Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 84.0%

Time: 17.7s

Alternatives: 10

Speedup: 53.0×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 16.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 84.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t\_2\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_3\right) \cdot t\_3}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{-1 + {i}^{2} \cdot \left(4 + \frac{\frac{{\left(\alpha + \beta\right)}^{2}}{i} - \left(\alpha + \beta\right) \cdot -4}{i}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2)))
   (if (<= (/ (/ (* (+ (* alpha beta) t_3) t_3) t_1) (+ t_1 -1.0)) INFINITY)
     (/
      (*
       t_3
       (/ (fma i t_2 (* alpha beta)) (pow (fma i 2.0 (+ alpha beta)) 2.0)))
      (+
       -1.0
       (*
        (pow i 2.0)
        (+
         4.0
         (/ (- (/ (pow (+ alpha beta) 2.0) i) (* (+ alpha beta) -4.0)) i)))))
     (+ (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* -0.0625 (/ beta i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double tmp;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (t_3 * (fma(i, t_2, (alpha * beta)) / pow(fma(i, 2.0, (alpha + beta)), 2.0))) / (-1.0 + (pow(i, 2.0) * (4.0 + (((pow((alpha + beta), 2.0) / i) - ((alpha + beta) * -4.0)) / i))));
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_3) * t_3) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(t_3 * Float64(fma(i, t_2, Float64(alpha * beta)) / (fma(i, 2.0, Float64(alpha + beta)) ^ 2.0))) / Float64(-1.0 + Float64((i ^ 2.0) * Float64(4.0 + Float64(Float64(Float64((Float64(alpha + beta) ^ 2.0) / i) - Float64(Float64(alpha + beta) * -4.0)) / i)))));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) + Float64(-0.0625 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$3 * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(N[Power[i, 2.0], $MachinePrecision] * N[(4.0 + N[(N[(N[(N[Power[N[(alpha + beta), $MachinePrecision], 2.0], $MachinePrecision] / i), $MachinePrecision] - N[(N[(alpha + beta), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 41.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. pow2 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-undefine 99.7%

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

      12. pow2 99.7%

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

      13. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in i around -inf 99.8%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 4.5%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 69.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) - 0.0625 \cdot \frac{\beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 65.1%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i}} \]

      2. distribute-rgt-out-- 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{\beta \cdot \left(0.5 - 0.25\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      3. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot \color{blue}{0.25}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      4. *-commutative 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{0.25 \cdot \beta}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      5. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\beta}{i} \]

   7. Simplified 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + -0.0625 \cdot \frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{-1 + {i}^{2} \cdot \left(4 + \frac{\frac{{\left(\alpha + \beta\right)}^{2}}{i} - \left(\alpha + \beta\right) \cdot -4}{i}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t\_3\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_4\right) \cdot t\_4}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{t\_4 \cdot \frac{\mathsf{fma}\left(i, t\_3, \alpha \cdot \beta\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3)))
   (if (<= (/ (/ (* (+ (* alpha beta) t_4) t_4) t_1) t_2) INFINITY)
     (/
      (*
       t_4
       (/ (fma i t_3 (* alpha beta)) (pow (fma i 2.0 (+ alpha beta)) 2.0)))
      t_2)
     (+ (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* -0.0625 (/ beta i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i + (alpha + beta);
	double t_4 = i * t_3;
	double tmp;
	if ((((((alpha * beta) + t_4) * t_4) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (t_4 * (fma(i, t_3, (alpha * beta)) / pow(fma(i, 2.0, (alpha + beta)), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(i + Float64(alpha + beta))
	t_4 = Float64(i * t_3)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_4) * t_4) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(t_4 * Float64(fma(i, t_3, Float64(alpha * beta)) / (fma(i, 2.0, Float64(alpha + beta)) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) + Float64(-0.0625 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(t$95$4 * N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 41.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. pow2 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-undefine 99.7%

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

      12. pow2 99.7%

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

      13. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 4.5%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 69.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) - 0.0625 \cdot \frac{\beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 65.1%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i}} \]

      2. distribute-rgt-out-- 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{\beta \cdot \left(0.5 - 0.25\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      3. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot \color{blue}{0.25}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      4. *-commutative 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{0.25 \cdot \beta}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      5. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\beta}{i} \]

   7. Simplified 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + -0.0625 \cdot \frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

