Octave 3.8, jcobi/1

Percentage Accurate: 74.7% → 99.7%

Time: 9.2s

Alternatives: 20

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: 74.7% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, \mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (pow
    (/
     alpha
     (fma (* 0.5 (/ (fma -2.0 beta -2.0) alpha)) (- beta -2.0) (+ 1.0 beta)))
    -1.0)
   (fma
    (/ beta (- (+ alpha beta) -2.0))
    0.5
    (*
     (fma
      (/ alpha (- (pow (+ alpha beta) 2.0) 4.0))
      (- (+ alpha beta) 2.0)
      -1.0)
     (- 0.5)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = pow((alpha / fma((0.5 * (fma(-2.0, beta, -2.0) / alpha)), (beta - -2.0), (1.0 + beta))), -1.0);
	} else {
		tmp = fma((beta / ((alpha + beta) - -2.0)), 0.5, (fma((alpha / (pow((alpha + beta), 2.0) - 4.0)), ((alpha + beta) - 2.0), -1.0) * -0.5));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(alpha / fma(Float64(0.5 * Float64(fma(-2.0, beta, -2.0) / alpha)), Float64(beta - -2.0), Float64(1.0 + beta))) ^ -1.0;
	else
		tmp = fma(Float64(beta / Float64(Float64(alpha + beta) - -2.0)), 0.5, Float64(fma(Float64(alpha / Float64((Float64(alpha + beta) ^ 2.0) - 4.0)), Float64(Float64(alpha + beta) - 2.0), -1.0) * Float64(-0.5)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[Power[N[(alpha / N[(N[(0.5 * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] * N[(beta - -2.0), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(beta / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(alpha / N[(N[Power[N[(alpha + beta), $MachinePrecision], 2.0], $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] - 2.0), $MachinePrecision] + -1.0), $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. flip-+ N/A

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

      11. associate-/r/ N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, -\color{blue}{\mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right)} \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, \mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{0.5}{\left(\alpha + \beta\right) - -2}, -0.5 \cdot \mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (pow
    (/
     alpha
     (fma (* 0.5 (/ (fma -2.0 beta -2.0) alpha)) (- beta -2.0) (+ 1.0 beta)))
    -1.0)
   (fma
    beta
    (/ 0.5 (- (+ alpha beta) -2.0))
    (*
     -0.5
     (fma
      (/ alpha (- (pow (+ alpha beta) 2.0) 4.0))
      (- (+ alpha beta) 2.0)
      -1.0)))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = pow((alpha / fma((0.5 * (fma(-2.0, beta, -2.0) / alpha)), (beta - -2.0), (1.0 + beta))), -1.0);
	} else {
		tmp = fma(beta, (0.5 / ((alpha + beta) - -2.0)), (-0.5 * fma((alpha / (pow((alpha + beta), 2.0) - 4.0)), ((alpha + beta) - 2.0), -1.0)));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(alpha / fma(Float64(0.5 * Float64(fma(-2.0, beta, -2.0) / alpha)), Float64(beta - -2.0), Float64(1.0 + beta))) ^ -1.0;
	else
		tmp = fma(beta, Float64(0.5 / Float64(Float64(alpha + beta) - -2.0)), Float64(-0.5 * fma(Float64(alpha / Float64((Float64(alpha + beta) ^ 2.0) - 4.0)), Float64(Float64(alpha + beta) - 2.0), -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[Power[N[(alpha / N[(N[(0.5 * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] * N[(beta - -2.0), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(beta * N[(0.5 / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(alpha / N[(N[Power[N[(alpha + beta), $MachinePrecision], 2.0], $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] - 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

   8. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. lift-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(\beta, \frac{0.5}{\left(\alpha + \beta\right) - -2}, -0.5 \cdot \mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \color{blue}{\left(\alpha + \beta\right)} - 2, -1\right)\right) \]

