Octave 3.8, jcobi/1

Percentage Accurate: 73.5% → 99.8%

Time: 10.1s

Alternatives: 11

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: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta \cdot \left(1 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\alpha + \left(\beta + 2\right)}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.95)
   (/ (+ 1.0 (- (* beta (- 1.0 (/ beta alpha))) (/ 2.0 alpha))) alpha)
   (pow
    (pow (fma (- alpha beta) (/ -0.5 (+ alpha (+ beta 2.0))) 0.5) 3.0)
    0.3333333333333333)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.95) {
		tmp = (1.0 + ((beta * (1.0 - (beta / alpha))) - (2.0 / alpha))) / alpha;
	} else {
		tmp = pow(pow(fma((alpha - beta), (-0.5 / (alpha + (beta + 2.0))), 0.5), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.95)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta * Float64(1.0 - Float64(beta / alpha))) - Float64(2.0 / alpha))) / alpha);
	else
		tmp = (fma(Float64(alpha - beta), Float64(-0.5 / Float64(alpha + Float64(beta + 2.0))), 0.5) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.95], N[(N[(1.0 + N[(N[(beta * N[(1.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[Power[N[Power[N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(alpha + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.94999999999999996

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

      2. sub-neg 8.0%

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

      3. +-commutative 8.0%

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

      4. neg-sub0 8.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 8.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 8.0%

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

      7. distribute-frac-neg 8.0%

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

      8. +-commutative 8.0%

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

      9. sub-neg 8.0%

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

      10. div-sub 8.0%

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

      11. sub-neg 8.0%

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

      12. metadata-eval 8.0%

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

      13. neg-mul-1 8.0%

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

      14. *-commutative 8.0%

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

      15. +-commutative 8.0%

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

      16. associate-/l/ 8.0%

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

      17. associate-*l/ 8.0%

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

   3. Simplified 7.6%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 96.9%

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

   6. Taylor expanded in beta around 0 99.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in beta around inf 99.8%

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

       1. mul-1-neg 99.8%

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

   11. Simplified 99.8%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 99.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right)}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right) \cdot \left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right)\right) \cdot \left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 100.0%

         \[\leadsto {\color{blue}{\left({\left(0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. +-commutative 100.0%

         \[\leadsto {\left({\color{blue}{\left(\left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)} + 0.5\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. fma-define 100.0%

         \[\leadsto {\left({\color{blue}{\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right)\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. associate-+r+ 100.0%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\color{blue}{\left(\beta + \alpha\right) + 2}}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333} \]

      7. +-commutative 100.0%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\color{blue}{\left(\alpha + \beta\right)} + 2}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333} \]

      8. associate-+l+ 100.0%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\color{blue}{\alpha + \left(\beta + 2\right)}}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\alpha + \left(\beta + 2\right)}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta \cdot \left(1 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(\alpha - \beta, \frac{-0.5}{\alpha + \left(\beta + 2\right)}, 0.5\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta \cdot \left(1 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.95d0)) then
        tmp = (1.0d0 + ((beta * (1.0d0 - (beta / alpha))) - (2.0d0 / alpha))) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -0.95:
		tmp = (1.0 + ((beta * (1.0 - (beta / alpha))) - (2.0 / alpha))) / alpha
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.95)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta * Float64(1.0 - Float64(beta / alpha))) - Float64(2.0 / alpha))) / alpha);
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	tmp = 0.0;
	if (t_0 <= -0.95)
		tmp = (1.0 + ((beta * (1.0 - (beta / alpha))) - (2.0 / alpha))) / alpha;
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.95], N[(N[(1.0 + N[(N[(beta * N[(1.0 - N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.94999999999999996

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

      2. sub-neg 8.0%

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

      3. +-commutative 8.0%

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

      4. neg-sub0 8.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 8.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 8.0%

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

      7. distribute-frac-neg 8.0%

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

      8. +-commutative 8.0%

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

      9. sub-neg 8.0%

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

      10. div-sub 8.0%

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

      11. sub-neg 8.0%

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

      12. metadata-eval 8.0%

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

      13. neg-mul-1 8.0%

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

      14. *-commutative 8.0%

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

      15. +-commutative 8.0%

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

      16. associate-/l/ 8.0%

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

      17. associate-*l/ 8.0%

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

   3. Simplified 7.6%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 96.9%

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

   6. Taylor expanded in beta around 0 99.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in beta around inf 99.8%

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

       1. mul-1-neg 99.8%

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

   11. Simplified 99.8%

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

   if -0.94999999999999996 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.95:\\ \;\;\;\;\frac{1 + \left(\beta \cdot \left(1 - \frac{\beta}{\alpha}\right) - \frac{2}{\alpha}\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-0.99999995d0)) then
        tmp = (beta + 1.0d0) / alpha
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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%

