Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.8%

Time: 10.8s

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: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
   (+ (/ (- 1.0 (/ 2.0 alpha)) alpha) (/ beta alpha))
   (fma (/ (- beta alpha) (+ beta (+ alpha 2.0))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = ((1.0 - (2.0 / alpha)) / alpha) + (beta / alpha);
	} else {
		tmp = fma(((beta - alpha) / (beta + (alpha + 2.0))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(1.0 - Float64(2.0 / alpha)) / alpha) + Float64(beta / alpha));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(1.0 - N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(beta / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998000000000054

   1. Initial program 5.9%

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

   3. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 99.9%

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

   6. Taylor expanded in beta around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\beta, \frac{1}{\alpha} - \frac{\color{blue}{3}}{{\alpha}^{2}}, \frac{1}{\alpha}\right) - \frac{2}{{\alpha}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\beta, \frac{1}{\alpha} - \color{blue}{\frac{3}{{\alpha}^{2}}}, \frac{1}{\alpha}\right) - \frac{2}{{\alpha}^{2}} \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

   9. Taylor expanded in alpha around inf

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

       1. lower-/.f64 99.9

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

   11. Simplified 99.9%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate--l+ N/A

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

       6. +-commutative N/A

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

       7. lower-+.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/r* N/A

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

       12. lift-/.f64 N/A

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

       13. div-sub N/A

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

       14. lift--.f64 N/A

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

       15. lift-/.f64 N/A

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

       16. lift-/.f64 N/A

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

       17. un-div-inv N/A

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

       18. lower-/.f64 100.0

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

   13. Applied egg-rr 100.0%

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

   if -0.999998000000000054 < (/.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{1 - \frac{2}{\alpha}}{\alpha} + \frac{\beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -1.0)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.2) (/ 1.0 (+ alpha 2.0)) (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.2) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta - alpha) / ((beta + alpha) + 2.0d0)
    if (t_0 <= (-1.0d0)) then
        tmp = (beta + 1.0d0) / alpha
    else if (t_0 <= 0.2d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    else
        tmp = 1.0d0 + ((-1.0d0) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.2) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (beta - alpha) / ((beta + alpha) + 2.0)
	tmp = 0
	if t_0 <= -1.0:
		tmp = (beta + 1.0) / alpha
	elif t_0 <= 0.2:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = 1.0 + (-1.0 / beta)
	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 <= -1.0)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.2)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	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 <= -1.0)
		tmp = (beta + 1.0) / alpha;
	elseif (t_0 <= 0.2)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = 1.0 + (-1.0 / beta);
	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, -1.0], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -1

   1. Initial program 4.8%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.9

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

   5. Simplified 99.9%

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

   if -1 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 98.7

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Simplified 99.7%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 98.5

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

   10. Simplified 98.5%

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

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{3 + \alpha}{\beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

         \[\leadsto 1 + \frac{\mathsf{fma}\left(\alpha, -0.5, \color{blue}{-1.5}\right)}{\beta} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{\mathsf{fma}\left(\alpha, -0.5, -1.5\right)}{\beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{3}{2} \cdot 1}{\beta}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{3}{2}}}{\beta} \]

      4. lower-/.f64 99.9

         \[\leadsto 1 - \color{blue}{\frac{1.5}{\beta}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1.5}{\beta}} \]

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -1:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.2)
       (fma alpha (fma alpha 0.125 -0.25) 0.5)
       (+ 1.0 (/ -1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.2) {
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	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 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.2)
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	else
		tmp = Float64(1.0 + Float64(-1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(alpha * N[(alpha * 0.125 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 77.6

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

   10. Simplified 77.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 98.5

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

   7. Simplified 98.5%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{1}{8} \cdot \alpha - \frac{1}{4}, \frac{1}{2}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{1}{8} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{1}{8} + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \]

      6. lower-fma.f64 97.6

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.125, -0.25\right)}, 0.5\right) \]

   10. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)} \]

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{3 + \alpha}{\beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

         \[\leadsto 1 + \frac{\mathsf{fma}\left(\alpha, -0.5, \color{blue}{-1.5}\right)}{\beta} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{\mathsf{fma}\left(\alpha, -0.5, -1.5\right)}{\beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{3}{2} \cdot 1}{\beta}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{3}{2}}}{\beta} \]

      4. lower-/.f64 99.9

         \[\leadsto 1 - \color{blue}{\frac{1.5}{\beta}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1.5}{\beta}} \]

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.2) (fma alpha (fma alpha 0.125 -0.25) 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.2) {
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	} else {
		tmp = 1.0;
	}
	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 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.2)
		tmp = fma(alpha, fma(alpha, 0.125, -0.25), 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(alpha * N[(alpha * 0.125 + -0.25), $MachinePrecision] + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 77.6

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

   10. Simplified 77.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 98.5

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

   7. Simplified 98.5%

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

   8. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \left(\frac{1}{8} \cdot \alpha - \frac{1}{4}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \frac{1}{8} \cdot \alpha - \frac{1}{4}, \frac{1}{2}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\frac{1}{8} \cdot \alpha + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\alpha \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\alpha, \alpha \cdot \frac{1}{8} + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \]

