Octave 3.8, jcobi/3

Percentage Accurate: 94.4% → 99.8%

Time: 9.2s

Alternatives: 20

Speedup: 2.6×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

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

Initial Program: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot 1\\ \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{t\_0}}{t\_0}}{t\_0 + 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 1.0))))
   (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) t_0) t_0) (+ t_0 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Fortran

real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * 1.0d0)
    code = (((((alpha + beta) + (beta * alpha)) + 1.0d0) / t_0) / t_0) / (t_0 + 1.0d0)
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + (2.0 * 1.0);
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
}

Python

def code(alpha, beta):
	t_0 = (alpha + beta) + (2.0 * 1.0)
	return (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0)

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * 1.0))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) + Float64(beta * alpha)) + 1.0) / t_0) / t_0) / Float64(t_0 + 1.0))
end

MATLAB

function tmp = code(alpha, beta)
	t_0 = (alpha + beta) + (2.0 * 1.0);
	tmp = (((((alpha + beta) + (beta * alpha)) + 1.0) / t_0) / t_0) / (t_0 + 1.0);
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \frac{\frac{1}{t\_0 \cdot \frac{\frac{t\_0}{\alpha - -1}}{\beta - -1}}}{\left(\left(\beta + \alpha\right) - -1\right) + 2} \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (/
    (/ 1.0 (* t_0 (/ (/ t_0 (- alpha -1.0)) (- beta -1.0))))
    (+ (- (+ beta alpha) -1.0) 2.0))))

C

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

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    t_0 = (beta + alpha) + 2.0d0
    code = (1.0d0 / (t_0 * ((t_0 / (alpha - (-1.0d0))) / (beta - (-1.0d0))))) / (((beta + alpha) - (-1.0d0)) + 2.0d0)
end function

Java

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

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	return (1.0 / (t_0 * ((t_0 / (alpha - -1.0)) / (beta - -1.0)))) / (((beta + alpha) - -1.0) + 2.0)

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	return Float64(Float64(1.0 / Float64(t_0 * Float64(Float64(t_0 / Float64(alpha - -1.0)) / Float64(beta - -1.0)))) / Float64(Float64(Float64(beta + alpha) - -1.0) + 2.0))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = (1.0 / (t_0 * ((t_0 / (alpha - -1.0)) / (beta - -1.0)))) / (((beta + alpha) - -1.0) + 2.0);
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(1.0 / N[(t$95$0 * N[(N[(t$95$0 / N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision] / N[(beta - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] - -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 95.2

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 95.2

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

   10. lift-*.f64 N/A

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

   11. metadata-eval 95.2

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

4. Applied rewrites 95.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 95.2

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

6. Applied rewrites 94.8%

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

8. Applied rewrites 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lower-+.f64 99.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.9

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

   20. lift-+.f64 N/A

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

   21. +-commutative N/A

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

   22. lift-+.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{-1}{\frac{t\_0}{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)} \cdot t\_0}}{\left(-1 - \left(\beta + \alpha\right)\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+17)
     (/
      (/ -1.0 (* (/ t_0 (* (- -1.0 beta) (- -1.0 alpha))) t_0))
      (- (- -1.0 (+ beta alpha)) 2.0))
     (/ (/ (- alpha -1.0) t_0) (- t_0 -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = (-1.0 / ((t_0 / ((-1.0 - beta) * (-1.0 - alpha))) * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 5d+17) then
        tmp = ((-1.0d0) / ((t_0 / (((-1.0d0) - beta) * ((-1.0d0) - alpha))) * t_0)) / (((-1.0d0) - (beta + alpha)) - 2.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (t_0 - (-1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = (-1.0 / ((t_0 / ((-1.0 - beta) * (-1.0 - alpha))) * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 5e+17:
		tmp = (-1.0 / ((t_0 / ((-1.0 - beta) * (-1.0 - alpha))) * t_0)) / ((-1.0 - (beta + alpha)) - 2.0)
	else:
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+17)
		tmp = Float64(Float64(-1.0 / Float64(Float64(t_0 / Float64(Float64(-1.0 - beta) * Float64(-1.0 - alpha))) * t_0)) / Float64(Float64(-1.0 - Float64(beta + alpha)) - 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 - -1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 5e+17)
		tmp = (-1.0 / ((t_0 / ((-1.0 - beta) * (-1.0 - alpha))) * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	else
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+17], N[(N[(-1.0 / N[(N[(t$95$0 / N[(N[(-1.0 - beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e17

