Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.8%

Time: 10.8s

Alternatives: 10

Speedup: 0.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.7% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 - \beta}{\frac{\alpha}{\frac{2 + \beta \cdot 2}{\alpha}}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
   (/
    (+
     (/ (- -2.0 beta) (/ alpha (/ (+ 2.0 (* beta 2.0)) alpha)))
     (/ (+ beta (- beta -2.0)) alpha))
    2.0)
   (/ (fma (- beta alpha) (/ 1.0 (+ beta (+ alpha 2.0))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (((-2.0 - beta) / (alpha / ((2.0 + (beta * 2.0)) / alpha))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = fma((beta - alpha), (1.0 / (beta + (alpha + 2.0))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) / Float64(alpha / Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha))) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(beta + Float64(alpha + 2.0))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] / N[(alpha / N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99980000000000002

   1. Initial program 6.4%

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

      1. div-sub 6.4%

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

      2. associate-+l- 9.3%

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

      3. +-commutative 9.3%

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

      4. associate-+l+ 9.3%

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

      5. +-commutative 9.3%

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

      6. associate-+l+ 9.3%

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

   4. Applied egg-rr 9.3%

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

   5. Taylor expanded in alpha around -inf 96.8%

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

   7. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. times-frac 100.0%

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

      4. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. associate-/l* 100.0%

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

       3. clear-num 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

       6. associate-+l+ 100.0%

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

       7. *-un-lft-identity 100.0%

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

       8. *-un-lft-identity 100.0%

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

       9. distribute-rgt-out 100.0%

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

       10. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 - \beta}{\frac{\alpha}{\frac{2 + \beta \cdot 2}{\alpha}}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\beta + \left(\alpha + 2\right)}, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 - \beta}{\frac{\alpha}{\frac{2 + \beta \cdot 2}{\alpha}}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
     (/
      (+
       (/ (- -2.0 beta) (/ alpha (/ (+ 2.0 (* beta 2.0)) alpha)))
       (/ (+ beta (- beta -2.0)) alpha))
      2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (((-2.0 - beta) / (alpha / ((2.0 + (beta * 2.0)) / alpha))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / 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) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = ((((-2.0d0) - beta) / (alpha / ((2.0d0 + (beta * 2.0d0)) / alpha))) + ((beta + (beta - (-2.0d0))) / alpha)) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / 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) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = (((-2.0 - beta) / (alpha / ((2.0 + (beta * 2.0)) / alpha))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = (((-2.0 - beta) / (alpha / ((2.0 + (beta * 2.0)) / alpha))) + ((beta + (beta - -2.0)) / alpha)) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(Float64(-2.0 - beta) / Float64(alpha / Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha))) + Float64(Float64(beta + Float64(beta - -2.0)) / alpha)) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / 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) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = (((-2.0 - beta) / (alpha / ((2.0 + (beta * 2.0)) / alpha))) + ((beta + (beta - -2.0)) / alpha)) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(N[(-2.0 - beta), $MachinePrecision] / N[(alpha / N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + N[(beta - -2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99980000000000002

   1. Initial program 6.4%

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

      1. div-sub 6.4%

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

      2. associate-+l- 9.3%

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

      3. +-commutative 9.3%

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

      4. associate-+l+ 9.3%

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

      5. +-commutative 9.3%

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

      6. associate-+l+ 9.3%

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

   4. Applied egg-rr 9.3%

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

   5. Taylor expanded in alpha around -inf 96.8%

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

   7. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. times-frac 100.0%

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

      4. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. associate-/l* 100.0%

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

       3. clear-num 100.0%

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

       4. +-commutative 100.0%

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

       5. +-commutative 100.0%

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

       6. associate-+l+ 100.0%

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

       7. *-un-lft-identity 100.0%

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

       8. *-un-lft-identity 100.0%

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

       9. distribute-rgt-out 100.0%

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

       10. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{-2 - \beta}{\frac{\alpha}{\frac{2 + \beta \cdot 2}{\alpha}}} + \frac{\beta + \left(\beta - -2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ t_1 := \frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{if}\;\beta \leq -5.6 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\beta \leq -8 \cdot 10^{-283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq 1.7:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 alpha) 2.0)) (t_1 (/ (+ 1.0 (* alpha -0.5)) 2.0)))
   (if (<= beta -5.6e-239)
     t_1
     (if (<= beta -8e-283)
       t_0
       (if (<= beta 9.2e-70)
         t_1
         (if (<= beta 2e-33)
           t_0
           (if (<= beta 1.7) t_1 (+ 1.0 (/ -1.0 beta)))))))))

