0.5, If[LessEqual[beta, 2.7e-127], t$95$1, If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < -3.4000000000000002e-174 or 2.7e-127 < beta < 2

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   5. Taylor expanded in beta around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   if -3.4000000000000002e-174 < beta < -1.5e-207 or -5.50000000000000018e-303 < beta < 2.04999999999999994e-293 or 6.99999999999999975e-174 < beta < 2.7e-127

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in alpha around inf 77.6%

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

   5. Taylor expanded in beta around 0 77.6%

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

   if -1.5e-207 < beta < -5.50000000000000018e-303 or 2.04999999999999994e-293 < beta < 6.99999999999999975e-174

   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. Taylor expanded in alpha around 0 81.9%

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

      1. div-inv 81.9%

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

      2. fma-def 81.9%

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

      3. +-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

   7. Taylor expanded in beta around 0 81.9%

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

   if 2 < beta

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in beta around inf 87.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq -5.5 \cdot 10^{-303}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.05 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{-174}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq -5.7 \cdot 10^{-303}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 -3.4e-174)
     t_0
     (if (<= beta -1.5e-207)
       t_1
       (if (<= beta -5.7e-303)
         (/ (- 1.0 (* alpha 0.5)) 2.0)
         (if (<= beta 3.8e-289)
           t_1
           (if (<= beta 4.2e-174)
             0.5
             (if (<= beta 2.1e-128) 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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -1.5e-207) {
		tmp = t_1;
	} else if (beta <= -5.7e-303) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 3.8e-289) {
		tmp = t_1;
	} else if (beta <= 4.2e-174) {
		tmp = 0.5;
	} else if (beta <= 2.1e-128) {
		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 <= (-3.4d-174)) then
        tmp = t_0
    else if (beta <= (-1.5d-207)) then
        tmp = t_1
    else if (beta <= (-5.7d-303)) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else if (beta <= 3.8d-289) then
        tmp = t_1
    else if (beta <= 4.2d-174) then
        tmp = 0.5d0
    else if (beta <= 2.1d-128) 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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -1.5e-207) {
		tmp = t_1;
	} else if (beta <= -5.7e-303) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 3.8e-289) {
		tmp = t_1;
	} else if (beta <= 4.2e-174) {
		tmp = 0.5;
	} else if (beta <= 2.1e-128) {
		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 <= -3.4e-174:
		tmp = t_0
	elif beta <= -1.5e-207:
		tmp = t_1
	elif beta <= -5.7e-303:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	elif beta <= 3.8e-289:
		tmp = t_1
	elif beta <= 4.2e-174:
		tmp = 0.5
	elif beta <= 2.1e-128:
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -1.5e-207)
		tmp = t_1;
	elseif (beta <= -5.7e-303)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	elseif (beta <= 3.8e-289)
		tmp = t_1;
	elseif (beta <= 4.2e-174)
		tmp = 0.5;
	elseif (beta <= 2.1e-128)
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -1.5e-207)
		tmp = t_1;
	elseif (beta <= -5.7e-303)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	elseif (beta <= 3.8e-289)
		tmp = t_1;
	elseif (beta <= 4.2e-174)
		tmp = 0.5;
	elseif (beta <= 2.1e-128)
		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, -3.4e-174], t$95$0, If[LessEqual[beta, -1.5e-207], t$95$1, If[LessEqual[beta, -5.7e-303], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 3.8e-289], t$95$1, If[LessEqual[beta, 4.2e-174], 0.5, If[LessEqual[beta, 2.1e-128], t$95$1, If[LessEqual[beta, 2.0], t$95$0, 1.0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if beta < -3.4000000000000002e-174 or 2.1000000000000001e-128 < beta < 2

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   5. Taylor expanded in beta around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   if -3.4000000000000002e-174 < beta < -1.5e-207 or -5.69999999999999981e-303 < beta < 3.80000000000000009e-289 or 4.20000000000000021e-174 < beta < 2.1000000000000001e-128

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in alpha around inf 77.6%

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

   5. Taylor expanded in beta around 0 77.6%

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

   if -1.5e-207 < beta < -5.69999999999999981e-303

   1. Initial program 88.2%

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

      1. +-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in beta around 0 88.2%

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

      1. +-commutative 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in alpha around 0 84.3%

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

   if 3.80000000000000009e-289 < beta < 4.20000000000000021e-174

   1. Initial program 79.8%

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

      1. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in alpha around 0 79.6%

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

      1. div-inv 79.6%

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

      2. fma-def 79.6%

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

      3. +-commutative 79.6%

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in beta around 0 79.6%

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

   if 2 < beta

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in beta around inf 87.4%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq -5.7 \cdot 10^{-303}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 70.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq -6.8 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq -5.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{-175}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \end{array} \]

