Octave 3.8, jcobi/1

Percentage Accurate: 74.2% → 99.9%

Time: 9.4s

Alternatives: 9

Speedup: 0.7×

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: 74.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \beta, -1\right), \frac{\beta + \left(\beta + 2\right)}{\alpha}, 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{t\_0} - \frac{\alpha}{t\_0}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ beta (+ alpha 2.0))))
   (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99996)
     (/
      (+ beta (fma (fma -0.5 beta -1.0) (/ (+ beta (+ beta 2.0)) alpha) 1.0))
      alpha)
     (/ (+ 1.0 (- (/ beta t_0) (/ alpha t_0))) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = beta + (alpha + 2.0);
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99996) {
		tmp = (beta + fma(fma(-0.5, beta, -1.0), ((beta + (beta + 2.0)) / alpha), 1.0)) / alpha;
	} else {
		tmp = (1.0 + ((beta / t_0) - (alpha / t_0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(beta + Float64(alpha + 2.0))
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.99996)
		tmp = Float64(Float64(beta + fma(fma(-0.5, beta, -1.0), Float64(Float64(beta + Float64(beta + 2.0)) / alpha), 1.0)) / alpha);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(beta / t_0) - Float64(alpha / t_0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.99996], N[(N[(beta + N[(N[(-0.5 * beta + -1.0), $MachinePrecision] * N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 + N[(N[(beta / t$95$0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99995999999999996

   1. Initial program 6.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. flip-- N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. sqr-neg N/A

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

      9. difference-of-squares N/A

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

      10. sub-neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 2.8

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

      21. lift-+.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. associate-+l+ N/A

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

      25. lower-+.f64 N/A

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

      26. lower-+.f64 2.8

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

      27. lift-+.f64 N/A

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

   4. Applied rewrites 2.8%

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

   5. Taylor expanded in alpha around inf

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

   7. Applied rewrites 99.8%

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

   if -0.99995999999999996 < (/.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. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \beta, -1\right), \frac{\beta + \left(\beta + 2\right)}{\alpha}, 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\frac{\beta}{\beta + \left(\alpha + 2\right)} - \frac{\alpha}{\beta + \left(\alpha + 2\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \beta, -1\right), \frac{\beta + \left(\beta + 2\right)}{\alpha}, 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{-\beta}, \beta\right)}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.99996)
   (/
    (+ beta (fma (fma -0.5 beta -1.0) (/ (+ beta (+ beta 2.0)) alpha) 1.0))
    alpha)
   (fma
    (/ (fma beta (/ alpha (- beta)) beta) (+ beta (+ alpha 2.0)))
    0.5
    0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.99996) {
		tmp = (beta + fma(fma(-0.5, beta, -1.0), ((beta + (beta + 2.0)) / alpha), 1.0)) / alpha;
	} else {
		tmp = fma((fma(beta, (alpha / -beta), beta) / (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)) <= -0.99996)
		tmp = Float64(Float64(beta + fma(fma(-0.5, beta, -1.0), Float64(Float64(beta + Float64(beta + 2.0)) / alpha), 1.0)) / alpha);
	else
		tmp = fma(Float64(fma(beta, Float64(alpha / Float64(-beta)), beta) / 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], -0.99996], N[(N[(beta + N[(N[(-0.5 * beta + -1.0), $MachinePrecision] * N[(N[(beta + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta * N[(alpha / (-beta)), $MachinePrecision] + beta), $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))) < -0.99995999999999996

   1. Initial program 6.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. flip-- N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. sqr-neg N/A

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

      9. difference-of-squares N/A

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

      10. sub-neg N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 2.8

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

      21. lift-+.f64 N/A

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. associate-+l+ N/A

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

      25. lower-+.f64 N/A

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

      26. lower-+.f64 2.8

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

      27. lift-+.f64 N/A

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

   4. Applied rewrites 2.8%

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

   5. Taylor expanded in alpha around inf

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

   7. Applied rewrites 99.8%

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

   if -0.99995999999999996 < (/.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 rewrites 99.8%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.99996:\\ \;\;\;\;\frac{\beta + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \beta, -1\right), \frac{\beta + \left(\beta + 2\right)}{\alpha}, 1\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{-\beta}, \beta\right)}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{-\beta}, \beta\right)}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
   (/ (+ beta 1.0) alpha)
   (fma
    (/ (fma beta (/ alpha (- beta)) beta) (+ beta (+ alpha 2.0)))
    0.5
    0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((fma(beta, (alpha / -beta), beta) / (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)) <= -0.999999)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(fma(beta, Float64(alpha / Float64(-beta)), beta) / 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], -0.999999], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta * N[(alpha / (-beta)), $MachinePrecision] + beta), $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))) < -0.999998999999999971

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. 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 rewrites 99.7%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\beta, \frac{\alpha}{-\beta}, \beta\right)}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\beta + \left(\alpha + 2\right)}{\beta - \alpha}}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
   (/ (+ beta 1.0) alpha)
   (fma (/ 1.0 (/ (+ beta (+ alpha 2.0)) (- beta alpha))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((1.0 / ((beta + (alpha + 2.0)) / (beta - alpha))), 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)) <= -0.999999)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(1.0 / Float64(Float64(beta + Float64(alpha + 2.0)) / Float64(beta - alpha))), 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], -0.999999], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(1.0 / N[(N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $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))) < -0.999998999999999971

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. 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 rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. *-inverses N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. clear-num N/A

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

      12. lower-/.f64 N/A

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

      13. clear-num N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

   6. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\beta + \left(\alpha + 2\right)}{\beta - \alpha}}}, 0.5, 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\beta + \left(\alpha + 2\right)}{\beta - \alpha}}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
   (/ (+ 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)) <= -0.999999) {
		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)) <= -0.999999)
		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], -0.999999], 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))) < -0.999998999999999971

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. 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 rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)}, 0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.999999)
   (/ (+ beta 1.0) alpha)
   (fma (- beta alpha) (/ 0.5 (+ beta (+ alpha 2.0))) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.999999) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = fma((beta - alpha), (0.5 / (beta + (alpha + 2.0))), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(beta + alpha) + 2.0)) <= -0.999999)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	else
		tmp = fma(Float64(beta - alpha), Float64(0.5 / Float64(beta + Float64(alpha + 2.0))), 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], -0.999999], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta - alpha), $MachinePrecision] * N[(0.5 / N[(beta + N[(alpha + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.999998999999999971

   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 99.7

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

   5. Applied rewrites 99.7%

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

   if -0.999998999999999971 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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. 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 rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \leq -0.999999:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{0.5}{\beta + \left(\alpha + 2\right)}, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ beta alpha) 2.0)) -0.5)
   (/ (+ beta 1.0) alpha)
   (fma 0.5 (/ beta (+ beta 2.0)) 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(0.5, (beta / (beta + 2.0)), 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(0.5, Float64(beta / Float64(beta + 2.0)), 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[(0.5 * N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] + 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 10.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   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. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.2%

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

Alternative 8: 62.7% accurate, 0.9× speedup

Math

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

FPCore

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

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 = 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)) <= (-0.5d0)) then
        tmp = (beta + 1.0d0) / alpha
    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)) <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / ((beta + alpha) + 2.0)) <= -0.5:
		tmp = (beta + 1.0) / alpha
	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)) <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	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)) <= -0.5)
		tmp = (beta + 1.0) / alpha;
	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], -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], 1.0]

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 10.2%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   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. Taylor expanded in beta around inf

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

   5. Applied rewrites 48.1%

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

4. Final simplification 62.0%

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

Alternative 9: 36.6% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 74.0%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 35.8%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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))