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

      2. sub-neg 7.0%

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

      3. +-commutative 7.0%

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

      4. neg-sub0 7.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 7.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 7.0%

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

      7. distribute-frac-neg 7.0%

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

      8. +-commutative 7.0%

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

      9. sub-neg 7.0%

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

      10. div-sub 7.0%

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

      11. sub-neg 7.0%

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

      12. metadata-eval 7.0%

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

      13. neg-mul-1 7.0%

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

      14. *-commutative 7.0%

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

      15. +-commutative 7.0%

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

      16. associate-/l/ 7.0%

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

      17. associate-*l/ 7.0%

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

   3. Simplified 6.8%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 96.9%

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

   6. Taylor expanded in beta around 0 99.9%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 99.9%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in alpha around inf 99.1%

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

   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%

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

Alternative 4: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 67000000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 67000000000000.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta (+ alpha 2.0)))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 67000000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 67000000000000.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + (alpha + 2.0d0))))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 67000000000000.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 67000000000000.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 67000000000000.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + Float64(alpha + 2.0)))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 67000000000000.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + (alpha + 2.0))));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 67000000000000.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 6.7e13

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. neg-sub0 99.5%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 99.5%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 99.5%

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

      7. distribute-frac-neg 99.5%

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

      8. +-commutative 99.5%

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

      9. sub-neg 99.5%

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

      10. div-sub 99.5%

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

      11. sub-neg 99.5%

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

      12. metadata-eval 99.5%

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

      13. neg-mul-1 99.5%

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

      14. *-commutative 99.5%

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

      15. +-commutative 99.5%

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

      16. associate-/l/ 99.5%

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

      17. associate-*l/ 99.5%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   if 6.7e13 < alpha

   1. Initial program 22.1%

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

      1. +-commutative 22.1%

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

      2. sub-neg 22.1%

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

      3. +-commutative 22.1%

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

      4. neg-sub0 22.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 22.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 22.1%

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

      7. distribute-frac-neg 22.1%

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

      8. +-commutative 22.1%

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

      9. sub-neg 22.1%

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

      10. div-sub 22.1%

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

      11. sub-neg 22.1%

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

      12. metadata-eval 22.1%

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

      13. neg-mul-1 22.1%

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

      14. *-commutative 22.1%

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

      15. +-commutative 22.1%

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

      16. associate-/l/ 22.1%

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

      17. associate-*l/ 22.1%

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

   3. Simplified 21.9%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 80.9%

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

   6. Taylor expanded in beta around 0 83.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 83.5%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 83.5%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 83.5%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in alpha around inf 84.1%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 67000000000000:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 20.0)
   (+ 0.5 (* (- alpha beta) (/ -0.5 (+ beta 2.0))))
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 20.0d0) then
        tmp = 0.5d0 + ((alpha - beta) * ((-0.5d0) / (beta + 2.0d0)))
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 20.0:
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)))
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 20.0)
		tmp = Float64(0.5 + Float64(Float64(alpha - beta) * Float64(-0.5 / Float64(beta + 2.0))));
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 20.0)
		tmp = 0.5 + ((alpha - beta) * (-0.5 / (beta + 2.0)));
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 20.0], N[(0.5 + N[(N[(alpha - beta), $MachinePrecision] * N[(-0.5 / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 20

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 100.0%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. div-sub 100.0%

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

      11. sub-neg 100.0%

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

      12. metadata-eval 100.0%

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

      13. neg-mul-1 100.0%

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

      14. *-commutative 100.0%

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

      15. +-commutative 100.0%

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

      16. associate-/l/ 100.0%

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

      17. associate-*l/ 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around 0 99.6%

      \[\leadsto 0.5 + \left(\alpha - \beta\right) \cdot \color{blue}{\frac{-0.5}{2 + \beta}} \]
   6. Step-by-step derivation

      1. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 20 < alpha

   1. Initial program 24.8%

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

      1. +-commutative 24.8%

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

      2. sub-neg 24.8%

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

      3. +-commutative 24.8%

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

      4. neg-sub0 24.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 24.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 24.8%

