Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 84.7%

Time: 15.7s

Alternatives: 6

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+192} \lor \neg \left(\beta \leq 1.6 \cdot 10^{+211}\right):\\ \;\;\;\;{\left(\frac{\sqrt{i + \alpha} \cdot \sqrt{i}}{\beta}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot {\left(\sqrt[3]{\frac{\beta + \alpha}{i}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+143)
   0.0625
   (if (or (<= beta 3.9e+192) (not (<= beta 1.6e+211)))
     (pow (/ (* (sqrt (+ i alpha)) (sqrt i)) beta) 2.0)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 0.125 (pow (cbrt (/ (+ beta alpha) i)) 3.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+143) {
		tmp = 0.0625;
	} else if ((beta <= 3.9e+192) || !(beta <= 1.6e+211)) {
		tmp = pow(((sqrt((i + alpha)) * sqrt(i)) / beta), 2.0);
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * pow(cbrt(((beta + alpha) / i)), 3.0));
	}
	return tmp;
}

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+143) {
		tmp = 0.0625;
	} else if ((beta <= 3.9e+192) || !(beta <= 1.6e+211)) {
		tmp = Math.pow(((Math.sqrt((i + alpha)) * Math.sqrt(i)) / beta), 2.0);
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * Math.pow(Math.cbrt(((beta + alpha) / i)), 3.0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.8e+143)
		tmp = 0.0625;
	elseif ((beta <= 3.9e+192) || !(beta <= 1.6e+211))
		tmp = Float64(Float64(sqrt(Float64(i + alpha)) * sqrt(i)) / beta) ^ 2.0;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * (cbrt(Float64(Float64(beta + alpha) / i)) ^ 3.0)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.8e+143], 0.0625, If[Or[LessEqual[beta, 3.9e+192], N[Not[LessEqual[beta, 1.6e+211]], $MachinePrecision]], N[Power[N[(N[(N[Sqrt[N[(i + alpha), $MachinePrecision]], $MachinePrecision] * N[Sqrt[i], $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[Power[N[Power[N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if beta < 3.8e143

   1. Initial program 13.0%

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

      1. associate-/l/ 11.5%

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

      2. associate-*l* 11.4%

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

      3. times-frac 19.4%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in i around inf 80.6%

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

   if 3.8e143 < beta < 3.8999999999999998e192 or 1.59999999999999988e211 < 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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in beta around inf 21.5%

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

      1. associate-/l* 23.5%

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

      2. +-commutative 23.5%

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

   6. Simplified 23.5%

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

      1. associate-/r/ 23.5%

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

      2. +-commutative 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. add-sqr-sqrt 23.5%

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

      2. pow2 23.5%

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

      3. associate-*l/ 21.5%

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

      4. sqrt-div 21.5%

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

      5. +-commutative 21.5%

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

      6. unpow2 21.5%

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

      7. sqrt-prod 45.1%

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

      8. add-sqr-sqrt 45.2%

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

   10. Applied egg-rr 45.2%

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

       1. pow1/2 45.2%

          \[\leadsto {\left(\frac{\color{blue}{{\left(i \cdot \left(i + \alpha\right)\right)}^{0.5}}}{\beta}\right)}^{2} \]

       2. *-commutative 45.2%

          \[\leadsto {\left(\frac{{\color{blue}{\left(\left(i + \alpha\right) \cdot i\right)}}^{0.5}}{\beta}\right)}^{2} \]

       3. unpow-prod-down 68.7%

          \[\leadsto {\left(\frac{\color{blue}{{\left(i + \alpha\right)}^{0.5} \cdot {i}^{0.5}}}{\beta}\right)}^{2} \]

       4. pow1/2 68.7%

          \[\leadsto {\left(\frac{\color{blue}{\sqrt{i + \alpha}} \cdot {i}^{0.5}}{\beta}\right)}^{2} \]

       5. pow1/2 68.7%

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

   12. Applied egg-rr 68.7%

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

   if 3.8999999999999998e192 < beta < 1.59999999999999988e211

   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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 76.9%

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

      1. add-cube-cbrt 63.3%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\alpha + \beta}{i}} \cdot \sqrt[3]{\frac{\alpha + \beta}{i}}\right) \cdot \sqrt[3]{\frac{\alpha + \beta}{i}}\right)} \]

      2. pow3 63.3%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\sqrt[3]{\frac{\alpha + \beta}{i}}\right)}^{3}} \]

   6. Applied egg-rr 63.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+192} \lor \neg \left(\beta \leq 1.6 \cdot 10^{+211}\right):\\ \;\;\;\;{\left(\frac{\sqrt{i + \alpha} \cdot \sqrt{i}}{\beta}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot {\left(\sqrt[3]{\frac{\beta + \alpha}{i}}\right)}^{3}\\ \end{array} \]

