Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 83.1%

Time: 21.2s

Alternatives: 7

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: 16.0% 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: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := \alpha + \left(i + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{i \cdot \left(t\_4 \cdot \frac{\mathsf{fma}\left(\alpha, \beta, i \cdot t\_4\right)}{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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))))
        (t_4 (+ alpha (+ i beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       i
       (*
        t_4
        (/ (fma alpha beta (* i t_4)) (pow (fma i 2.0 (+ alpha beta)) 2.0))))
      t_2)
     (-
      (+ 0.0625 (* 0.0625 (+ 1.0 (+ (* 2.0 (/ (+ alpha beta) i)) -1.0))))
      (* 0.125 (/ 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 t_4 = alpha + (i + beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (i * (t_4 * (fma(alpha, beta, (i * t_4)) / pow(fma(i, 2.0, (alpha + beta)), 2.0)))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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)))
	t_4 = Float64(alpha + Float64(i + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(i * Float64(t_4 * Float64(fma(alpha, beta, Float64(i * t_4)) / (fma(i, 2.0, Float64(alpha + beta)) ^ 2.0)))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(alpha + beta) / i)) + -1.0)))) - Float64(0.125 * 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[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$4 * N[(N[(alpha * beta + N[(i * t$95$4), $MachinePrecision]), $MachinePrecision] / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(1.0 + N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * 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 48.8%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+r+ 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-undefine 99.8%

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

      8. pow2 99.8%

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

      9. *-commutative 99.8%

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

      10. pow2 99.8%

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

      11. associate-+r+ 99.8%

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

      12. +-commutative 99.8%

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

      13. pow2 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l* 99.9%

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

      2. associate-+r+ 99.9%

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

      3. associate-+r+ 99.9%

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

   6. Simplified 99.9%

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

   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%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in i around inf 82.6%

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

      1. expm1-log1p-u 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. distribute-lft-out 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   8. Step-by-step derivation

      1. expm1-undefine 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)} - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-expm1-u 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. log1p-undefine 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. rem-exp-log 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      5. expm1-log1p-u 82.6%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right)}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      6. associate-/l* 82.6%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{2 \cdot \frac{\alpha + \beta}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   9. Applied egg-rr 82.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(1 + 2 \cdot \frac{\alpha + \beta}{i}\right) - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   10. Step-by-step derivation

       1. associate--l+ 82.6%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

       2. *-commutative 82.6%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\color{blue}{\frac{\alpha + \beta}{i} \cdot 2} - 1\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   11. Simplified 82.6%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   12. Taylor expanded in alpha around 0 76.4%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

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

Alternative 2: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := {\left(\beta + i \cdot 2\right)}^{2}\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;i \cdot \left(\frac{i}{t\_3} \cdot \frac{{\left(i + \beta\right)}^{2}}{t\_3 + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (+ i (+ alpha beta))))
        (t_3 (pow (+ beta (* i 2.0)) 2.0)))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (* i (* (/ i t_3) (/ (pow (+ i beta) 2.0) (+ t_3 -1.0))))
     (-
      (+ 0.0625 (* 0.0625 (+ 1.0 (+ (* 2.0 (/ (+ alpha beta) i)) -1.0))))
      (* 0.125 (/ 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 = pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = i * ((i / t_3) * (pow((i + beta), 2.0) / (t_3 + -1.0)));
	} else {
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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 = i * (i + (alpha + beta));
	double t_3 = Math.pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = i * ((i / t_3) * (Math.pow((i + beta), 2.0) / (t_3 + -1.0)));
	} else {
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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 = math.pow((beta + (i * 2.0)), 2.0)
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= math.inf:
		tmp = i * ((i / t_3) * (math.pow((i + beta), 2.0) / (t_3 + -1.0)))
	else:
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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(beta + Float64(i * 2.0)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(i * Float64(Float64(i / t_3) * Float64((Float64(i + beta) ^ 2.0) / Float64(t_3 + -1.0))));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(alpha + beta) / i)) + -1.0)))) - Float64(0.125 * 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 = i * (i + (alpha + beta));
	t_3 = (beta + (i * 2.0)) ^ 2.0;
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Inf)
		tmp = i * ((i / t_3) * (((i + beta) ^ 2.0) / (t_3 + -1.0)));
	else
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(i * N[(N[(i / t$95$3), $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(1.0 + N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * 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 48.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in alpha around 0 41.4%