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.8

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 99.9%

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

   if 5e17 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{-1}{\frac{\left(\beta + \alpha\right) + 2}{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)} \cdot \left(\left(\beta + \alpha\right) + 2\right)}}{\left(-1 - \left(\beta + \alpha\right)\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0}}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, 2, 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 9e+23)
     (/
      (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_0)
      (fma (+ beta alpha) (+ 3.0 (+ beta alpha)) (fma (+ beta alpha) 2.0 6.0)))
     (/ (/ (- alpha -1.0) t_0) (- t_0 -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 9e+23) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_0) / fma((beta + alpha), (3.0 + (beta + alpha)), fma((beta + alpha), 2.0, 6.0));
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 9e+23)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_0) / fma(Float64(beta + alpha), Float64(3.0 + Float64(beta + alpha)), fma(Float64(beta + alpha), 2.0, 6.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 - -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9e+23], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] * N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * 2.0 + 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.99999999999999958e23

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.8

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, 2, \color{blue}{6}\right)\right)} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) + 1}{\left(\beta + \alpha\right) + 2}}{\color{blue}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, 2, 6\right)\right)}} \]

   if 8.99999999999999958e23 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\beta + \alpha\right) + 2}}{\mathsf{fma}\left(\beta + \alpha, 3 + \left(\beta + \alpha\right), \mathsf{fma}\left(\beta + \alpha, 2, 6\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\left(-1 - \beta\right) \cdot \left(\alpha - -1\right)}{t\_0 \cdot t\_0}}{\left(-1 - \left(\beta + \alpha\right)\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+17)
     (/
      (/ (* (- -1.0 beta) (- alpha -1.0)) (* t_0 t_0))
      (- (- -1.0 (+ beta alpha)) 2.0))
     (/ (/ (- alpha -1.0) t_0) (- t_0 -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = (((-1.0 - beta) * (alpha - -1.0)) / (t_0 * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 5d+17) then
        tmp = ((((-1.0d0) - beta) * (alpha - (-1.0d0))) / (t_0 * t_0)) / (((-1.0d0) - (beta + alpha)) - 2.0d0)
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (t_0 - (-1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = (((-1.0 - beta) * (alpha - -1.0)) / (t_0 * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 5e+17:
		tmp = (((-1.0 - beta) * (alpha - -1.0)) / (t_0 * t_0)) / ((-1.0 - (beta + alpha)) - 2.0)
	else:
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+17)
		tmp = Float64(Float64(Float64(Float64(-1.0 - beta) * Float64(alpha - -1.0)) / Float64(t_0 * t_0)) / Float64(Float64(-1.0 - Float64(beta + alpha)) - 2.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 - -1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 5e+17)
		tmp = (((-1.0 - beta) * (alpha - -1.0)) / (t_0 * t_0)) / ((-1.0 - (beta + alpha)) - 2.0);
	else
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+17], N[(N[(N[(N[(-1.0 - beta), $MachinePrecision] * N[(alpha - -1.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e17

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.8

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 99.7%

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

   if 5e17 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{\left(-1 - \beta\right) \cdot \left(\alpha - -1\right)}{\left(\left(\beta + \alpha\right) + 2\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}}{\left(-1 - \left(\beta + \alpha\right)\right) - 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{t\_0}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 9e+23)
     (/
      (/ (- (fma beta alpha (+ beta alpha)) -1.0) t_0)
      (* (+ 3.0 (+ beta alpha)) t_0))
     (/ (/ (- alpha -1.0) t_0) (- t_0 -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 9e+23) {
		tmp = ((fma(beta, alpha, (beta + alpha)) - -1.0) / t_0) / ((3.0 + (beta + alpha)) * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 9e+23)
		tmp = Float64(Float64(Float64(fma(beta, alpha, Float64(beta + alpha)) - -1.0) / t_0) / Float64(Float64(3.0 + Float64(beta + alpha)) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 - -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 9e+23], N[(N[(N[(N[(beta * alpha + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.99999999999999958e23