   10. Applied rewrites 99.9%

       \[\leadsto \mathsf{fma}\left(\beta, \frac{0.5}{\left(\alpha + \beta\right) - -2}, -0.5 \cdot \color{blue}{\mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{0.5}{\left(\alpha + \beta\right) - -2}, -0.5 \cdot \mathsf{fma}\left(\frac{\alpha}{{\left(\alpha + \beta\right)}^{2} - 4}, \left(\alpha + \beta\right) - 2, -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) - -2\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{t\_0}, 0.5, \left(\frac{\alpha}{t\_0} - 1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (+ alpha beta) -2.0)))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
     (pow
      (/
       alpha
       (fma (* 0.5 (/ (fma -2.0 beta -2.0) alpha)) (- beta -2.0) (+ 1.0 beta)))
      -1.0)
     (fma (/ beta t_0) 0.5 (* (- (/ alpha t_0) 1.0) (- 0.5))))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) - -2.0;
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = pow((alpha / fma((0.5 * (fma(-2.0, beta, -2.0) / alpha)), (beta - -2.0), (1.0 + beta))), -1.0);
	} else {
		tmp = fma((beta / t_0), 0.5, (((alpha / t_0) - 1.0) * -0.5));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) - -2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(alpha / fma(Float64(0.5 * Float64(fma(-2.0, beta, -2.0) / alpha)), Float64(beta - -2.0), Float64(1.0 + beta))) ^ -1.0;
	else
		tmp = fma(Float64(beta / t_0), 0.5, Float64(Float64(Float64(alpha / t_0) - 1.0) * Float64(-0.5)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[Power[N[(alpha / N[(N[(0.5 * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] * N[(beta - -2.0), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(beta / t$95$0), $MachinePrecision] * 0.5 + N[(N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;{\left(\frac{\alpha}{\mathsf{fma}\left(0.5 \cdot \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, \beta - -2, 1 + \beta\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, \left(\frac{\alpha}{\left(\alpha + \beta\right) - -2} - 1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{-2 - \left(\alpha + \beta\right)}{\alpha - \beta}\right)}^{-1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (/
    (fma (fma 0.5 beta 1.0) (/ (fma -2.0 beta -2.0) alpha) (+ 1.0 beta))
    alpha)
   (/ (+ (pow (/ (- -2.0 (+ alpha beta)) (- alpha beta)) -1.0) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = fma(fma(0.5, beta, 1.0), (fma(-2.0, beta, -2.0) / alpha), (1.0 + beta)) / alpha;
	} else {
		tmp = (pow(((-2.0 - (alpha + beta)) / (alpha - beta)), -1.0) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(fma(fma(0.5, beta, 1.0), Float64(fma(-2.0, beta, -2.0) / alpha), Float64(1.0 + beta)) / alpha);
	else
		tmp = Float64(Float64((Float64(Float64(-2.0 - Float64(alpha + beta)) / Float64(alpha - beta)) ^ -1.0) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(0.5 * beta + 1.0), $MachinePrecision] * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[Power[N[(N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha - beta), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. neg-mul-1 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. neg-mul-1 N/A

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

      18. remove-double-neg N/A

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

      19. neg-mul-1 N/A

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

      20. sub-neg N/A

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

      21. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{-2 - \left(\alpha + \beta\right)}{\alpha - \beta}\right)}^{-1} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(-0.5 * N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(1.0 - N[Power[beta, -1.0], $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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 98.0

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

   9. Applied rewrites 98.0%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 96.6%

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

4. Final simplification 97.8%

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

Alternative 6: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \alpha, 0.125\right), \alpha, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ (+ alpha beta) 2.0))))
   (if (<= t_0 -0.5)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.5)
       (fma (fma (fma -0.0625 alpha 0.125) alpha -0.25) alpha 0.5)
       (- 1.0 (pow beta -1.0))))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / ((alpha + beta) + 2.0);
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.5) {
		tmp = fma(fma(fma(-0.0625, alpha, 0.125), alpha, -0.25), alpha, 0.5);
	} else {
		tmp = 1.0 - pow(beta, -1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(fma(fma(-0.0625, alpha, 0.125), alpha, -0.25), alpha, 0.5);
	else
		tmp = Float64(1.0 - (beta ^ -1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(N[(-0.0625 * alpha + 0.125), $MachinePrecision] * alpha + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(1.0 - N[Power[beta, -1.0], $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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{1}{8} + \frac{-1}{16} \cdot \alpha\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \alpha, 0.125\right), \alpha, -0.25\right), \color{blue}{\alpha}, 0.5\right) \]

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 96.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.5:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0625, \alpha, 0.125\right), \alpha, -0.25\right), \alpha, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {\beta}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 0.5);
	else
		tmp = Float64(1.0 - (beta ^ -1.0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 0.5), $MachinePrecision], N[(1.0 - N[Power[beta, -1.0], $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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.4%

