Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 84.5%

Time: 19.5s

Alternatives: 11

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: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t\_2\\ t_4 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{t\_4} \cdot \frac{t\_3}{t\_4}}{\mathsf{fma}\left(\beta, \beta + \mathsf{fma}\left(2, \alpha, i \cdot 4\right), {\left(\alpha + i \cdot 2\right)}^{2}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (/
      (* (/ (fma i t_2 (* alpha beta)) t_4) (/ t_3 t_4))
      (+
       (fma
        beta
        (+ beta (fma 2.0 alpha (* i 4.0)))
        (pow (+ alpha (* i 2.0)) 2.0))
       -1.0))
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((fma(i, t_2, (alpha * beta)) / t_4) * (t_3 / t_4)) / (fma(beta, (beta + fma(2.0, alpha, (i * 4.0))), pow((alpha + (i * 2.0)), 2.0)) + -1.0);
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(fma(i, t_2, Float64(alpha * beta)) / t_4) * Float64(t_3 / t_4)) / Float64(fma(beta, Float64(beta + fma(2.0, alpha, Float64(i * 4.0))), (Float64(alpha + Float64(i * 2.0)) ^ 2.0)) + -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + 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] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(t$95$3 / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(beta * N[(beta + N[(2.0 * alpha + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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 42.4%

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

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. *-commutative 42.4%

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

      5. fma-undefine 42.4%

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

      6. times-frac 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. *-commutative 99.6%

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

      13. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in beta around 0 99.5%

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

      1. fma-define 99.5%

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

      2. fma-define 99.5%

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

   7. Simplified 99.5%

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

   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. 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* 6.6%

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

      3. +-commutative 6.6%

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

      4. +-commutative 6.6%

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

      5. +-commutative 6.6%

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

      6. associate-+l+ 6.6%

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

      7. +-commutative 6.6%

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

      8. associate-*l* 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in i around inf 72.1%

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

      1. expm1-log1p-u 57.3%

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

      2. expm1-undefine 57.2%

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

      3. associate-*r/ 57.2%

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

      4. distribute-lft-out 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. expm1-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in i around 0 72.7%

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

   11. Taylor expanded in alpha around 0 63.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(\beta, \beta + \mathsf{fma}\left(2, \alpha, i \cdot 4\right), {\left(\alpha + i \cdot 2\right)}^{2}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t\_2\\ t_4 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{t\_4} \cdot \frac{t\_3}{t\_4}}{\left(i \cdot \left(i \cdot 4 + \left(\alpha + \beta\right) \cdot 4\right) + {\left(\alpha + \beta\right)}^{2}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (/
      (* (/ (fma i t_2 (* alpha beta)) t_4) (/ t_3 t_4))
      (+
       (+ (* i (+ (* i 4.0) (* (+ alpha beta) 4.0))) (pow (+ alpha beta) 2.0))
       -1.0))
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((fma(i, t_2, (alpha * beta)) / t_4) * (t_3 / t_4)) / (((i * ((i * 4.0) + ((alpha + beta) * 4.0))) + pow((alpha + beta), 2.0)) + -1.0);
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(fma(i, t_2, Float64(alpha * beta)) / t_4) * Float64(t_3 / t_4)) / Float64(Float64(Float64(i * Float64(Float64(i * 4.0) + Float64(Float64(alpha + beta) * 4.0))) + (Float64(alpha + beta) ^ 2.0)) + -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + 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] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(t$95$3 / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * N[(N[(i * 4.0), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(alpha + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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 42.4%

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

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. *-commutative 42.4%

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

      5. fma-undefine 42.4%

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

      6. times-frac 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. *-commutative 99.6%

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

      13. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in i around 0 99.6%

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

   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. 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* 6.6%

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

      3. +-commutative 6.6%

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

      4. +-commutative 6.6%

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

      5. +-commutative 6.6%

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

      6. associate-+l+ 6.6%

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

      7. +-commutative 6.6%

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

      8. associate-*l* 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in i around inf 72.1%

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

      1. expm1-log1p-u 57.3%

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

      2. expm1-undefine 57.2%

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

      3. associate-*r/ 57.2%

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

      4. distribute-lft-out 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. expm1-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in i around 0 72.7%