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 -3.4e-174)
     t_0
     (if (<= beta -6.8e-208)
       t_1
       (if (<= beta -5.5e-303)
         (/ (- 1.0 (* alpha 0.5)) 2.0)
         (if (<= beta 2.3e-292)
           t_1
           (if (<= beta 6e-175)
             0.5
             (if (<= beta 2.1e-128)
               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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -6.8e-208) {
		tmp = t_1;
	} else if (beta <= -5.5e-303) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 2.3e-292) {
		tmp = t_1;
	} else if (beta <= 6e-175) {
		tmp = 0.5;
	} else if (beta <= 2.1e-128) {
		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 <= (-3.4d-174)) then
        tmp = t_0
    else if (beta <= (-6.8d-208)) then
        tmp = t_1
    else if (beta <= (-5.5d-303)) then
        tmp = (1.0d0 - (alpha * 0.5d0)) / 2.0d0
    else if (beta <= 2.3d-292) then
        tmp = t_1
    else if (beta <= 6d-175) then
        tmp = 0.5d0
    else if (beta <= 2.1d-128) 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 <= -3.4e-174) {
		tmp = t_0;
	} else if (beta <= -6.8e-208) {
		tmp = t_1;
	} else if (beta <= -5.5e-303) {
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	} else if (beta <= 2.3e-292) {
		tmp = t_1;
	} else if (beta <= 6e-175) {
		tmp = 0.5;
	} else if (beta <= 2.1e-128) {
		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 <= -3.4e-174:
		tmp = t_0
	elif beta <= -6.8e-208:
		tmp = t_1
	elif beta <= -5.5e-303:
		tmp = (1.0 - (alpha * 0.5)) / 2.0
	elif beta <= 2.3e-292:
		tmp = t_1
	elif beta <= 6e-175:
		tmp = 0.5
	elif beta <= 2.1e-128:
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -6.8e-208)
		tmp = t_1;
	elseif (beta <= -5.5e-303)
		tmp = Float64(Float64(1.0 - Float64(alpha * 0.5)) / 2.0);
	elseif (beta <= 2.3e-292)
		tmp = t_1;
	elseif (beta <= 6e-175)
		tmp = 0.5;
	elseif (beta <= 2.1e-128)
		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 <= -3.4e-174)
		tmp = t_0;
	elseif (beta <= -6.8e-208)
		tmp = t_1;
	elseif (beta <= -5.5e-303)
		tmp = (1.0 - (alpha * 0.5)) / 2.0;
	elseif (beta <= 2.3e-292)
		tmp = t_1;
	elseif (beta <= 6e-175)
		tmp = 0.5;
	elseif (beta <= 2.1e-128)
		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, -3.4e-174], t$95$0, If[LessEqual[beta, -6.8e-208], t$95$1, If[LessEqual[beta, -5.5e-303], N[(N[(1.0 - N[(alpha * 0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 2.3e-292], t$95$1, If[LessEqual[beta, 6e-175], 0.5, If[LessEqual[beta, 2.1e-128], 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 5 regimes

2. if beta < -3.4000000000000002e-174 or 2.1000000000000001e-128 < beta < 2

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in alpha around 0 66.6%

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

   5. Taylor expanded in beta around 0 63.9%

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

      1. *-commutative 63.9%

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

   7. Simplified 63.9%

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

   if -3.4000000000000002e-174 < beta < -6.8e-208 or -5.50000000000000018e-303 < beta < 2.2999999999999999e-292 or 6e-175 < beta < 2.1000000000000001e-128

   1. Initial program 29.2%

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

      1. +-commutative 29.2%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in alpha around inf 77.6%

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

   5. Taylor expanded in beta around 0 77.6%

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

   if -6.8e-208 < beta < -5.50000000000000018e-303

   1. Initial program 88.2%

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

      1. +-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in beta around 0 88.2%

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

      1. +-commutative 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in alpha around 0 84.3%

