Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.3%

Time: 10.3s

Alternatives: 9

Speedup: 0.7×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% 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: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      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. sub-neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+r+ N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-2neg N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      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. sub-neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. associate-+r+ N/A

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

   6. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f64 N/A

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

      14. lower-+.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.5)
     (/ (+ beta 1.0) alpha)
     (if (<= t_0 0.01) (fma alpha -0.25 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (beta + 1.0) / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(beta + 1.0) / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(beta + 1.0), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(alpha * -0.25 + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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))) < 0.0100000000000000002

   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. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 97.5

          \[\leadsto 0.5 + 0.5 \cdot \frac{\alpha}{\color{blue}{-2} - \alpha} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot \frac{\alpha}{-2 - \alpha}} \]

   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 97.5%

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

   if 0.0100000000000000002 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (- beta alpha) (+ 2.0 (+ beta alpha)))))
   (if (<= t_0 -0.5)
     (/ 1.0 alpha)
     (if (<= t_0 0.01) (fma alpha -0.25 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (beta - alpha) / (2.0 + (beta + alpha));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.01) {
		tmp = fma(alpha, -0.25, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.01)
		tmp = fma(alpha, -0.25, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(alpha * -0.25 + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.6%

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

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 5.5

          \[\leadsto 0.5 + 0.5 \cdot \frac{\alpha}{\color{blue}{-2} - \alpha} \]

   5. Applied rewrites 5.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot \frac{\alpha}{-2 - \alpha}} \]

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 73.3%

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

   if -0.5 < (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) < 0.0100000000000000002

   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. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 97.5

          \[\leadsto 0.5 + 0.5 \cdot \frac{\alpha}{\color{blue}{-2} - \alpha} \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot \frac{\alpha}{-2 - \alpha}} \]

   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 97.5%

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

   if 0.0100000000000000002 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq -0.5:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\alpha, -0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.5%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      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. sub-neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 N/A

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

      21. lower-neg.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

4. Final simplification 99.8%

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

Alternative 6: 98.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.6%

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

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in alpha around 0

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

   7. Applied rewrites 99.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 99.3

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

   9. Applied rewrites 99.3%

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

4. Final simplification 99.0%

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

Alternative 7: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.6%

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

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 98.9%

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

Alternative 8: 71.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.01:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (- beta alpha) (+ 2.0 (+ beta alpha))) 0.01) 0.5 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= 0.01) {
		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) / (2.0d0 + (beta + alpha))) <= 0.01d0) 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) / (2.0 + (beta + alpha))) <= 0.01) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((beta - alpha) / (2.0 + (beta + alpha))) <= 0.01:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(beta - alpha) / Float64(2.0 + Float64(beta + alpha))) <= 0.01)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((beta - alpha) / (2.0 + (beta + alpha))) <= 0.01)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.01], 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))) < 0.0100000000000000002

   1. Initial program 62.9%

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

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. metadata-eval 60.6

          \[\leadsto 0.5 + 0.5 \cdot \frac{\alpha}{\color{blue}{-2} - \alpha} \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{0.5 + 0.5 \cdot \frac{\alpha}{-2 - \alpha}} \]

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 60.4%

      \[\leadsto 0.5 \]

   if 0.0100000000000000002 < (/.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. Taylor expanded in beta around inf

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

   5. Applied rewrites 95.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{2 + \left(\beta + \alpha\right)} \leq 0.01:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.6% accurate, 35.0× speedup

Math

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

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

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

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 74.3%

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

3. Taylor expanded in beta around inf

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

5. Applied rewrites 38.9%

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

Reproduce

herbie shell --seed 2024227 
(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))