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

Alternative 3: 83.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_3\right) \cdot t\_3}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{{i}^{2} \cdot \frac{{\left(i + \beta\right)}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* (+ (* alpha beta) t_3) t_3) t_1) t_2) INFINITY)
     (/
      (* (pow i 2.0) (/ (pow (+ i beta) 2.0) (pow (+ beta (* i 2.0)) 2.0)))
      t_2)
     (+ (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* -0.0625 (/ beta i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (pow(i, 2.0) * (pow((i + beta), 2.0) / pow((beta + (i * 2.0)), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow(i, 2.0) * (Math.pow((i + beta), 2.0) / Math.pow((beta + (i * 2.0)), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = t_1 + -1.0
	t_3 = i * (i + (alpha + beta))
	tmp = 0
	if (((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= math.inf:
		tmp = (math.pow(i, 2.0) * (math.pow((i + beta), 2.0) / math.pow((beta + (i * 2.0)), 2.0))) / t_2
	else:
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64((i ^ 2.0) * Float64((Float64(i + beta) ^ 2.0) / (Float64(beta + Float64(i * 2.0)) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) + Float64(-0.0625 * Float64(beta / i)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = t_1 + -1.0;
	t_3 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Inf)
		tmp = ((i ^ 2.0) * (((i + beta) ^ 2.0) / ((beta + (i * 2.0)) ^ 2.0))) / t_2;
	else
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[Power[i, 2.0], $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 41.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. pow2 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-undefine 99.7%

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

      12. pow2 99.7%

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

      13. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in alpha around 0 33.1%

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

      1. associate-/l* 88.1%

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

   7. Simplified 88.1%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 4.5%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 69.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) - 0.0625 \cdot \frac{\beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 65.1%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i}} \]

      2. distribute-rgt-out-- 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{\beta \cdot \left(0.5 - 0.25\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      3. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot \color{blue}{0.25}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      4. *-commutative 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{0.25 \cdot \beta}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      5. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\beta}{i} \]

   7. Simplified 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + -0.0625 \cdot \frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{{i}^{2} \cdot \frac{{\left(i + \beta\right)}^{2}}{{\left(\beta + i \cdot 2\right)}^{2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + t\_3\right) \cdot t\_3}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \frac{i \cdot \left(i + \beta\right)}{{\left(\beta + i \cdot 2\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* (+ (* alpha beta) t_3) t_3) t_1) t_2) INFINITY)
     (/ (* t_3 (/ (* i (+ i beta)) (pow (+ beta (* i 2.0)) 2.0))) t_2)
     (+ (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* -0.0625 (/ beta i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (t_3 * ((i * (i + beta)) / pow((beta + (i * 2.0)), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (t_3 * ((i * (i + beta)) / Math.pow((beta + (i * 2.0)), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = t_1 + -1.0
	t_3 = i * (i + (alpha + beta))
	tmp = 0
	if (((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= math.inf:
		tmp = (t_3 * ((i * (i + beta)) / math.pow((beta + (i * 2.0)), 2.0))) / t_2
	else:
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(t_3 * Float64(Float64(i * Float64(i + beta)) / (Float64(beta + Float64(i * 2.0)) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) + Float64(-0.0625 * Float64(beta / i)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = t_1 + -1.0;
	t_3 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((((alpha * beta) + t_3) * t_3) / t_1) / t_2) <= Inf)
		tmp = (t_3 * ((i * (i + beta)) / ((beta + (i * 2.0)) ^ 2.0))) / t_2;
	else
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(t$95$3 * N[(N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 41.7%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-undefine 99.7%

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

      8. pow2 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-undefine 99.7%

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

      12. pow2 99.7%

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

      13. +-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in alpha around 0 88.5%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf 4.5%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 69.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) - 0.0625 \cdot \frac{\beta}{i}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 65.1%

         \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i}} \]

      2. distribute-rgt-out-- 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{\beta \cdot \left(0.5 - 0.25\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      3. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot \color{blue}{0.25}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      4. *-commutative 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{0.25 \cdot \beta}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

      5. metadata-eval 65.1%

         \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\beta}{i} \]

   7. Simplified 65.1%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + -0.0625 \cdot \frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{i \cdot \left(i + \beta\right)}{{\left(\beta + i \cdot 2\right)}^{2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{\left(\alpha \cdot \beta + t\_2\right) \cdot t\_2}{t\_1}}{t\_1 + -1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + 0.25 \cdot \left(0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\alpha + \beta\right) \cdot 0.25\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta))))
        (t_3 (/ (/ (* (+ (* alpha beta) t_2) t_2) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.1)
     t_3
     (-
      (/
       (+
        (* i 0.0625)
        (*
         0.25
         (- (* 0.25 (+ (* alpha 2.0) (* beta 2.0))) (* (+ alpha beta) 0.25))))
       i)
      (* 0.0625 (/ (+ alpha beta) i))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (((i * 0.0625d0) + (0.25d0 * ((0.25d0 * ((alpha * 2.0d0) + (beta * 2.0d0))) - ((alpha + beta) * 0.25d0)))) / i) - (0.0625d0 * ((alpha + beta) / i))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	t_3 = ((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(Float64(Float64(alpha * beta) + t_2) * t_2) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(0.25 * Float64(Float64(0.25 * Float64(Float64(alpha * 2.0) + Float64(beta * 2.0))) - Float64(Float64(alpha + beta) * 0.25)))) / i) - Float64(0.0625 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = ((((alpha * beta) + t_2) * t_2) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(alpha * beta), $MachinePrecision] + t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(0.25 * N[(N[(0.25 * N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(alpha + beta), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.0625 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.6%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.8%