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

   11. Taylor expanded in alpha around 0 63.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(i \cdot \left(i \cdot 4 + \left(\alpha + \beta\right) \cdot 4\right) + {\left(\alpha + \beta\right)}^{2}\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t\_3\\ t_5 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_4 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t\_3, \alpha \cdot \beta\right)}{t\_5} \cdot \frac{t\_4}{t\_5}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (+ alpha beta)))
        (t_4 (* i t_3))
        (t_5 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/ (* (/ (fma i t_3 (* alpha beta)) t_5) (/ t_4 t_5)) t_2)
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 + (alpha + beta);
	double t_4 = i * t_3;
	double t_5 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = ((fma(i, t_3, (alpha * beta)) / t_5) * (t_4 / t_5)) / t_2;
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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(alpha + beta))
	t_4 = Float64(i * t_3)
	t_5 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(fma(i, t_3, Float64(alpha * beta)) / t_5) * Float64(t_4 / t_5)) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] * N[(t$95$4 / t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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 42.4%

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

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. *-commutative 42.4%

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

      5. fma-undefine 42.4%

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

      6. times-frac 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. *-commutative 99.6%

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

      13. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   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. 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* 6.6%

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

      3. +-commutative 6.6%

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

      4. +-commutative 6.6%

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

      5. +-commutative 6.6%

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

      6. associate-+l+ 6.6%

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

      7. +-commutative 6.6%

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

      8. associate-*l* 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in i around inf 72.1%

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

      1. expm1-log1p-u 57.3%

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

      2. expm1-undefine 57.2%

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

      3. associate-*r/ 57.2%

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

      4. distribute-lft-out 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. expm1-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in i around 0 72.7%

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

   11. Taylor expanded in alpha around 0 63.6%

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

4. Final simplification 78.8%

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

Alternative 4: 84.4% accurate, 0.1× speedup

Math

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

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)
     (/
      (*
       (fma i t_4 (* alpha beta))
       (* i (/ t_4 (pow (fma i 2.0 (+ alpha beta)) 2.0))))
      t_2)
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 = (fma(i, t_4, (alpha * beta)) * (i * (t_4 / pow(fma(i, 2.0, (alpha + beta)), 2.0)))) / t_2;
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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(fma(i, t_4, Float64(alpha * beta)) * Float64(i * Float64(t_4 / (fma(i, 2.0, Float64(alpha + beta)) ^ 2.0)))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[(i * N[(t$95$4 / N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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 42.4%

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

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. *-commutative 42.4%

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

      5. fma-undefine 42.4%

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

      6. *-un-lft-identity 42.4%

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

      7. times-frac 99.6%

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

      8. +-commutative 99.6%

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

      9. pow2 99.6%

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

      10. +-commutative 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. /-rgt-identity 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+r+ 99.6%

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

      4. associate-/l* 99.4%

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

      5. +-commutative 99.4%

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

      6. associate-+r+ 99.4%

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

   6. Simplified 99.4%

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

   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%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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* 6.6%

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

      3. +-commutative 6.6%

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

      4. +-commutative 6.6%

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

      5. +-commutative 6.6%

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

      6. associate-+l+ 6.6%

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

      7. +-commutative 6.6%

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

      8. associate-*l* 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in i around inf 72.1%

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

      1. expm1-log1p-u 57.3%

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

      2. expm1-undefine 57.2%

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

      3. associate-*r/ 57.2%

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

      4. distribute-lft-out 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. expm1-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in i around 0 72.7%

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

   11. Taylor expanded in alpha around 0 63.6%

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

4. Final simplification 78.7%

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

Alternative 5: 84.1% accurate, 0.3× speedup

Math

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

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)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       (/ t_3 (fma i 2.0 (+ alpha beta)))
       (/ (* i (+ i beta)) (+ beta (* i 2.0))))
      t_2)
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = ((t_3 / fma(i, 2.0, (alpha + beta))) * ((i * (i + beta)) / (beta + (i * 2.0)))) / t_2;
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(t_3 / fma(i, 2.0, Float64(alpha + beta))) * Float64(Float64(i * Float64(i + beta)) / Float64(beta + Float64(i * 2.0)))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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]}, 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[(N[(t$95$3 / N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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 42.4%

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

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. +-commutative 42.4%

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

      4. *-commutative 42.4%

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

      5. fma-undefine 42.4%

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

      6. times-frac 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. +-commutative 99.6%

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

      12. *-commutative 99.6%

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

      13. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in alpha around 0 89.7%

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

   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. 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* 6.6%