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 8.99999999999999958e23 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\beta, \alpha, \beta + \alpha\right) - -1}{\left(\beta + \alpha\right) + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+17)
     (/
      (* (- -1.0 beta) (- -1.0 alpha))
      (* (* (+ 3.0 (+ beta alpha)) t_0) t_0))
     (/ (/ (- alpha -1.0) t_0) (- t_0 -1.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 5d+17) then
        tmp = (((-1.0d0) - beta) * ((-1.0d0) - alpha)) / (((3.0d0 + (beta + alpha)) * t_0) * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / t_0) / (t_0 - (-1.0d0))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 5e+17:
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0)
	else:
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+17)
		tmp = Float64(Float64(Float64(-1.0 - beta) * Float64(-1.0 - alpha)) / Float64(Float64(Float64(3.0 + Float64(beta + alpha)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / t_0) / Float64(t_0 - -1.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 5e+17)
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	else
		tmp = ((alpha - -1.0) / t_0) / (t_0 - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+17], N[(N[(N[(-1.0 - beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e17

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.8

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 91.1%

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

   if 5e17 < beta

   1. Initial program 84.0%

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

   3. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-in N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\left(\beta + \alpha\right) + 2}}{\left(\left(\beta + \alpha\right) + 2\right) - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0)))
   (if (<= beta 5e+17)
     (/
      (* (- -1.0 beta) (- -1.0 alpha))
      (* (* (+ 3.0 (+ beta alpha)) t_0) t_0))
     (/ (/ (- alpha -1.0) beta) (+ (- (+ beta alpha) -1.0) 2.0)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (beta + alpha) + 2.0d0
    if (beta <= 5d+17) then
        tmp = (((-1.0d0) - beta) * ((-1.0d0) - alpha)) / (((3.0d0 + (beta + alpha)) * t_0) * t_0)
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (((beta + alpha) - (-1.0d0)) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double t_0 = (beta + alpha) + 2.0;
	double tmp;
	if (beta <= 5e+17) {
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	t_0 = (beta + alpha) + 2.0
	tmp = 0
	if beta <= 5e+17:
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0)
	else:
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	tmp = 0.0
	if (beta <= 5e+17)
		tmp = Float64(Float64(Float64(-1.0 - beta) * Float64(-1.0 - alpha)) / Float64(Float64(Float64(3.0 + Float64(beta + alpha)) * t_0) * t_0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(Float64(Float64(beta + alpha) - -1.0) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	t_0 = (beta + alpha) + 2.0;
	tmp = 0.0;
	if (beta <= 5e+17)
		tmp = ((-1.0 - beta) * (-1.0 - alpha)) / (((3.0 + (beta + alpha)) * t_0) * t_0);
	else
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[beta, 5e+17], N[(N[(N[(-1.0 - beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] - -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5e17

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.8

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 91.1%

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

   if 5e17 < beta

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 84.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.0

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\left(-1 - \beta\right) \cdot \left(-1 - \alpha\right)}{\left(\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3e+23)
   (/
    (/ (- beta -1.0) (+ beta 2.0))
    (* (+ 3.0 (+ beta alpha)) (+ (+ beta alpha) 2.0)))
   (/ (/ (- alpha -1.0) beta) (+ (- (+ beta alpha) -1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+23) {
		tmp = ((beta - -1.0) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * ((beta + alpha) + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (beta <= 3d+23) then
        tmp = ((beta - (-1.0d0)) / (beta + 2.0d0)) / ((3.0d0 + (beta + alpha)) * ((beta + alpha) + 2.0d0))
    else
        tmp = ((alpha - (-1.0d0)) / beta) / (((beta + alpha) - (-1.0d0)) + 2.0d0)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3e+23) {
		tmp = ((beta - -1.0) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * ((beta + alpha) + 2.0));
	} else {
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	tmp = 0
	if beta <= 3e+23:
		tmp = ((beta - -1.0) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * ((beta + alpha) + 2.0))
	else:
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3e+23)
		tmp = Float64(Float64(Float64(beta - -1.0) / Float64(beta + 2.0)) / Float64(Float64(3.0 + Float64(beta + alpha)) * Float64(Float64(beta + alpha) + 2.0)));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(Float64(Float64(beta + alpha) - -1.0) + 2.0));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (beta <= 3e+23)
		tmp = ((beta - -1.0) / (beta + 2.0)) / ((3.0 + (beta + alpha)) * ((beta + alpha) + 2.0));
	else
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3e+23], N[(N[(N[(beta - -1.0), $MachinePrecision] / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] - -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.0000000000000001e23