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+192}:\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \frac{i}{\beta}\right) \cdot \left(i + \alpha\right)\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))))
   (if (<= beta 4.7e+143)
     0.0625
     (if (<= beta 3.4e+192)
       (* (* (/ 1.0 beta) (/ i beta)) (+ i alpha))
       (if (<= beta 3.1e+218) (- (+ 0.0625 t_0) t_0) (pow (/ i beta) 2.0))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 4.7e+143) {
		tmp = 0.0625;
	} else if (beta <= 3.4e+192) {
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	} else if (beta <= 3.1e+218) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = pow((i / beta), 2.0);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.125d0 * (beta / i)
    if (beta <= 4.7d+143) then
        tmp = 0.0625d0
    else if (beta <= 3.4d+192) then
        tmp = ((1.0d0 / beta) * (i / beta)) * (i + alpha)
    else if (beta <= 3.1d+218) then
        tmp = (0.0625d0 + t_0) - t_0
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 4.7e+143) {
		tmp = 0.0625;
	} else if (beta <= 3.4e+192) {
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	} else if (beta <= 3.1e+218) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	tmp = 0
	if beta <= 4.7e+143:
		tmp = 0.0625
	elif beta <= 3.4e+192:
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha)
	elif beta <= 3.1e+218:
		tmp = (0.0625 + t_0) - t_0
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (beta <= 4.7e+143)
		tmp = 0.0625;
	elseif (beta <= 3.4e+192)
		tmp = Float64(Float64(Float64(1.0 / beta) * Float64(i / beta)) * Float64(i + alpha));
	elseif (beta <= 3.1e+218)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = 0.0;
	if (beta <= 4.7e+143)
		tmp = 0.0625;
	elseif (beta <= 3.4e+192)
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	elseif (beta <= 3.1e+218)
		tmp = (0.0625 + t_0) - t_0;
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.7e+143], 0.0625, If[LessEqual[beta, 3.4e+192], N[(N[(N[(1.0 / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.1e+218], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.7e143

   1. Initial program 13.0%

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

      1. associate-/l/ 11.5%

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

      2. associate-*l* 11.4%

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

      3. times-frac 19.4%

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

   3. Simplified 38.2%

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

   4. Taylor expanded in i around inf 80.6%

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

   if 4.7e143 < beta < 3.39999999999999996e192

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 14.2%

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

   4. Taylor expanded in beta around inf 14.6%

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

      1. associate-/l* 16.5%

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

      2. +-commutative 16.5%

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

   6. Simplified 16.5%

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

      1. associate-/r/ 16.5%

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

      2. +-commutative 16.5%

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

   8. Applied egg-rr 16.5%

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

      1. *-un-lft-identity 16.5%

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

      2. unpow2 16.5%

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

      3. times-frac 50.1%

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

   10. Applied egg-rr 50.1%

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

   if 3.39999999999999996e192 < beta < 3.1000000000000002e218

   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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 11.4%

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

   4. Taylor expanded in i around inf 80.6%

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

   5. Taylor expanded in alpha around 0 70.9%

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

   6. Taylor expanded in alpha around 0 80.6%

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

   if 3.1000000000000002e218 < 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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in beta around inf 24.3%

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

      1. associate-/l* 26.4%

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

      2. +-commutative 26.4%

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

   6. Simplified 26.4%

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

      1. associate-/r/ 26.4%

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

      2. +-commutative 26.4%

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

   8. Applied egg-rr 26.4%

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

      1. add-sqr-sqrt 26.4%

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

      2. pow2 26.4%

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

      3. associate-*l/ 24.3%

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

      4. sqrt-div 24.3%

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

      5. +-commutative 24.3%

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

      6. unpow2 24.3%

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

      7. sqrt-prod 46.9%

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

      8. add-sqr-sqrt 47.1%

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

   10. Applied egg-rr 47.1%

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

   11. Taylor expanded in i around inf 69.1%

       \[\leadsto {\left(\frac{\color{blue}{i}}{\beta}\right)}^{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+192}:\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \frac{i}{\beta}\right) \cdot \left(i + \alpha\right)\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+218}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]