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

      3. +-commutative 6.6%

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

      4. +-commutative 6.6%

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

      5. +-commutative 6.6%

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

      6. associate-+l+ 6.6%

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

      7. +-commutative 6.6%

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

      8. associate-*l* 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in i around inf 72.1%

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

      1. expm1-log1p-u 57.3%

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

      2. expm1-undefine 57.2%

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

      3. associate-*r/ 57.2%

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

      4. distribute-lft-out 57.2%

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

   7. Applied egg-rr 57.2%

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

      1. expm1-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in i around 0 72.7%

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

   11. Taylor expanded in alpha around 0 63.6%

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

4. Final simplification 74.6%

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

Alternative 6: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := i \cdot \left(i + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq 0.1:\\ \;\;\;\;\frac{\frac{t\_4 \cdot \left(\alpha \cdot \beta + t\_4\right)}{t\_1}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot 0.0625 + \left(\alpha + \beta\right) \cdot 0.125}{i} - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

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 (* i (+ i beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) 0.1)
     (/ (/ (* t_4 (+ (* alpha beta) t_4)) t_1) t_2)
     (-
      (/ (+ (* i 0.0625) (* (+ alpha beta) 0.125)) i)
      (* 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 = i * (i + beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 0.1) {
		tmp = ((t_4 * ((alpha * beta) + t_4)) / t_1) / t_2;
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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) :: t_4
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = t_1 + (-1.0d0)
    t_3 = i * (i + (alpha + beta))
    t_4 = i * (i + beta)
    if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 0.1d0) then
        tmp = ((t_4 * ((alpha * beta) + t_4)) / t_1) / t_2
    else
        tmp = (((i * 0.0625d0) + ((alpha + beta) * 0.125d0)) / i) - (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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = i * (i + beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 0.1) {
		tmp = ((t_4 * ((alpha * beta) + t_4)) / t_1) / t_2;
	} else {
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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 = t_1 + -1.0
	t_3 = i * (i + (alpha + beta))
	t_4 = i * (i + beta)
	tmp = 0
	if (((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 0.1:
		tmp = ((t_4 * ((alpha * beta) + t_4)) / t_1) / t_2
	else:
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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(i * Float64(i + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= 0.1)
		tmp = Float64(Float64(Float64(t_4 * Float64(Float64(alpha * beta) + t_4)) / t_1) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(i * 0.0625) + Float64(Float64(alpha + beta) * 0.125)) / i) - 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 = t_1 + -1.0;
	t_3 = i * (i + (alpha + beta));
	t_4 = i * (i + beta);
	tmp = 0.0;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 0.1)
		tmp = ((t_4 * ((alpha * beta) + t_4)) / t_1) / t_2;
	else
		tmp = (((i * 0.0625) + ((alpha + beta) * 0.125)) / i) - (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[(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[(i * 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], 0.1], N[(N[(N[(t$95$4 * N[(N[(alpha * beta), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(i * 0.0625), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / i), $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

   3. Taylor expanded in alpha around 0 93.6%

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

   4. Taylor expanded in alpha around 0 93.5%

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

   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 0.9%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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* 5.6%

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

      3. +-commutative 5.6%

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

      4. +-commutative 5.6%

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

      5. +-commutative 5.6%

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

      6. associate-+l+ 5.6%

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

      7. +-commutative 5.6%

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

      8. associate-*l* 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in i around inf 73.0%