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 83.7

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

   7. Applied rewrites 83.7%

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

   if 3.0000000000000001e23 < beta

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 84.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.0

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 89.2

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

   7. Applied rewrites 89.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\beta - -1}{\beta + 2}}{\left(3 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\beta + \alpha\right) + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\beta} \cdot \left(-1 - \alpha\right)}{3 + \left(\beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (/ (* (/ -1.0 beta) (- -1.0 alpha)) (+ 3.0 (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else {
		tmp = ((-1.0 / beta) * (-1.0 - alpha)) / (3.0 + (beta + alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	else
		tmp = Float64(Float64(Float64(-1.0 / beta) * Float64(-1.0 - alpha)) / Float64(3.0 + Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.9], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / beta), $MachinePrecision] * N[(-1.0 - alpha), $MachinePrecision]), $MachinePrecision] / N[(3.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 3.89999999999999991 < beta

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 84.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.4

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   7. Applied rewrites 87.9%

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

   9. Applied rewrites 87.9%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. associate-+l+ N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f64 87.9

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 87.9

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

   11. Applied rewrites 87.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\beta} \cdot \left(-1 - \alpha\right)}{3 + \left(\beta + \alpha\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (/ (/ (- alpha -1.0) beta) (+ (- (+ beta alpha) -1.0) 2.0))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (((beta + alpha) - -1.0) + 2.0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(Float64(Float64(beta + alpha) - -1.0) + 2.0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.9], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(N[(beta + alpha), $MachinePrecision] - -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 3.89999999999999991 < beta

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 84.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.4

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\left(\left(\beta + \alpha\right) - -1\right) + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.9)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (/ (/ (- alpha -1.0) beta) (+ 3.0 beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.9) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / (3.0 + beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.9)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / Float64(3.0 + beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.9], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / N[(3.0 + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.89999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 3.89999999999999991 < beta

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 84.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 84.4

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-+.f64 87.8

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

   10. Applied rewrites 87.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.9:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{3 + \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 5.2)
   (/ (- alpha -1.0) (fma (fma (+ 7.0 alpha) alpha 16.0) alpha 12.0))
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 5.2) {
		tmp = (alpha - -1.0) / fma(fma((7.0 + alpha), alpha, 16.0), alpha, 12.0);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 5.2)
		tmp = Float64(Float64(alpha - -1.0) / fma(fma(Float64(7.0 + alpha), alpha, 16.0), alpha, 12.0));
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 5.2], N[(N[(alpha - -1.0), $MachinePrecision] / N[(N[(N[(7.0 + alpha), $MachinePrecision] * alpha + 16.0), $MachinePrecision] * alpha + 12.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.20000000000000018

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1 + \alpha}{12 + \color{blue}{\alpha \cdot \left(16 + \alpha \cdot \left(7 + \alpha\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.9%

      \[\leadsto \frac{1 + \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \color{blue}{\alpha}, 12\right)} \]

   if 5.20000000000000018 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 87.7%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2:\\ \;\;\;\;\frac{\alpha - -1}{\mathsf{fma}\left(\mathsf{fma}\left(7 + \alpha, \alpha, 16\right), \alpha, 12\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ (/ (- alpha -1.0) beta) beta)))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = ((alpha - -1.0) / beta) / beta;
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(Float64(Float64(alpha - -1.0) / beta) / beta);
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(N[(N[(alpha - -1.0), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.5 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 87.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha - -1}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.5)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ (- alpha -1.0) (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.5) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = (alpha - -1.0) / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.5)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(Float64(alpha - -1.0) / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.5], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(N[(alpha - -1.0), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.5