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in beta around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 96.6%

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

4. Final simplification 97.5%

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

Alternative 8: 92.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = alpha ^ -1.0;
	elseif (t_0 <= 0.5)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 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 7.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 81.4%

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

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.4%

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

   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 94.8%

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

4. Final simplification 92.3%

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

Alternative 9: 91.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = alpha ^ -1.0;
	elseif (t_0 <= 0.5)
		tmp = fma(-0.25, alpha, 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[Power[alpha, -1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(-0.25 * alpha + 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 7.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 81.4%

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

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.4%

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

   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 94.8%

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

4. Final simplification 92.3%

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

Alternative 10: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) - -2\\ \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{t\_0}, 0.5, \left(\frac{\alpha}{t\_0} - 1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (+ alpha beta) -2.0)))
   (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
     (/
      (fma (fma 0.5 beta 1.0) (/ (fma -2.0 beta -2.0) alpha) (+ 1.0 beta))
      alpha)
     (fma (/ beta t_0) 0.5 (* (- (/ alpha t_0) 1.0) (- 0.5))))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) - -2.0;
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = fma(fma(0.5, beta, 1.0), (fma(-2.0, beta, -2.0) / alpha), (1.0 + beta)) / alpha;
	} else {
		tmp = fma((beta / t_0), 0.5, (((alpha / t_0) - 1.0) * -0.5));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) - -2.0)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(fma(fma(0.5, beta, 1.0), Float64(fma(-2.0, beta, -2.0) / alpha), Float64(1.0 + beta)) / alpha);
	else
		tmp = fma(Float64(beta / t_0), 0.5, Float64(Float64(Float64(alpha / t_0) - 1.0) * Float64(-0.5)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]}, If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(0.5 * beta + 1.0), $MachinePrecision] * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(beta / t$95$0), $MachinePrecision] * 0.5 + N[(N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] * (-0.5)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, \left(\frac{\alpha}{\left(\alpha + \beta\right) - -2} - 1\right) \cdot \left(-0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(-0.5 * N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 * alpha + -1.0), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

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

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 100.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in beta around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 98.0

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

   9. Applied rewrites 98.0%

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   9. Taylor expanded in beta around inf

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-lft-in N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 97.0

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

   11. Applied rewrites 97.0%

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

Alternative 12: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{-2 - \left(\alpha + \beta\right)}, \beta, \mathsf{fma}\left(\frac{\alpha}{\left(\alpha + \beta\right) - -2}, -0.5, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (/
    (fma (fma 0.5 beta 1.0) (/ (fma -2.0 beta -2.0) alpha) (+ 1.0 beta))
    alpha)
   (fma
    (/ -0.5 (- -2.0 (+ alpha beta)))
    beta
    (fma (/ alpha (- (+ alpha beta) -2.0)) -0.5 0.5))))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = fma(fma(0.5, beta, 1.0), (fma(-2.0, beta, -2.0) / alpha), (1.0 + beta)) / alpha;
	} else {
		tmp = fma((-0.5 / (-2.0 - (alpha + beta))), beta, fma((alpha / ((alpha + beta) - -2.0)), -0.5, 0.5));
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(fma(fma(0.5, beta, 1.0), Float64(fma(-2.0, beta, -2.0) / alpha), Float64(1.0 + beta)) / alpha);
	else
		tmp = fma(Float64(-0.5 / Float64(-2.0 - Float64(alpha + beta))), beta, fma(Float64(alpha / Float64(Float64(alpha + beta) - -2.0)), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(0.5 * beta + 1.0), $MachinePrecision] * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(-0.5 / N[(-2.0 - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(alpha / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.9