C

double code(double alpha, double beta) {
	double t_0 = (2.0 / alpha) / 2.0;
	double t_1 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (beta <= -5.6e-239) {
		tmp = t_1;
	} else if (beta <= -8e-283) {
		tmp = t_0;
	} else if (beta <= 9.2e-70) {
		tmp = t_1;
	} else if (beta <= 2e-33) {
		tmp = t_0;
	} else if (beta <= 1.7) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 / alpha) / 2.0d0
    t_1 = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    if (beta <= (-5.6d-239)) then
        tmp = t_1
    else if (beta <= (-8d-283)) then
        tmp = t_0
    else if (beta <= 9.2d-70) then
        tmp = t_1
    else if (beta <= 2d-33) then
        tmp = t_0
    else if (beta <= 1.7d0) then
        tmp = t_1
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (2.0 / alpha) / 2.0;
	double t_1 = (1.0 + (alpha * -0.5)) / 2.0;
	double tmp;
	if (beta <= -5.6e-239) {
		tmp = t_1;
	} else if (beta <= -8e-283) {
		tmp = t_0;
	} else if (beta <= 9.2e-70) {
		tmp = t_1;
	} else if (beta <= 2e-33) {
		tmp = t_0;
	} else if (beta <= 1.7) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (2.0 / alpha) / 2.0
	t_1 = (1.0 + (alpha * -0.5)) / 2.0
	tmp = 0
	if beta <= -5.6e-239:
		tmp = t_1
	elif beta <= -8e-283:
		tmp = t_0
	elif beta <= 9.2e-70:
		tmp = t_1
	elif beta <= 2e-33:
		tmp = t_0
	elif beta <= 1.7:
		tmp = t_1
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(2.0 / alpha) / 2.0)
	t_1 = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0)
	tmp = 0.0
	if (beta <= -5.6e-239)
		tmp = t_1;
	elseif (beta <= -8e-283)
		tmp = t_0;
	elseif (beta <= 9.2e-70)
		tmp = t_1;
	elseif (beta <= 2e-33)
		tmp = t_0;
	elseif (beta <= 1.7)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (2.0 / alpha) / 2.0;
	t_1 = (1.0 + (alpha * -0.5)) / 2.0;
	tmp = 0.0;
	if (beta <= -5.6e-239)
		tmp = t_1;
	elseif (beta <= -8e-283)
		tmp = t_0;
	elseif (beta <= 9.2e-70)
		tmp = t_1;
	elseif (beta <= 2e-33)
		tmp = t_0;
	elseif (beta <= 1.7)
		tmp = t_1;
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[beta, -5.6e-239], t$95$1, If[LessEqual[beta, -8e-283], t$95$0, If[LessEqual[beta, 9.2e-70], t$95$1, If[LessEqual[beta, 2e-33], t$95$0, If[LessEqual[beta, 1.7], t$95$1, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < -5.60000000000000025e-239 or -7.99999999999999957e-283 < beta < 9.20000000000000002e-70 or 2.0000000000000001e-33 < beta < 1.69999999999999996

   1. Initial program 78.0%

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

   3. Taylor expanded in beta around 0 75.9%

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

      1. +-commutative 75.9%

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

   5. Simplified 75.9%

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

   6. Taylor expanded in alpha around 0 74.9%

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

      1. *-commutative 74.9%

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

   8. Simplified 74.9%

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

   if -5.60000000000000025e-239 < beta < -7.99999999999999957e-283 or 9.20000000000000002e-70 < beta < 2.0000000000000001e-33

   1. Initial program 31.1%

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

   3. Taylor expanded in alpha around inf 75.3%

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

   4. Taylor expanded in beta around 0 75.3%

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

   if 1.69999999999999996 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in alpha around 0 84.8%