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.5 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha - -1}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.2)
   (fma
    (fma
     (fma 0.024691358024691357 alpha -0.011574074074074073)
     alpha
     -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.2) {
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.2)
		tmp = fma(fma(fma(0.024691358024691357, alpha, -0.011574074074074073), alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.2], N[(N[(N[(0.024691358024691357 * alpha + -0.011574074074074073), $MachinePrecision] * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2000000000000002

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\alpha \cdot \left(\frac{2}{81} \cdot \alpha - \frac{5}{432}\right) - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.024691358024691357, \alpha, -0.011574074074074073\right), \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.2000000000000002 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 81.4%

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

Alternative 16: 91.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 3.1)
   (fma
    (fma -0.011574074074074073 alpha -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ 1.0 (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 3.1) {
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = 1.0 / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 3.1)
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(1.0 / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 3.1], N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(1.0 / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.10000000000000009

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 89.8

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 3.10000000000000009 < beta

   1. Initial program 84.4%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 81.4%

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

Alternative 17: 74.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 7.6e+65)
   (fma
    (fma -0.011574074074074073 alpha -0.027777777777777776)
    alpha
    0.08333333333333333)
   (/ alpha (* beta beta))))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	double tmp;
	if (beta <= 7.6e+65) {
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	} else {
		tmp = alpha / (beta * beta);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	tmp = 0.0
	if (beta <= 7.6e+65)
		tmp = fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
	else
		tmp = Float64(alpha / Float64(beta * beta));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := If[LessEqual[beta, 7.6e+65], N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision], N[(alpha / N[(beta * beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.60000000000000022e65

   1. Initial program 99.8%

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

   3. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]

   if 7.60000000000000022e65 < beta

   1. Initial program 82.3%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.1%

      \[\leadsto \frac{\alpha}{\color{blue}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma
  (fma -0.011574074074074073 alpha -0.027777777777777776)
  alpha
  0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(fma(-0.011574074074074073, alpha, -0.027777777777777776), alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(N[(-0.011574074074074073 * alpha + -0.027777777777777776), $MachinePrecision] * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 66.7

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{12} + \color{blue}{\alpha \cdot \left(\frac{-5}{432} \cdot \alpha - \frac{1}{36}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 47.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.011574074074074073, \alpha, -0.027777777777777776\right), \color{blue}{\alpha}, 0.08333333333333333\right) \]
9. Add Preprocessing

Alternative 19: 45.1% accurate, 12.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \mathsf{fma}\left(-0.027777777777777776, \alpha, 0.08333333333333333\right) \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta)
 :precision binary64
 (fma -0.027777777777777776 alpha 0.08333333333333333))

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return fma(-0.027777777777777776, alpha, 0.08333333333333333);
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return fma(-0.027777777777777776, alpha, 0.08333333333333333)
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := N[(-0.027777777777777776 * alpha + 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 66.7

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{12} + \color{blue}{\frac{-1}{36} \cdot \alpha} \]
7. Step-by-step derivation

8. Applied rewrites 47.4%

   \[\leadsto \mathsf{fma}\left(-0.027777777777777776, \color{blue}{\alpha}, 0.08333333333333333\right) \]
9. Add Preprocessing

Alternative 20: 44.8% accurate, 84.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0.08333333333333333 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta) :precision binary64 0.08333333333333333)

C

assert(alpha < beta);
double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 0.08333333333333333d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta) {
	return 0.08333333333333333;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta):
	return 0.08333333333333333

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta)
	return 0.08333333333333333
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta)
	tmp = 0.08333333333333333;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_] := 0.08333333333333333

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in beta around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 66.7

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in alpha around 0

   \[\leadsto \frac{1}{12} \]
7. Step-by-step derivation

8. Applied rewrites 47.3%

   \[\leadsto 0.08333333333333333 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))