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

      1. expm1-log1p-u 62.6%

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

      2. expm1-undefine 62.6%

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

      3. associate-*r/ 62.6%

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

      4. distribute-lft-out 62.6%

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

   7. Applied egg-rr 62.6%

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

      1. expm1-define 62.6%

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

   9. Simplified 62.6%

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

   10. Taylor expanded in i around 0 73.5%

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

   11. Taylor expanded in alpha around 0 66.9%

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

4. Final simplification 71.5%

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

Alternative 7: 77.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 17.9%

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

   1. associate-/l/ 15.7%

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

   2. associate-/l* 20.3%

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

   3. +-commutative 20.3%

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

   4. +-commutative 20.3%

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

   5. +-commutative 20.3%

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

   6. associate-+l+ 20.3%

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

   7. +-commutative 20.3%

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

   8. associate-*l* 20.2%

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

3. Simplified 20.2%

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

5. Taylor expanded in i around inf 74.6%

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

   1. expm1-log1p-u 66.0%

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

   2. expm1-undefine 66.0%

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

   3. associate-*r/ 66.0%

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

   4. distribute-lft-out 66.0%

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

7. Applied egg-rr 66.0%

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

   1. expm1-define 66.0%

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

9. Simplified 66.0%

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

10. Taylor expanded in i around 0 75.0%

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

11. Taylor expanded in alpha around 0 69.6%

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

12. Final simplification 69.6%

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

Alternative 8: 77.9% accurate, 3.5× speedup

Math

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

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) (* beta 0.125)) (* (+ alpha beta) 0.125)) i))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 17.9%

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

   1. associate-/l/ 15.7%

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

   2. associate-/l* 20.3%

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

   3. +-commutative 20.3%

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

   4. +-commutative 20.3%

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

   5. +-commutative 20.3%

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

   6. associate-+l+ 20.3%

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

   7. +-commutative 20.3%

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

   8. associate-*l* 20.2%

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

3. Simplified 20.2%

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

5. Taylor expanded in i around inf 74.6%

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

6. Taylor expanded in alpha around 0 69.1%

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

7. Taylor expanded in i around 0 69.1%

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

8. Final simplification 69.1%

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

Alternative 9: 77.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 17.9%

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

   1. associate-/l/ 15.7%

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

   2. associate-/l* 20.3%

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

   3. +-commutative 20.3%

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

   4. +-commutative 20.3%

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

   5. +-commutative 20.3%

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

   6. associate-+l+ 20.3%

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

   7. +-commutative 20.3%

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

   8. associate-*l* 20.2%

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

3. Simplified 20.2%

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

5. Taylor expanded in i around inf 74.6%

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

6. Taylor expanded in alpha around 0 69.1%

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

Alternative 10: 74.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\beta - \left(\alpha + \beta\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.6e+222) 0.0625 (/ (* 0.125 (- beta (+ alpha beta))) i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.6e+222) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (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) :: tmp
    if (beta <= 7.6d+222) then
        tmp = 0.0625d0
    else
        tmp = (0.125d0 * (beta - (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 tmp;
	if (beta <= 7.6e+222) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (alpha + beta))) / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.6e+222:
		tmp = 0.0625
	else:
		tmp = (0.125 * (beta - (alpha + beta))) / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.6e+222)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(0.125 * Float64(beta - Float64(alpha + beta))) / 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 <= 7.6e+222)
		tmp = 0.0625;
	else
		tmp = (0.125 * (beta - (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_] := If[LessEqual[beta, 7.6e+222], 0.0625, N[(N[(0.125 * N[(beta - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.60000000000000035e222

   1. Initial program 19.8%

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

      1. associate-/l/ 17.4%

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

      2. associate-/l* 21.6%

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

      3. +-commutative 21.6%

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

      4. +-commutative 21.6%

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

      5. +-commutative 21.6%

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

      6. associate-+l+ 21.6%

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

      7. +-commutative 21.6%

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

      8. associate-*l* 21.6%

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

   3. Simplified 21.6%

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

   5. Taylor expanded in i around inf 72.8%

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

   if 7.60000000000000035e222 < 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* 8.0%

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

      3. +-commutative 8.0%

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

      4. +-commutative 8.0%

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

      5. +-commutative 8.0%

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

      6. associate-+l+ 8.0%

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

      7. +-commutative 8.0%

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

      8. associate-*l* 8.0%

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

   3. Simplified 8.0%

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

   5. Taylor expanded in i around inf 43.8%

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

   6. Taylor expanded in alpha around 0 39.7%

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

   7. Taylor expanded in i around 0 35.3%

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

      1. distribute-lft-out-- 35.3%

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

   9. Simplified 35.3%

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

Alternative 11: 71.5% 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 17.9%

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

   1. associate-/l/ 15.7%

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

   2. associate-/l* 20.3%

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

   3. +-commutative 20.3%

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

   4. +-commutative 20.3%

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

   5. +-commutative 20.3%

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

   6. associate-+l+ 20.3%

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

   7. +-commutative 20.3%

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

   8. associate-*l* 20.2%

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

3. Simplified 20.2%

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

5. Taylor expanded in i around inf 66.5%

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

Reproduce

herbie shell --seed 2024152 
(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)))