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

      1. times-frac 92.2%

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

      2. sub-neg 92.2%

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

      3. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   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%

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

   3. Simplified 6.8%

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

   5. Taylor expanded in i around inf 82.6%

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

      1. expm1-log1p-u 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. distribute-lft-out 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 72.9%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   8. Step-by-step derivation

      1. expm1-undefine 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)} - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-expm1-u 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. log1p-undefine 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. rem-exp-log 72.9%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      5. expm1-log1p-u 82.6%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right)}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      6. associate-/l* 82.6%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{2 \cdot \frac{\alpha + \beta}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   9. Applied egg-rr 82.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(1 + 2 \cdot \frac{\alpha + \beta}{i}\right) - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   10. Step-by-step derivation

       1. associate--l+ 82.6%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

       2. *-commutative 82.6%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\color{blue}{\frac{\alpha + \beta}{i} \cdot 2} - 1\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   11. Simplified 82.6%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   12. Taylor expanded in alpha around 0 76.4%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;i \cdot \left(\frac{i}{{\left(\beta + i \cdot 2\right)}^{2}} \cdot \frac{{\left(i + \beta\right)}^{2}}{{\left(\beta + i \cdot 2\right)}^{2} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (+ i (+ alpha beta))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.1)
     t_3
     (-
      (+ 0.0625 (* 0.0625 (+ 1.0 (+ (* 2.0 (/ (+ alpha beta) i)) -1.0))))
      (* 0.125 (/ 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 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * (1.0d0 + ((2.0d0 * ((alpha + beta) / i)) + (-1.0d0))))) - (0.125d0 * (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 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(alpha + beta) / i)) + -1.0)))) - Float64(0.125 * 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 = i * (i + (alpha + beta));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $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[(0.0625 + N[(0.0625 * N[(1.0 + N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * 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))) < 0.10000000000000001

   1. Initial program 99.8%

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

   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 0.5%

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

   3. Simplified 23.2%

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

   5. Taylor expanded in i around inf 83.1%

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

      1. expm1-log1p-u 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. distribute-lft-out 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 75.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   8. Step-by-step derivation

      1. expm1-undefine 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)} - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-expm1-u 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. log1p-undefine 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. rem-exp-log 75.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      5. expm1-log1p-u 83.1%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right)}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      6. associate-/l* 83.1%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{2 \cdot \frac{\alpha + \beta}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   9. Applied egg-rr 83.1%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(1 + 2 \cdot \frac{\alpha + \beta}{i}\right) - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
   10. Step-by-step derivation

       1. associate--l+ 83.1%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

       2. *-commutative 83.1%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\color{blue}{\frac{\alpha + \beta}{i} \cdot 2} - 1\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   11. Simplified 83.1%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   12. Taylor expanded in alpha around 0 78.0%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.0625 (+ 1.0 (+ (* 2.0 (/ (+ alpha beta) i)) -1.0))))
  (* 0.125 (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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 + (0.0625d0 * (1.0d0 + ((2.0d0 * ((alpha + beta) / i)) + (-1.0d0))))) - (0.125d0 * (beta / i))
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (beta / i));
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (beta / i))

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(alpha + beta) / i)) + -1.0)))) - Float64(0.125 * Float64(beta / i)))
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (1.0 + ((2.0 * ((alpha + beta) / i)) + -1.0)))) - (0.125 * (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[(0.0625 * N[(1.0 + N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.1%

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

3. Simplified 33.6%

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

5. Taylor expanded in i around inf 82.5%

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

   1. expm1-log1p-u 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   2. distribute-lft-out 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied egg-rr 75.7%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
8. Step-by-step derivation