Alternative 3: 78.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+201}\right):\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \frac{i}{\beta}\right) \cdot \left(i + \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.2e+143)
   0.0625
   (if (or (<= beta 5e+192) (not (<= beta 2.4e+201)))
     (* (* (/ 1.0 beta) (/ i beta)) (+ i alpha))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+143) {
		tmp = 0.0625;
	} else if ((beta <= 5e+192) || !(beta <= 2.4e+201)) {
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 5.2d+143) then
        tmp = 0.0625d0
    else if ((beta <= 5d+192) .or. (.not. (beta <= 2.4d+201))) then
        tmp = ((1.0d0 / beta) * (i / beta)) * (i + alpha)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.2e+143) {
		tmp = 0.0625;
	} else if ((beta <= 5e+192) || !(beta <= 2.4e+201)) {
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.2e+143:
		tmp = 0.0625
	elif (beta <= 5e+192) or not (beta <= 2.4e+201):
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.2e+143)
		tmp = 0.0625;
	elseif ((beta <= 5e+192) || !(beta <= 2.4e+201))
		tmp = Float64(Float64(Float64(1.0 / beta) * Float64(i / beta)) * Float64(i + alpha));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.2e+143)
		tmp = 0.0625;
	elseif ((beta <= 5e+192) || ~((beta <= 2.4e+201)))
		tmp = ((1.0 / beta) * (i / beta)) * (i + alpha);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 5.2e+143], 0.0625, If[Or[LessEqual[beta, 5e+192], N[Not[LessEqual[beta, 2.4e+201]], $MachinePrecision]], N[(N[(N[(1.0 / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] * N[(i + alpha), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.1999999999999998e143 or 5.00000000000000033e192 < beta < 2.39999999999999993e201

   1. Initial program 12.8%

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

      1. associate-/l/ 11.2%

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

      2. associate-*l* 11.2%

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

      3. times-frac 19.0%

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

   3. Simplified 37.5%

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

   4. Taylor expanded in i around inf 80.6%

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

   if 5.1999999999999998e143 < beta < 5.00000000000000033e192 or 2.39999999999999993e201 < 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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 7.0%

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

   4. Taylor expanded in beta around inf 22.7%

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

      1. associate-/l* 24.6%

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

      2. +-commutative 24.6%

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

   6. Simplified 24.6%

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

      1. associate-/r/ 24.6%

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

      2. +-commutative 24.6%

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

   8. Applied egg-rr 24.6%

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

      1. *-un-lft-identity 24.6%

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

      2. unpow2 24.6%

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

      3. times-frac 47.4%

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

   10. Applied egg-rr 47.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.2 \cdot 10^{+143}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+192} \lor \neg \left(\beta \leq 2.4 \cdot 10^{+201}\right):\\ \;\;\;\;\left(\frac{1}{\beta} \cdot \frac{i}{\beta}\right) \cdot \left(i + \alpha\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4: 77.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \left(0.0625 + t_0\right) - t_0 \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i)))) (- (+ 0.0625 t_0) t_0)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_0;
}

Fortran

NOTE: alpha and beta 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
    t_0 = 0.125d0 * (beta / i)
    code = (0.0625d0 + t_0) - t_0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	return (0.0625 + t_0) - t_0

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	return Float64(Float64(0.0625 + t_0) - t_0)
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = (0.0625 + t_0) - t_0;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]

Derivation

1. Initial program 10.7%

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

   1. associate-/l/ 9.4%

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

   2. associate-*l* 9.4%

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

   3. times-frac 15.9%

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

3. Simplified 32.5%

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

4. Taylor expanded in i around inf 80.4%

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

5. Taylor expanded in alpha around 0 74.2%

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

6. Taylor expanded in alpha around 0 75.5%

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

7. Final simplification 75.5%

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

Alternative 5: 73.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+263}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\beta - \left(\beta + \alpha\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+263) 0.0625 (/ (* 0.125 (- beta (+ beta alpha))) i)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+263) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 2d+263) then
        tmp = 0.0625d0
    else
        tmp = (0.125d0 * (beta - (beta + alpha))) / i
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+263) {
		tmp = 0.0625;
	} else {
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+263:
		tmp = 0.0625
	else:
		tmp = (0.125 * (beta - (beta + alpha))) / i
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+263)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(0.125 * Float64(beta - Float64(beta + alpha))) / i);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+263)
		tmp = 0.0625;
	else
		tmp = (0.125 * (beta - (beta + alpha))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+263], 0.0625, N[(N[(0.125 * N[(beta - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.00000000000000003e263

   1. Initial program 11.3%

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

      1. associate-/l/ 9.9%

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

      2. associate-*l* 9.9%

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

      3. times-frac 16.8%

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

   3. Simplified 34.3%

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

   4. Taylor expanded in i around inf 75.9%

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

   if 2.00000000000000003e263 < 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* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 45.4%

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

   5. Taylor expanded in alpha around 0 45.4%

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

   6. Taylor expanded in i around 0 45.4%

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

      1. distribute-lft-out-- 45.4%

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

   8. Simplified 45.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+263}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\beta - \left(\beta + \alpha\right)\right)}{i}\\ \end{array} \]

Alternative 6: 71.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta 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;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.7%

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

   1. associate-/l/ 9.4%

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

   2. associate-*l* 9.4%

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

   3. times-frac 15.9%

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

3. Simplified 32.5%

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

4. Taylor expanded in i around inf 72.3%

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

5. Final simplification 72.3%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023311 
(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)))