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

   4. Taylor expanded in beta around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in beta around 0 84.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -5.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq -8 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 1.7:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + \beta \cdot 0.5}{2}\\ t_1 := \frac{\frac{2}{\alpha}}{2}\\ \mathbf{if}\;\beta \leq -4 \cdot 10^{-262}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\beta \leq -6 \cdot 10^{-283}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

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 -4e-262)
     t_0
     (if (<= beta -6e-283)
       t_1
       (if (<= beta 9.2e-70)
         (/ (+ 1.0 (* alpha -0.5)) 2.0)
         (if (<= beta 3e-33)
           t_1
           (if (<= beta 2.0) t_0 (+ 1.0 (/ -1.0 beta)))))))))

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 <= -4e-262) {
		tmp = t_0;
	} else if (beta <= -6e-283) {
		tmp = t_1;
	} else if (beta <= 9.2e-70) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else if (beta <= 3e-33) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: 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 <= (-4d-262)) then
        tmp = t_0
    else if (beta <= (-6d-283)) then
        tmp = t_1
    else if (beta <= 9.2d-70) then
        tmp = (1.0d0 + (alpha * (-0.5d0))) / 2.0d0
    else if (beta <= 3d-33) then
        tmp = t_1
    else if (beta <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    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 <= -4e-262) {
		tmp = t_0;
	} else if (beta <= -6e-283) {
		tmp = t_1;
	} else if (beta <= 9.2e-70) {
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	} else if (beta <= 3e-33) {
		tmp = t_1;
	} else if (beta <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 <= -4e-262:
		tmp = t_0
	elif beta <= -6e-283:
		tmp = t_1
	elif beta <= 9.2e-70:
		tmp = (1.0 + (alpha * -0.5)) / 2.0
	elif beta <= 3e-33:
		tmp = t_1
	elif beta <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0 + (-1.0 / beta)
	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 <= -4e-262)
		tmp = t_0;
	elseif (beta <= -6e-283)
		tmp = t_1;
	elseif (beta <= 9.2e-70)
		tmp = Float64(Float64(1.0 + Float64(alpha * -0.5)) / 2.0);
	elseif (beta <= 3e-33)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	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 <= -4e-262)
		tmp = t_0;
	elseif (beta <= -6e-283)
		tmp = t_1;
	elseif (beta <= 9.2e-70)
		tmp = (1.0 + (alpha * -0.5)) / 2.0;
	elseif (beta <= 3e-33)
		tmp = t_1;
	elseif (beta <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0 + (-1.0 / beta);
	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, -4e-262], t$95$0, If[LessEqual[beta, -6e-283], t$95$1, If[LessEqual[beta, 9.2e-70], N[(N[(1.0 + N[(alpha * -0.5), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[beta, 3e-33], t$95$1, If[LessEqual[beta, 2.0], t$95$0, N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < -4.00000000000000005e-262 or 3.0000000000000002e-33 < beta < 2

   1. Initial program 78.8%

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

   3. Taylor expanded in alpha around 0 78.2%

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

   4. Taylor expanded in beta around 0 76.7%

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

      1. *-commutative 76.7%

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

   6. Simplified 76.7%

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

   if -4.00000000000000005e-262 < beta < -5.99999999999999992e-283 or 9.20000000000000002e-70 < beta < 3.0000000000000002e-33