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

   8. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 99.9

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift--.f64 N/A

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

      17. lift-+.f64 99.9

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

      18. lift-*.f64 N/A

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

      19. lift--.f64 N/A

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

      20. sub-neg N/A

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

      21. metadata-eval N/A

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

   10. Applied rewrites 99.9%

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

Alternative 13: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \beta, 1\right), \frac{\mathsf{fma}\left(-2, \beta, -2\right)}{\alpha}, 1 + \beta\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (/
    (fma (fma 0.5 beta 1.0) (/ (fma -2.0 beta -2.0) alpha) (+ 1.0 beta))
    alpha)
   (fma (/ (- alpha beta) (- -2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = fma(fma(0.5, beta, 1.0), (fma(-2.0, beta, -2.0) / alpha), (1.0 + beta)) / alpha;
	} else {
		tmp = fma(((alpha - beta) / (-2.0 - (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(fma(fma(0.5, beta, 1.0), Float64(fma(-2.0, beta, -2.0) / alpha), Float64(1.0 + beta)) / alpha);
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(0.5 * beta + 1.0), $MachinePrecision] * N[(N[(-2.0 * beta + -2.0), $MachinePrecision] / alpha), $MachinePrecision] + N[(1.0 + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $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))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot \beta\right) + \frac{1}{2} \cdot \frac{-1 \cdot {\left(2 + \beta\right)}^{2} - \beta \cdot \left(2 + \beta\right)}{\alpha}}{\alpha}} \]
   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. Applied rewrites 99.9%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 99.9%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. 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 99.9%

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

Alternative 14: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{\left(\alpha + \beta\right) - -2}, 0.5, \frac{\mathsf{fma}\left(0.5, \beta, 1\right)}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (fma (/ beta (- (+ alpha beta) -2.0)) 0.5 (/ (fma 0.5 beta 1.0) alpha))
   (fma (/ (- alpha beta) (- -2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = fma((beta / ((alpha + beta) - -2.0)), 0.5, (fma(0.5, beta, 1.0) / alpha));
	} else {
		tmp = fma(((alpha - beta) / (-2.0 - (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = fma(Float64(beta / Float64(Float64(alpha + beta) - -2.0)), 0.5, Float64(fma(0.5, beta, 1.0) / alpha));
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(beta / N[(N[(alpha + beta), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(0.5 * beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $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))) < -0.99990000000000001

   1. Initial program 6.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 9.3

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 9.3

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

   4. Applied rewrites 9.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. associate-+r- N/A

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

      15. lift-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. div-inv N/A

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

   6. Applied rewrites 9.3%

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

   7. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. 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 99.9%

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

Alternative 15: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.5)
		tmp = fma(fma(0.125, alpha, -0.25), alpha, 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(N[(0.125 * alpha + -0.25), $MachinePrecision] * alpha + 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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

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

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 97.4%

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

   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 94.8%

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

Alternative 16: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} \leq -0.9999:\\ \;\;\;\;\frac{\left(\beta - -2\right) + \beta}{\alpha} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{-2 - \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ (+ alpha beta) 2.0)) -0.9999)
   (* (/ (+ (- beta -2.0) beta) alpha) 0.5)
   (fma (/ (- alpha beta) (- -2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.9999) {
		tmp = (((beta - -2.0) + beta) / alpha) * 0.5;
	} else {
		tmp = fma(((alpha - beta) / (-2.0 - (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) <= -0.9999)
		tmp = Float64(Float64(Float64(Float64(beta - -2.0) + beta) / alpha) * 0.5);
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(-2.0 - Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.9999], N[(N[(N[(N[(beta - -2.0), $MachinePrecision] + beta), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(alpha - beta), $MachinePrecision] / N[(-2.0 - N[(alpha + beta), $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))) < -0.99990000000000001

   1. Initial program 6.8%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if -0.99990000000000001 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64)))

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. 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 99.9%

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

Alternative 17: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.5) {
		tmp = (((beta - -2.0) + beta) / alpha) * 0.5;
	} 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(alpha + beta) + 2.0)) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(beta - -2.0) + beta) / alpha) * 0.5);
	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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(beta - -2.0), $MachinePrecision] + beta), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $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 7.9%

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

   3. Taylor expanded in alpha around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. metadata-eval 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   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-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 18: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= -0.5) {
		tmp = (1.0 + beta) / 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(alpha + beta) + 2.0)) <= -0.5)
		tmp = Float64(Float64(1.0 + beta) / 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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(1.0 + beta), $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 7.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   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-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 19: 71.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / ((alpha + beta) + 2.0)) <= 0.5)
		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[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], 0.5], 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))) < 0.5

   1. Initial program 65.9%

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

   3. Taylor expanded in beta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto 0.5 \]

   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 94.8%

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

Alternative 20: 36.5% 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 74.4%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 34.2%

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

Reproduce

herbie shell --seed 2024302 
(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))