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

      7. distribute-frac-neg 24.8%

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

      8. +-commutative 24.8%

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

      9. sub-neg 24.8%

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

      10. div-sub 24.8%

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

      11. sub-neg 24.8%

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

      12. metadata-eval 24.8%

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

      13. neg-mul-1 24.8%

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

      14. *-commutative 24.8%

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

      15. +-commutative 24.8%

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

      16. associate-/l/ 24.8%

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

      17. associate-*l/ 24.8%

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

   3. Simplified 24.4%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 79.6%

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

   6. Taylor expanded in beta around 0 82.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 82.1%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 82.1%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in alpha around inf 82.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= alpha 20.0)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (+ beta 1.0) alpha)))

C

double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (alpha <= 20.0d0) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = (beta + 1.0d0) / alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (alpha <= 20.0) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = (beta + 1.0) / alpha;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if alpha <= 20.0:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = (beta + 1.0) / alpha
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (alpha <= 20.0)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(beta + 1.0) / alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (alpha <= 20.0)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = (beta + 1.0) / alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[alpha, 20.0], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 20

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in alpha around 0 99.0%

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

      1. +-commutative 99.0%

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

   7. Simplified 99.0%

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

   if 20 < alpha

   1. Initial program 24.8%

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

      1. +-commutative 24.8%

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

      2. sub-neg 24.8%

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

      3. +-commutative 24.8%

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

      4. neg-sub0 24.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 24.8%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 24.8%

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

      7. distribute-frac-neg 24.8%

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

      8. +-commutative 24.8%

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

      9. sub-neg 24.8%

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

      10. div-sub 24.8%

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

      11. sub-neg 24.8%

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

      12. metadata-eval 24.8%

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

      13. neg-mul-1 24.8%

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

      14. *-commutative 24.8%

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

      15. +-commutative 24.8%

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

      16. associate-/l/ 24.8%

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

      17. associate-*l/ 24.8%

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

   3. Simplified 24.4%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in alpha around inf 79.6%

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

   6. Taylor expanded in beta around 0 82.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}}}{\alpha} \]
   7. Step-by-step derivation

      1. associate--l+ 82.1%

         \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(\left(1 + -1 \cdot \frac{\beta}{\alpha}\right) - 3 \cdot \frac{1}{\alpha}\right) - 2 \cdot \frac{1}{\alpha}\right)}}{\alpha} \]

      2. associate--l+ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{\beta}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right)} - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      3. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \beta}{\alpha}} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      4. mul-1-neg 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{\color{blue}{-\beta}}{\alpha} - 3 \cdot \frac{1}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      5. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \color{blue}{\frac{3 \cdot 1}{\alpha}}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      6. metadata-eval 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{\color{blue}{3}}{\alpha}\right)\right) - 2 \cdot \frac{1}{\alpha}\right)}{\alpha} \]

      7. associate-*r/ 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \color{blue}{\frac{2 \cdot 1}{\alpha}}\right)}{\alpha} \]

      8. metadata-eval 82.1%

         \[\leadsto \frac{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{\color{blue}{2}}{\alpha}\right)}{\alpha} \]

   8. Simplified 82.1%

      \[\leadsto \frac{\color{blue}{1 + \left(\beta \cdot \left(1 + \left(\frac{-\beta}{\alpha} - \frac{3}{\alpha}\right)\right) - \frac{2}{\alpha}\right)}}{\alpha} \]

   9. Taylor expanded in alpha around inf 82.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 20:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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. Step-by-step derivation

      1. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in beta around 0 66.4%

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

      1. *-commutative 66.4%

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

   10. 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. Step-by-step derivation

      1. +-commutative 84.3%

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

   7. Simplified 84.3%

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

   8. 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 8: 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. Step-by-step derivation

      1. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in beta around 0 66.4%

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

      1. *-commutative 66.4%

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

   10. 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%

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

      2. sub-neg 86.3%

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

      3. +-commutative 86.3%

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

      4. neg-sub0 86.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 86.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 86.3%