   1. Initial program 29.1%

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

   3. Taylor expanded in alpha around inf 77.5%

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

   4. Taylor expanded in beta around 0 77.5%

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

   if -5.99999999999999992e-283 < beta < 9.20000000000000002e-70

   1. Initial program 76.4%

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

   3. Taylor expanded in beta around 0 76.4%

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

      1. +-commutative 76.4%

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

   5. Simplified 76.4%

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

   6. Taylor expanded in alpha around 0 75.4%

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

      1. *-commutative 75.4%

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

   8. Simplified 75.4%

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

   if 2 < beta

   1. Initial program 87.8%

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

   3. Taylor expanded in alpha around 0 84.8%

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

   4. Taylor expanded in beta around inf 84.8%

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

      1. associate-*r/ 84.8%

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

      2. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in beta around 0 84.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq -4 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{elif}\;\beta \leq -6 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1 + \alpha \cdot -0.5}{2}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{elif}\;\beta \leq 2:\\ \;\;\;\;\frac{1 + \beta \cdot 0.5}{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} + \left(1 - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.9998)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ (/ beta t_0) (- 1.0 (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / 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) / ((beta + alpha) + 2.0d0)) <= (-0.9998d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = ((beta / t_0) + (1.0d0 - (alpha / 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) / ((beta + alpha) + 2.0)) <= -0.9998) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = beta + (alpha + 2.0)
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.9998:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.9998)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_0) + Float64(1.0 - Float64(alpha / 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) / ((beta + alpha) + 2.0)) <= -0.9998)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = ((beta / t_0) + (1.0 - (alpha / t_0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] + N[(1.0 - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2)) < -0.99980000000000002

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf 99.8%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

      1. div-sub 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(\alpha + 2\right)} + \left(1 - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.9998)
     (/ (/ (+ 2.0 (* 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 <= -0.9998) {
		tmp = ((2.0 + (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 <= (-0.9998d0)) then
        tmp = ((2.0d0 + (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 <= -0.9998) {
		tmp = ((2.0 + (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 <= -0.9998:
		tmp = ((2.0 + (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 <= -0.9998)
		tmp = Float64(Float64(Float64(2.0 + 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 <= -0.9998)
		tmp = ((2.0 + (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, -0.9998], N[(N[(N[(2.0 + 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)) < -0.99980000000000002

   1. Initial program 6.4%

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

   3. Taylor expanded in alpha around inf 99.8%

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

   if -0.99980000000000002 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) 2))

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 9.5e+14)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 9.5e+14) {
		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 <= 9.5d+14) 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 <= 9.5e+14) {
		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 <= 9.5e+14:
		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 <= 9.5e+14)
		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 <= 9.5e+14)
		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, 9.5e+14], 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 < 9.5e14

   1. Initial program 99.6%

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

   3. Taylor expanded in alpha around 0 98.9%

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

   if 9.5e14 < alpha

   1. Initial program 24.1%

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

   3. Taylor expanded in alpha around inf 82.5%

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

   4. Taylor expanded in beta around 0 67.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 9.5e+14)
   (/ (+ 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 <= 9.5e+14) {
		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 <= 9.5d+14) 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 <= 9.5e+14) {
		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 <= 9.5e+14:
		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 <= 9.5e+14)
		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 <= 9.5e+14)
		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, 9.5e+14], 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 < 9.5e14

   1. Initial program 99.6%

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

   3. Taylor expanded in alpha around 0 98.9%

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

   if 9.5e14 < alpha

   1. Initial program 24.1%

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

   3. Taylor expanded in alpha around inf 82.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 1.4e+15) 1.0 (/ (/ 2.0 alpha) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 1.4e+15) {
		tmp = 1.0;
	} 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 <= 1.4d+15) then
        tmp = 1.0d0
    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 <= 1.4e+15) {
		tmp = 1.0;
	} else {
		tmp = (2.0 / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 1.4e+15:
		tmp = 1.0
	else:
		tmp = (2.0 / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 1.4e+15)
		tmp = 1.0;
	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 <= 1.4e+15)
		tmp = 1.0;
	else
		tmp = (2.0 / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 1.4e+15], 1.0, N[(N[(2.0 / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.4e15

   1. Initial program 99.6%

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

   3. Taylor expanded in alpha around 0 98.9%

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

   4. Taylor expanded in beta around inf 36.2%

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

      1. associate-*r/ 36.2%

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

      2. metadata-eval 36.2%

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

   6. Simplified 36.2%

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

   7. Taylor expanded in beta around inf 46.3%

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

   if 1.4e15 < alpha

   1. Initial program 24.1%

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

   3. Taylor expanded in alpha around inf 82.5%

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

   4. Taylor expanded in beta around 0 67.7%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 1.0;
}

Fortran

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

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in alpha around 0 75.1%

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

4. Taylor expanded in beta around inf 30.7%

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

   1. associate-*r/ 30.7%

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

   2. metadata-eval 30.7%

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

6. Simplified 30.7%

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

7. Taylor expanded in beta around inf 38.3%

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

8. Final simplification 38.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))