      6. lower-fma.f64 97.6

         \[\leadsto \mathsf{fma}\left(\alpha, \color{blue}{\mathsf{fma}\left(\alpha, 0.125, -0.25\right)}, 0.5\right) \]

   10. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)} \]

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

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

   5. Simplified 99.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, \mathsf{fma}\left(\alpha, 0.125, -0.25\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ beta alpha) 2.0))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.2) (fma alpha -0.25 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((beta + alpha) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.2) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = 1.0;
	}
	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 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.2)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.2], N[(alpha * -0.25 + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 8.3%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 8.3%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 8.3

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

   7. Simplified 8.3%

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

   8. Taylor expanded in alpha around inf

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

      1. lower-/.f64 77.6

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

   10. Simplified 77.6%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 98.5

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

   7. Simplified 98.5%

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

   8. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \alpha + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\alpha \cdot \frac{-1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 97.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.25, 0.5\right)} \]

   10. Simplified 97.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\alpha, -0.25, 0.5\right)} \]

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

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

   5. Simplified 99.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999998)
   (/ (+ beta 1.0) alpha)
   (fma (/ (- beta alpha) (+ beta (+ alpha 2.0))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999998) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (beta + (alpha + 2.0))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999998)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta + Float64(alpha + 2.0))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998000000000054

   1. Initial program 5.9%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 99.4

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

   5. Simplified 99.4%

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

   if -0.999998000000000054 < (/.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.5

   1. Initial program 8.3%

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

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 97.6

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

   5. Simplified 97.6%

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

   if -0.5 < (/.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 98.8

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

   7. Simplified 98.8%

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

4. Final simplification 98.5%

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

Alternative 8: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 0.2)
   (/ 1.0 (+ alpha 2.0))
   (+ 1.0 (/ -1.0 beta))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((beta - alpha) / ((beta + alpha) + 2.0d0)) <= 0.2d0) then
        tmp = 1.0d0 / (alpha + 2.0d0)
    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 - alpha) / ((beta + alpha) + 2.0)) <= 0.2) {
		tmp = 1.0 / (alpha + 2.0);
	} else {
		tmp = 1.0 + (-1.0 / beta);
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2:
		tmp = 1.0 / (alpha + 2.0)
	else:
		tmp = 1.0 + (-1.0 / beta)
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= 0.2)
		tmp = Float64(1.0 / Float64(alpha + 2.0));
	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 - alpha) / ((beta + alpha) + 2.0)) <= 0.2)
		tmp = 1.0 / (alpha + 2.0);
	else
		tmp = 1.0 + (-1.0 / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 0.2], N[(1.0 / N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. clear-num N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 68.9

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. lower-+.f64 N/A

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

      14. lower-+.f64 68.9

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

   4. Applied egg-rr 68.9%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   7. Simplified 98.7%

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

   8. Taylor expanded in beta around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l* N/A

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

      4. lft-mult-inverse N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 92.2

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

   10. Simplified 92.2%

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

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{3 + \alpha}{\beta}} \]
   4. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.9

         \[\leadsto 1 + \frac{\mathsf{fma}\left(\alpha, -0.5, \color{blue}{-1.5}\right)}{\beta} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{1 + \frac{\mathsf{fma}\left(\alpha, -0.5, -1.5\right)}{\beta}} \]

   6. Taylor expanded in alpha around 0

      \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{3}{2} \cdot \frac{1}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \color{blue}{\frac{\frac{3}{2} \cdot 1}{\beta}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \frac{\color{blue}{\frac{3}{2}}}{\beta} \]

      4. lower-/.f64 99.9

         \[\leadsto 1 - \color{blue}{\frac{1.5}{\beta}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{1.5}{\beta}} \]

   9. Taylor expanded in beta around 0

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

       1. lower-/.f64 100.0

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

   11. Simplified 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;\frac{1}{\alpha + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) 0.2) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2) {
		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 - alpha) / ((beta + alpha) + 2.0d0)) <= 0.2d0) 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 - alpha) / ((beta + alpha) + 2.0)) <= 0.2) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= 0.2)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= 0.2)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 0.2], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.20000000000000001

   1. Initial program 68.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. flip-+ N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in alpha around inf

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

   7. Simplified 65.9%

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

   if 0.20000000000000001 < (/.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. Taylor expanded in beta around inf

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

   5. Simplified 99.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq 0.2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% accurate, 35.0× 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 77.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. flip-+ N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

4. Applied egg-rr 49.7%

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

5. Taylor expanded in alpha around inf

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

7. Simplified 53.0%

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

Alternative 11: 3.7% accurate, 35.0× 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 77.4%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

4. Applied egg-rr 77.4%

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

5. Taylor expanded in beta around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 53.7

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

7. Simplified 53.7%

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

8. Taylor expanded in alpha around inf

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

10. Simplified 3.5%

    \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{1}, 0.5\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

    2. metadata-eval 3.5

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

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))