   1. expm1-undefine 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)} - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   2. log1p-expm1-u 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   3. log1p-undefine 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)}} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. rem-exp-log 75.7%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right)} - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   5. expm1-log1p-u 82.5%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right)}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   6. associate-/l* 82.5%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(1 + \color{blue}{2 \cdot \frac{\alpha + \beta}{i}}\right) - 1\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

9. Applied egg-rr 82.5%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(1 + 2 \cdot \frac{\alpha + \beta}{i}\right) - 1\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
10. Step-by-step derivation

    1. associate--l+ 82.5%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

    2. *-commutative 82.5%

       \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\color{blue}{\frac{\alpha + \beta}{i} \cdot 2} - 1\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

11. Simplified 82.5%

    \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

12. Taylor expanded in alpha around 0 78.1%

    \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(\frac{\alpha + \beta}{i} \cdot 2 - 1\right)\right)\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]

13. Final simplification 78.1%

    \[\leadsto \left(0.0625 + 0.0625 \cdot \left(1 + \left(2 \cdot \frac{\alpha + \beta}{i} + -1\right)\right)\right) - 0.125 \cdot \frac{\beta}{i} \]
14. Add Preprocessing

Alternative 5: 76.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{\beta \cdot -0.125 + 0.0625 \cdot \left(i + \beta \cdot 2\right)}{i} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (/ (+ (* beta -0.125) (* 0.0625 (+ i (* beta 2.0)))) i))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return ((beta * -0.125) + (0.0625 * (i + (beta * 2.0)))) / 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 = ((beta * (-0.125d0)) + (0.0625d0 * (i + (beta * 2.0d0)))) / i
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return ((beta * -0.125) + (0.0625 * (i + (beta * 2.0)))) / i;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return ((beta * -0.125) + (0.0625 * (i + (beta * 2.0)))) / i

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(Float64(beta * -0.125) + Float64(0.0625 * Float64(i + Float64(beta * 2.0)))) / i)
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = ((beta * -0.125) + (0.0625 * (i + (beta * 2.0)))) / 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[(beta * -0.125), $MachinePrecision] + N[(0.0625 * N[(i + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 14.1%

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

3. Simplified 33.6%

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

5. Taylor expanded in i around inf 82.5%

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

6. Taylor expanded in i around 0 82.5%

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

   1. cancel-sign-sub-inv 82.5%

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

   2. distribute-lft-in 82.5%

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

   3. distribute-lft-out 82.5%

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

   4. metadata-eval 82.5%

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

   5. *-commutative 82.5%

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

8. Simplified 82.5%

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

9. Taylor expanded in alpha around 0 78.4%

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

10. Final simplification 78.4%

    \[\leadsto \frac{\beta \cdot -0.125 + 0.0625 \cdot \left(i + \beta \cdot 2\right)}{i} \]
11. Add Preprocessing

Alternative 6: 72.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.1e+201) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.1e+201) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / 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 <= 2.1d+201) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / 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 <= 2.1e+201) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.1e+201:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.1e+201)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / 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 <= 2.1e+201)
		tmp = 0.0625;
	else
		tmp = 0.0 / 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, 2.1e+201], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0999999999999999e201

   1. Initial program 15.6%

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

   3. Simplified 35.7%

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

   5. Taylor expanded in i around inf 81.7%

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

   if 2.0999999999999999e201 < beta

   1. Initial program 0.0%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in i around inf 42.8%

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

   6. Taylor expanded in i around 0 42.8%

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

      1. cancel-sign-sub-inv 42.8%

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

      2. distribute-lft-in 42.8%

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

      3. distribute-lft-out 42.8%

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

      4. metadata-eval 42.8%

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

      5. *-commutative 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in i around 0 25.1%

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

       1. distribute-rgt-out 25.1%

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

       2. metadata-eval 25.1%

          \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \color{blue}{0}}{i} \]

       3. mul0-rgt 25.1%

          \[\leadsto \frac{\color{blue}{0}}{i} \]

   11. Simplified 25.1%

       \[\leadsto \frac{\color{blue}{0}}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.1% 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 14.1%

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

3. Simplified 33.6%

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

5. Taylor expanded in i around inf 76.0%

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

Reproduce

herbie shell --seed 2024101 
(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)))