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

   if 2.2999999999999999e-292 < beta < 6e-175

   1. Initial program 79.8%

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

      1. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in alpha around 0 79.6%

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

      1. div-inv 79.6%

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

      2. fma-def 79.6%

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

      3. +-commutative 79.6%

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

   6. Applied egg-rr 79.6%

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

   7. Taylor expanded in beta around 0 79.6%

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

   if 2 < beta

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in alpha around 0 89.3%

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

   5. Taylor expanded in beta around inf 88.8%

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

      1. associate-*r/ 88.8%

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

      2. metadata-eval 88.8%

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

   7. Simplified 88.8%

      \[\leadsto \frac{\color{blue}{2 - \frac{2}{\beta}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -3.4 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -6.8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq -5.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{1 - \alpha \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq 2.3 \cdot 10^{-292}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{-175}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 7: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq -6.2 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq -2.15 \cdot 10^{-253}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1.55 \cdot 10^{-264}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.92:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha -6.2e-33)
   1.0
   (if (<= alpha -2.15e-253)
     0.5
     (if (<= alpha -1.55e-264)
       1.0
       (if (<= alpha 0.92) 0.5 (/ (/ 2.0 alpha) 2.0))))))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= -6.2e-33) {
		tmp = 1.0;
	} else if (alpha <= -2.15e-253) {
		tmp = 0.5;
	} else if (alpha <= -1.55e-264) {
		tmp = 1.0;
	} else if (alpha <= 0.92) {
		tmp = 0.5;
	} 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 <= (-6.2d-33)) then
        tmp = 1.0d0
    else if (alpha <= (-2.15d-253)) then
        tmp = 0.5d0
    else if (alpha <= (-1.55d-264)) then
        tmp = 1.0d0
    else if (alpha <= 0.92d0) then
        tmp = 0.5d0
    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 <= -6.2e-33) {
		tmp = 1.0;
	} else if (alpha <= -2.15e-253) {
		tmp = 0.5;
	} else if (alpha <= -1.55e-264) {
		tmp = 1.0;
	} else if (alpha <= 0.92) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= -6.2e-33:
		tmp = 1.0
	elif alpha <= -2.15e-253:
		tmp = 0.5
	elif alpha <= -1.55e-264:
		tmp = 1.0
	elif alpha <= 0.92:
		tmp = 0.5
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= -6.2e-33)
		tmp = 1.0;
	elseif (alpha <= -2.15e-253)
		tmp = 0.5;
	elseif (alpha <= -1.55e-264)
		tmp = 1.0;
	elseif (alpha <= 0.92)
		tmp = 0.5;
	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 <= -6.2e-33)
		tmp = 1.0;
	elseif (alpha <= -2.15e-253)
		tmp = 0.5;
	elseif (alpha <= -1.55e-264)
		tmp = 1.0;
	elseif (alpha <= 0.92)
		tmp = 0.5;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, -6.2e-33], 1.0, If[LessEqual[alpha, -2.15e-253], 0.5, If[LessEqual[alpha, -1.55e-264], 1.0, If[LessEqual[alpha, 0.92], 0.5, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if alpha < -6.19999999999999994e-33 or -2.1500000000000001e-253 < alpha < -1.5500000000000001e-264

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in beta around inf 86.3%

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

   if -6.19999999999999994e-33 < alpha < -2.1500000000000001e-253 or -1.5500000000000001e-264 < 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. Taylor expanded in alpha around 0 99.4%

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

      1. div-inv 99.4%

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

      2. fma-def 99.4%

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

      3. +-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in beta around 0 69.9%

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

   if 0.92000000000000004 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in alpha around inf 85.9%

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

   5. Taylor expanded in beta around 0 73.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq -6.2 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq -2.15 \cdot 10^{-253}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq -1.55 \cdot 10^{-264}:\\ \;\;\;\;1\\ \mathbf{elif}\;\alpha \leq 0.92:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 88.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 8800:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 8800.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 8800.0) {
		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 <= 8800.0d0) 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 <= 8800.0) {
		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 <= 8800.0:
		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 <= 8800.0)
		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 <= 8800.0)
		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, 8800.0], 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 < 8800

   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 99.1%

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

   if 8800 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in alpha around inf 85.9%

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

   5. Taylor expanded in beta around 0 73.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 8800:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 10500:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 10500.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 10500.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (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 <= 10500.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (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 <= 10500.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 10500.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 10500.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 10500.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 10500.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 10500

   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 99.1%

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

   if 10500 < alpha

   1. Initial program 20.7%

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

      1. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   4. Taylor expanded in alpha around inf 85.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 10500:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 10: 72.9% 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 66.4%

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

      1. +-commutative 66.4%

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

   3. Simplified 66.4%

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

   4. Taylor expanded in alpha around 0 64.4%

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

      1. div-inv 64.4%

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

      2. fma-def 64.4%

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

      3. +-commutative 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in beta around 0 62.0%

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

   if 2 < beta

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   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 70.6%

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

Alternative 11: 50.2% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 0.5

Julia

function code(alpha, beta)
	return 0.5
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 0.5

Derivation

1. Initial program 74.3%

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

   1. +-commutative 74.3%

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

3. Simplified 74.3%

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

4. Taylor expanded in alpha around 0 72.9%

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

   1. div-inv 72.9%

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

   2. fma-def 72.9%

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

   3. +-commutative 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in beta around 0 46.8%

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

8. Final simplification 46.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023271 
(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))