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

   3. Taylor expanded in i around inf 25.7%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 74.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in i around 0 74.4%

      \[\leadsto \color{blue}{\frac{0.0625 \cdot i + 0.25 \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + 0.25 \cdot \left(0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\alpha + \beta\right) \cdot 0.25\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{i \cdot 0.0625 + 0.25 \cdot \left(0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\alpha + \beta\right) \cdot 0.25\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (-
  (/
   (+
    (* i 0.0625)
    (*
     0.25
     (- (* 0.25 (+ (* alpha 2.0) (* beta 2.0))) (* (+ alpha beta) 0.25))))
   i)
  (* 0.0625 (/ (+ alpha beta) i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((i * 0.0625d0) + (0.25d0 * ((0.25d0 * ((alpha * 2.0d0) + (beta * 2.0d0))) - ((alpha + beta) * 0.25d0)))) / i) - (0.0625d0 * ((alpha + beta) / i))
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i))

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * 0.0625) + Float64(0.25 * Float64(Float64(0.25 * Float64(Float64(alpha * 2.0) + Float64(beta * 2.0))) - Float64(Float64(alpha + beta) * 0.25)))) / i) - Float64(0.0625 * Float64(Float64(alpha + beta) / i)))
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (((i * 0.0625) + (0.25 * ((0.25 * ((alpha * 2.0) + (beta * 2.0))) - ((alpha + beta) * 0.25)))) / i) - (0.0625 * ((alpha + beta) / i));
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(0.25 * N[(N[(0.25 * N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(alpha + beta), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.0625 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.7%

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

3. Taylor expanded in i around inf 32.5%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Taylor expanded in i around inf 73.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in i around 0 73.2%

   \[\leadsto \color{blue}{\frac{0.0625 \cdot i + 0.25 \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]

6. Final simplification 73.2%

   \[\leadsto \frac{i \cdot 0.0625 + 0.25 \cdot \left(0.25 \cdot \left(\alpha \cdot 2 + \beta \cdot 2\right) - \left(\alpha + \beta\right) \cdot 0.25\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
7. Add Preprocessing

Alternative 7: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+246}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0.125 + \left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.4e+246)
   0.0625
   (/ (+ (* (+ alpha beta) 0.125) (* (+ alpha beta) -0.125)) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+246) {
		tmp = 0.0625;
	} else {
		tmp = (((alpha + beta) * 0.125) + ((alpha + beta) * -0.125)) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.4d+246) then
        tmp = 0.0625d0
    else
        tmp = (((alpha + beta) * 0.125d0) + ((alpha + beta) * (-0.125d0))) / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+246) {
		tmp = 0.0625;
	} else {
		tmp = (((alpha + beta) * 0.125) + ((alpha + beta) * -0.125)) / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.4e+246:
		tmp = 0.0625
	else:
		tmp = (((alpha + beta) * 0.125) + ((alpha + beta) * -0.125)) / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.4e+246)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(alpha + beta) * 0.125) + Float64(Float64(alpha + beta) * -0.125)) / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.4e+246)
		tmp = 0.0625;
	else
		tmp = (((alpha + beta) * 0.125) + ((alpha + beta) * -0.125)) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.4e+246], 0.0625, N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.39999999999999988e246

   1. Initial program 15.5%

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

      1. associate-/l/ 13.2%

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

      2. associate-/l* 16.5%

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

      3. +-commutative 16.5%

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

      4. +-commutative 16.5%

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

      5. +-commutative 16.5%

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

      6. associate-+l+ 16.5%

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

      7. +-commutative 16.5%

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

      8. associate-*l* 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in i around inf 71.5%

      \[\leadsto \color{blue}{0.0625} \]

   if 3.39999999999999988e246 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-/l* 7.1%

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

      3. +-commutative 7.1%

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

      4. +-commutative 7.1%

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

      5. +-commutative 7.1%

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

      6. associate-+l+ 7.1%

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

      7. +-commutative 7.1%

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

      8. associate-*l* 7.1%

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

   3. Simplified 7.1%

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

   5. Taylor expanded in i around inf 40.5%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   6. Taylor expanded in i around 0 40.5%

