Octave 3.8, jcobi/1

Percentage Accurate: 73.5% → 99.7%

Time: 9.0s

Alternatives: 8

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: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \beta \cdot 2\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\alpha \cdot \left(2 \cdot \frac{1}{t\_0} - -2 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{\alpha \cdot {t\_0}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* beta 2.0))))
   (if (<= beta 9e+15)
     (/
      1.0
      (*
       alpha
       (-
        (* 2.0 (/ 1.0 t_0))
        (*
         -2.0
         (/
          (+ (pow (+ beta 2.0) 2.0) (* beta (+ beta 2.0)))
          (* alpha (pow t_0 2.0)))))))
     (/ 1.0 (+ 1.0 (/ (+ 1.0 alpha) beta))))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (beta <= 9e+15) {
		tmp = 1.0 / (alpha * ((2.0 * (1.0 / t_0)) - (-2.0 * ((pow((beta + 2.0), 2.0) + (beta * (beta + 2.0))) / (alpha * pow(t_0, 2.0))))));
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / 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) :: tmp
    t_0 = 2.0d0 + (beta * 2.0d0)
    if (beta <= 9d+15) then
        tmp = 1.0d0 / (alpha * ((2.0d0 * (1.0d0 / t_0)) - ((-2.0d0) * ((((beta + 2.0d0) ** 2.0d0) + (beta * (beta + 2.0d0))) / (alpha * (t_0 ** 2.0d0))))))
    else
        tmp = 1.0d0 / (1.0d0 + ((1.0d0 + alpha) / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta * 2.0);
	double tmp;
	if (beta <= 9e+15) {
		tmp = 1.0 / (alpha * ((2.0 * (1.0 / t_0)) - (-2.0 * ((Math.pow((beta + 2.0), 2.0) + (beta * (beta + 2.0))) / (alpha * Math.pow(t_0, 2.0))))));
	} else {
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = 2.0 + (beta * 2.0)
	tmp = 0
	if beta <= 9e+15:
		tmp = 1.0 / (alpha * ((2.0 * (1.0 / t_0)) - (-2.0 * ((math.pow((beta + 2.0), 2.0) + (beta * (beta + 2.0))) / (alpha * math.pow(t_0, 2.0))))))
	else:
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta))
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta * 2.0))
	tmp = 0.0
	if (beta <= 9e+15)
		tmp = Float64(1.0 / Float64(alpha * Float64(Float64(2.0 * Float64(1.0 / t_0)) - Float64(-2.0 * Float64(Float64((Float64(beta + 2.0) ^ 2.0) + Float64(beta * Float64(beta + 2.0))) / Float64(alpha * (t_0 ^ 2.0)))))));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(1.0 + alpha) / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = 2.0 + (beta * 2.0);
	tmp = 0.0;
	if (beta <= 9e+15)
		tmp = 1.0 / (alpha * ((2.0 * (1.0 / t_0)) - (-2.0 * ((((beta + 2.0) ^ 2.0) + (beta * (beta + 2.0))) / (alpha * (t_0 ^ 2.0))))));
	else
		tmp = 1.0 / (1.0 + ((1.0 + alpha) / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 9e+15], N[(1.0 / N[(alpha * N[(N[(2.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(N[Power[N[(beta + 2.0), $MachinePrecision], 2.0], $MachinePrecision] + N[(beta * N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9e15

   1. Initial program 69.7%

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

      1. +-commutative 69.7%

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

   3. Simplified 69.7%

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

      1. div-inv 69.5%

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

      2. fma-define 69.0%

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

      3. associate-+l+ 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. clear-num 69.0%

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

      2. inv-pow 69.0%

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

      3. fma-undefine 69.5%

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

      4. un-div-inv 69.7%

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

   8. Applied egg-rr 69.7%

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

      1. unpow-1 69.7%

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

      2. +-commutative 69.7%

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

      3. +-commutative 69.7%

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

      4. +-commutative 69.7%

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

      5. associate-+r+ 69.7%

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

   10. Simplified 69.7%

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

   11. Taylor expanded in alpha around inf 99.7%

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

   if 9e15 < beta

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

   3. Simplified 87.0%

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

      1. div-inv 87.0%

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

      2. fma-define 87.0%

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

      3. associate-+l+ 87.0%

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

   6. Applied egg-rr 87.0%

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

      1. clear-num 87.0%

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

      2. inv-pow 87.0%

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

      3. fma-undefine 87.0%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

      1. unpow-1 87.0%

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

      2. +-commutative 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

      5. associate-+r+ 87.0%

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

   10. Simplified 87.0%

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

   11. Taylor expanded in alpha around inf 58.6%

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

   12. Taylor expanded in beta around inf 99.9%

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

       1. +-commutative 99.9%

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

       2. distribute-lft-in 99.9%

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

       3. rgt-mult-inverse 99.9%

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

       4. *-rgt-identity 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\alpha \cdot \left(2 \cdot \frac{1}{2 + \beta \cdot 2} - -2 \cdot \frac{{\left(\beta + 2\right)}^{2} + \beta \cdot \left(\beta + 2\right)}{\alpha \cdot {\left(2 + \beta \cdot 2\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1 + \alpha}{\beta}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.99999995)
     (/ (/ (+ 2.0 (* beta 2.0)) alpha) 2.0)
     (/ (+ 1.0 t_0) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + 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 + (beta + alpha))
    if (t_0 <= (-0.99999995d0)) then
        tmp = ((2.0d0 + (beta * 2.0d0)) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + 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 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.99999995) {
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / (2.0 + (beta + alpha))
	tmp = 0
	if t_0 <= -0.99999995:
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.99999995)
		tmp = Float64(Float64(Float64(2.0 + Float64(beta * 2.0)) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	tmp = 0.0;
	if (t_0 <= -0.99999995)
		tmp = ((2.0 + (beta * 2.0)) / alpha) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.99999995], N[(N[(N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999999949999999971