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

      7. distribute-frac-neg 86.3%

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

      8. +-commutative 86.3%

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

      9. sub-neg 86.3%

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

      10. div-sub 86.3%

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

      11. sub-neg 86.3%

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

      12. metadata-eval 86.3%

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

      13. neg-mul-1 86.3%

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

      14. *-commutative 86.3%

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

      15. +-commutative 86.3%

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

      16. associate-/l/ 86.3%

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

      17. associate-*l/ 86.3%

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

   3. Simplified 86.2%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 86.2%

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

      2. *-commutative 86.2%

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

      3. fma-define 86.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{\beta + \left(\alpha + 2\right)}, \alpha - \beta, 0.5\right)} \]

      4. associate-+r+ 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\left(\beta + \alpha\right) + 2}}, \alpha - \beta, 0.5\right) \]

      5. +-commutative 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\left(\alpha + \beta\right)} + 2}, \alpha - \beta, 0.5\right) \]

      6. associate-+l+ 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\alpha + \left(\beta + 2\right)}}, \alpha - \beta, 0.5\right) \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{\alpha + \left(\beta + 2\right)}, \alpha - \beta, 0.5\right)} \]

   7. Taylor expanded in beta around inf 84.2%

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

Alternative 9: 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. Step-by-step derivation

      1. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   8. 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%

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

      2. sub-neg 86.3%

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

      3. +-commutative 86.3%

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

      4. neg-sub0 86.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

      5. associate-+l- 86.3%

         \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

      6. sub0-neg 86.3%

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

      7. distribute-frac-neg 86.3%

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

      8. +-commutative 86.3%

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

      9. sub-neg 86.3%

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

      10. div-sub 86.3%

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

      11. sub-neg 86.3%

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

      12. metadata-eval 86.3%

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

      13. neg-mul-1 86.3%

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

      14. *-commutative 86.3%

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

      15. +-commutative 86.3%

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

      16. associate-/l/ 86.3%

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

      17. associate-*l/ 86.3%

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

   3. Simplified 86.2%

      \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative 86.2%

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

      2. *-commutative 86.2%

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

      3. fma-define 86.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{\beta + \left(\alpha + 2\right)}, \alpha - \beta, 0.5\right)} \]

      4. associate-+r+ 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\left(\beta + \alpha\right) + 2}}, \alpha - \beta, 0.5\right) \]

      5. +-commutative 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\left(\alpha + \beta\right)} + 2}, \alpha - \beta, 0.5\right) \]

      6. associate-+l+ 86.2%

         \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\color{blue}{\alpha + \left(\beta + 2\right)}}, \alpha - \beta, 0.5\right) \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{\alpha + \left(\beta + 2\right)}, \alpha - \beta, 0.5\right)} \]

   7. Taylor expanded in beta around inf 84.2%

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

Alternative 10: 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. Step-by-step derivation

   1. +-commutative 73.6%

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

7. Simplified 73.6%

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

8. Taylor expanded in beta around 0 47.7%

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

Alternative 11: 3.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 0.0)

C

double code(double alpha, double beta) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 0.0;
}

Python

def code(alpha, beta):
	return 0.0

Julia

function code(alpha, beta)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_] := 0.0

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%

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

   2. sub-neg 75.9%

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

   3. +-commutative 75.9%

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

   4. neg-sub0 75.9%

      \[\leadsto \frac{1 + \frac{\color{blue}{\left(0 - \alpha\right)} + \beta}{\left(\alpha + \beta\right) + 2}}{2} \]

   5. associate-+l- 75.9%

      \[\leadsto \frac{1 + \frac{\color{blue}{0 - \left(\alpha - \beta\right)}}{\left(\alpha + \beta\right) + 2}}{2} \]

   6. sub0-neg 75.9%

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

   7. distribute-frac-neg 75.9%

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

   8. +-commutative 75.9%

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

   9. sub-neg 75.9%

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

   10. div-sub 75.9%

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

   11. sub-neg 75.9%

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

   12. metadata-eval 75.9%

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

   13. neg-mul-1 75.9%

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

   14. *-commutative 75.9%

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

   15. +-commutative 75.9%

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

   16. associate-/l/ 75.9%

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

   17. associate-*l/ 75.9%

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

3. Simplified 75.8%

   \[\leadsto \color{blue}{0.5 + \left(\alpha - \beta\right) \cdot \frac{-0.5}{\beta + \left(\alpha + 2\right)}} \]
4. Add Preprocessing

5. Taylor expanded in alpha around inf 3.7%

   \[\leadsto 0.5 + \color{blue}{-0.5} \]
6. Step-by-step derivation

   1. metadata-eval 3.7%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 3.7%

   \[\leadsto \color{blue}{0} \]
8. 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))