      \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 40.5%

         \[\leadsto \frac{\color{blue}{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) + \left(-0.125\right) \cdot \left(\alpha + \beta\right)}}{i} \]

      2. distribute-lft-in 40.5%

         \[\leadsto \frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)} + \left(-0.125\right) \cdot \left(\alpha + \beta\right)}{i} \]

      3. associate-*r* 40.5%

         \[\leadsto \frac{\color{blue}{\left(0.0625 \cdot 2\right) \cdot \left(\alpha + \beta\right)} + \left(-0.125\right) \cdot \left(\alpha + \beta\right)}{i} \]

      4. metadata-eval 40.5%

         \[\leadsto \frac{\color{blue}{0.125} \cdot \left(\alpha + \beta\right) + \left(-0.125\right) \cdot \left(\alpha + \beta\right)}{i} \]

      5. metadata-eval 40.5%

         \[\leadsto \frac{0.125 \cdot \left(\alpha + \beta\right) + \color{blue}{-0.125} \cdot \left(\alpha + \beta\right)}{i} \]

   8. Simplified 40.5%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\alpha + \beta\right) + -0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+246}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0.125 + \left(\alpha + \beta\right) \cdot -0.125}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (+ (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* -0.0625 (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.25d0 * ((beta * 0.25d0) / i))) + ((-0.0625d0) * (beta / i))
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i))

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) + Float64(-0.0625 * Float64(beta / i)))
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) + (-0.0625 * (beta / i));
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.7%

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

3. Taylor expanded in i around inf 32.5%

   \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Taylor expanded in i around inf 73.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in alpha around 0 70.6%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) - 0.0625 \cdot \frac{\beta}{i}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 70.6%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \beta - 0.25 \cdot \beta}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i}} \]

   2. distribute-rgt-out-- 70.6%

      \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{\beta \cdot \left(0.5 - 0.25\right)}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

   3. metadata-eval 70.6%

      \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot \color{blue}{0.25}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

   4. *-commutative 70.6%

      \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\color{blue}{0.25 \cdot \beta}}{i}\right) + \left(-0.0625\right) \cdot \frac{\beta}{i} \]

   5. metadata-eval 70.6%

      \[\leadsto \left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + \color{blue}{-0.0625} \cdot \frac{\beta}{i} \]

7. Simplified 70.6%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.25 \cdot \beta}{i}\right) + -0.0625 \cdot \frac{\beta}{i}} \]

8. Final simplification 70.6%

   \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) + -0.0625 \cdot \frac{\beta}{i} \]
9. Add Preprocessing

Alternative 9: 72.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+246}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot 0.25}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+246) 0.0625 (/ (* alpha 0.25) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+246) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * 0.25) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.55d+246) then
        tmp = 0.0625d0
    else
        tmp = (alpha * 0.25d0) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+246) {
		tmp = 0.0625;
	} else {
		tmp = (alpha * 0.25) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.55e+246:
		tmp = 0.0625
	else:
		tmp = (alpha * 0.25) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+246)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha * 0.25) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.55e+246)
		tmp = 0.0625;
	else
		tmp = (alpha * 0.25) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.55e+246], 0.0625, N[(N[(alpha * 0.25), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.54999999999999994e246

   1. Initial program 15.5%

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

      1. associate-/l/ 13.2%

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

      2. associate-/l* 16.5%

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

      3. +-commutative 16.5%

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

      4. +-commutative 16.5%

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

      5. +-commutative 16.5%

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

      6. associate-+l+ 16.5%

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

      7. +-commutative 16.5%

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

      8. associate-*l* 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in i around inf 71.5%

      \[\leadsto \color{blue}{0.0625} \]

   if 1.54999999999999994e246 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in i around 0 7.1%

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

      1. associate-/l* 7.1%

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

   5. Simplified 7.1%

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

   6. Taylor expanded in i around inf 7.1%

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

   7. Taylor expanded in beta around inf 37.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\alpha}{\beta}} \]
   8. Step-by-step derivation

      1. associate-*r/ 37.7%

         \[\leadsto \color{blue}{\frac{0.25 \cdot \alpha}{\beta}} \]

   9. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \alpha}{\beta}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+246}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha \cdot 0.25}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 53.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 14.7%

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

   1. associate-/l/ 12.5%

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

   2. associate-/l* 16.0%

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

   3. +-commutative 16.0%

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

   4. +-commutative 16.0%

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

   5. +-commutative 16.0%

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

   6. associate-+l+ 16.0%

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

   7. +-commutative 16.0%

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

   8. associate-*l* 15.9%

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

3. Simplified 15.9%

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

5. Taylor expanded in i around inf 67.8%

   \[\leadsto \color{blue}{0.0625} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))