   1. Initial program 7.0%

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

      1. +-commutative 7.0%

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

   3. Simplified 7.0%

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

   5. Taylor expanded in alpha around inf 99.1%

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

   if -0.999999949999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.99999995:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if beta < 1

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

   3. Simplified 69.9%

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

      1. div-inv 69.8%

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

      2. fma-define 69.2%

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

      3. associate-+l+ 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. clear-num 69.2%

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

      2. inv-pow 69.2%

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

      3. fma-undefine 69.7%

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

      4. un-div-inv 69.9%

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

   8. Applied egg-rr 69.9%

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

      1. unpow-1 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-+r+ 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in alpha around inf 99.7%

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

   12. Taylor expanded in beta around 0 97.1%

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

       1. associate-*r/ 97.1%

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

       2. metadata-eval 97.1%

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

   14. Simplified 97.1%

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

   if 1 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. div-inv 86.2%

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

      2. fma-define 86.2%

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

      3. associate-+l+ 86.2%

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

   6. Applied egg-rr 86.2%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. fma-undefine 86.2%

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

      4. un-div-inv 86.3%

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

   8. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. +-commutative 86.3%

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

      3. +-commutative 86.3%

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

      4. +-commutative 86.3%

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

      5. associate-+r+ 86.3%

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

   10. Simplified 86.3%

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

   11. Taylor expanded in alpha around inf 59.5%

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

   12. Taylor expanded in beta around inf 99.9%

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

       1. +-commutative 99.9%

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

       2. distribute-lft-in 99.9%

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

       3. rgt-mult-inverse 99.9%

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

       4. *-rgt-identity 99.9%

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

   14. Simplified 99.9%

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

Alternative 4: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.5:\\ \;\;\;\;\frac{1}{\alpha \cdot \left(1 + \frac{2}{\alpha}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = 1.0 / (alpha * (1.0 + (2.0 / alpha)));
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 9.5d0) then
        tmp = 1.0d0 / (alpha * (1.0d0 + (2.0d0 / alpha)))
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 9.5) {
		tmp = 1.0 / (alpha * (1.0 + (2.0 / alpha)));
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 9.5:
		tmp = 1.0 / (alpha * (1.0 + (2.0 / alpha)))
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 9.5)
		tmp = Float64(1.0 / Float64(alpha * Float64(1.0 + Float64(2.0 / alpha))));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 9.5)
		tmp = 1.0 / (alpha * (1.0 + (2.0 / alpha)));
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 9.5], N[(1.0 / N[(alpha * N[(1.0 + N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.5

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

   3. Simplified 69.9%

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

      1. div-inv 69.8%

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

      2. fma-define 69.2%

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

      3. associate-+l+ 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. clear-num 69.2%

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

      2. inv-pow 69.2%

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

      3. fma-undefine 69.7%

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

      4. un-div-inv 69.9%

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

   8. Applied egg-rr 69.9%

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

      1. unpow-1 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-+r+ 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in alpha around inf 99.7%

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

   12. Taylor expanded in beta around 0 97.1%

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

       1. associate-*r/ 97.1%

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

       2. metadata-eval 97.1%

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

   14. Simplified 97.1%

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

   if 9.5 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in alpha around 0 84.3%

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

   6. Taylor expanded in beta around inf 84.3%

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

Alternative 5: 70.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) (- 1.0 (/ 1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0 + (beta * 0.25d0)
    else
        tmp = 1.0d0 - (1.0d0 / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0 - (1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5 + (beta * 0.25);
	else
		tmp = 1.0 - (1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in alpha around 0 67.4%

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

   6. Taylor expanded in beta around 0 66.4%

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

      1. *-commutative 66.4%

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

   8. Simplified 66.4%

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

   if 2 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in alpha around 0 84.3%

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

   6. Taylor expanded in beta around inf 84.3%

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

Alternative 6: 70.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;0.5 + \beta \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (+ 0.5 (* beta 0.25)) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5 + (beta * 0.25);
	} 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 + (beta * 0.25d0)
    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 + (beta * 0.25);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5 + (beta * 0.25)
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = Float64(0.5 + Float64(beta * 0.25));
	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 + (beta * 0.25);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.5 + N[(beta * 0.25), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in alpha around 0 67.4%

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

   6. Taylor expanded in beta around 0 66.4%

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

      1. *-commutative 66.4%

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

   8. Simplified 66.4%

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

   if 2 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. div-inv 86.2%

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

      2. fma-define 86.2%

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

      3. associate-+l+ 86.2%

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

   6. Applied egg-rr 86.2%

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

   7. Taylor expanded in beta around inf 84.2%

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

Alternative 7: 70.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 (if (<= beta 2.0) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if beta <= 2.0:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 2.0)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in alpha around 0 67.4%

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

   6. Taylor expanded in beta around 0 65.6%

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

   if 2 < beta

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. div-inv 86.2%

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

      2. fma-define 86.2%

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

      3. associate-+l+ 86.2%

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

   6. Applied egg-rr 86.2%

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

   7. Taylor expanded in beta around inf 84.2%

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

Alternative 8: 49.1% 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 75.9%

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

   1. +-commutative 75.9%

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

3. Simplified 75.9%

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

5. Taylor expanded in alpha around 0 73.6%

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

6. Taylor expanded in beta around 0 47.7%

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

Reproduce

herbie shell --seed 2024150 
(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))