Octave 3.8, jcobi/4

Percentage Accurate: 15.6% → 85.9%

Time: 23.2s

Alternatives: 8

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: 15.6% 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: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+141}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + i\right) \cdot \frac{i}{t\_0 + -1}}{t\_0 + 1}\\ \end{array} \end{array} \]

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 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 1.85e+141)
     0.0625
     (/ (* (+ alpha i) (/ i (+ t_0 -1.0))) (+ t_0 1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 1.85e+141) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) * (i / (t_0 + -1.0))) / (t_0 + 1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.85e+141)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) * Float64(i / Float64(t_0 + -1.0))) / Float64(t_0 + 1.0));
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.85e+141], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] * N[(i / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8500000000000001e141

   1. Initial program 19.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around inf 77.7%

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

   if 1.8500000000000001e141 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf 15.7%

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

   4. Taylor expanded in beta around inf 18.6%

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

      1. *-commutative 18.6%

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

      2. difference-of-sqr-1 18.6%

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

      3. times-frac 66.9%

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

      4. +-commutative 66.9%

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

      5. +-commutative 66.9%

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

      6. *-commutative 66.9%

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

      7. fma-define 66.9%

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

      8. +-commutative 66.9%

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

      9. +-commutative 66.9%

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

      10. *-commutative 66.9%

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

      11. fma-define 66.9%

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

      12. +-commutative 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. associate-*l/ 67.1%

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

      2. +-commutative 67.1%

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

      3. sub-neg 67.1%

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

      4. +-commutative 67.1%

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

      5. metadata-eval 67.1%

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

      6. +-commutative 67.1%

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

      7. +-commutative 67.1%

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

   8. Simplified 67.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+141}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + i\right) \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{t\_0 + 1} \cdot \frac{i}{t\_0 + -1}\\ \end{array} \end{array} \]

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 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 4.6e+141)
     0.0625
     (* (/ (+ alpha i) (+ t_0 1.0)) (/ i (+ t_0 -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 4.6e+141) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / (t_0 + 1.0)) * (i / (t_0 + -1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.6e+141)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(t_0 + 1.0)) * Float64(i / Float64(t_0 + -1.0)));
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.6e+141], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6000000000000003e141

   1. Initial program 19.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in i around inf 77.7%

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

   if 4.6000000000000003e141 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf 15.7%

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

   4. Taylor expanded in beta around inf 18.6%

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

      1. *-commutative 18.6%

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

      2. difference-of-sqr-1 18.6%

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

      3. times-frac 66.9%

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

      4. +-commutative 66.9%

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

      5. +-commutative 66.9%

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

      6. *-commutative 66.9%

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

      7. fma-define 66.9%

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

      8. +-commutative 66.9%

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

      9. +-commutative 66.9%

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

      10. *-commutative 66.9%

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

      11. fma-define 66.9%

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

      12. +-commutative 66.9%

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

   6. Applied egg-rr 66.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.5× speedup

Math

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

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 (+ (+ beta alpha) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ beta alpha))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* beta alpha))) t_1) (+ t_1 -1.0)))
        (t_4 (* 0.125 (/ beta i))))
   (if (<= t_3 0.1) t_3 (- (+ 0.0625 t_4) t_4))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	double t_4 = 0.125 * (beta / i);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + t_4) - t_4;
	}
	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 = (beta + alpha) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (beta + alpha))
    t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + (-1.0d0))
    t_4 = 0.125d0 * (beta / i)
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + t_4) - t_4
    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 = (beta + alpha) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (beta + alpha));
	double t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	double t_4 = 0.125 * (beta / i);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + t_4) - t_4;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = (beta + alpha) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (beta + alpha))
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0)
	t_4 = 0.125 * (beta / i)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + t_4) - t_4
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(beta + alpha)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(beta * alpha))) / t_1) / Float64(t_1 + -1.0))
	t_4 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + t_4) - t_4);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (beta + alpha) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (beta + alpha));
	t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + -1.0);
	t_4 = 0.125 * (beta / i);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + t_4) - t_4;
	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[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + t$95$4), $MachinePrecision] - t$95$4), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.6%

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

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

   1. Initial program 0.8%

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

   3. Simplified 32.4%

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

   5. Taylor expanded in i around inf 74.2%

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

   6. Taylor expanded in alpha around 0 69.7%

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

      1. associate-*r/ 69.7%

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

   8. Simplified 69.7%

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

   9. Taylor expanded in i around inf 69.7%

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

   10. Taylor expanded in alpha around 0 70.5%

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

4. Final simplification 74.9%

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

Alternative 4: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;i \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t\_0\right) - t\_0\\ \end{array} \end{array} \]

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 (* 0.125 (/ beta i))))
   (if (<= i 3.1e+25)
     (/
      (* i (+ alpha i))
      (+ (* (+ (+ beta alpha) (* i 2.0)) (+ beta (* i 2.0))) -1.0))
     (- (+ 0.0625 t_0) t_0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (i <= 3.1e+25) {
		tmp = (i * (alpha + i)) / ((((beta + alpha) + (i * 2.0)) * (beta + (i * 2.0))) + -1.0);
	} else {
		tmp = (0.0625 + t_0) - t_0;
	}
	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) :: tmp
    t_0 = 0.125d0 * (beta / i)
    if (i <= 3.1d+25) then
        tmp = (i * (alpha + i)) / ((((beta + alpha) + (i * 2.0d0)) * (beta + (i * 2.0d0))) + (-1.0d0))
    else
        tmp = (0.0625d0 + t_0) - t_0
    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 = 0.125 * (beta / i);
	double tmp;
	if (i <= 3.1e+25) {
		tmp = (i * (alpha + i)) / ((((beta + alpha) + (i * 2.0)) * (beta + (i * 2.0))) + -1.0);
	} else {
		tmp = (0.0625 + t_0) - t_0;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	tmp = 0
	if i <= 3.1e+25:
		tmp = (i * (alpha + i)) / ((((beta + alpha) + (i * 2.0)) * (beta + (i * 2.0))) + -1.0)
	else:
		tmp = (0.0625 + t_0) - t_0
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (i <= 3.1e+25)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(Float64(Float64(beta + alpha) + Float64(i * 2.0)) * Float64(beta + Float64(i * 2.0))) + -1.0));
	else
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = 0.0;
	if (i <= 3.1e+25)
		tmp = (i * (alpha + i)) / ((((beta + alpha) + (i * 2.0)) * (beta + (i * 2.0))) + -1.0);
	else
		tmp = (0.0625 + t_0) - t_0;
	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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3.1e+25], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] * N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 3.0999999999999998e25

   1. Initial program 44.1%

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

   3. Taylor expanded in beta around inf 39.4%

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

   4. Taylor expanded in beta around inf 49.6%

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

   5. Taylor expanded in alpha around 0 43.0%

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

      1. *-commutative 43.0%

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

   7. Simplified 43.0%

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

   if 3.0999999999999998e25 < i

   1. Initial program 13.7%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in i around inf 77.6%

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

   6. Taylor expanded in alpha around 0 74.0%

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

      1. associate-*r/ 74.0%

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

   8. Simplified 74.0%

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

   9. Taylor expanded in i around inf 74.0%

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

   10. Taylor expanded in alpha around 0 74.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\beta + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;i \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t\_0\right) - t\_0\\ \end{array} \end{array} \]

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 (* 0.125 (/ beta i))))
   (if (<= i 2.7e+25)
     (/ (* i (+ alpha i)) (+ (* beta (+ (+ beta alpha) (* i 2.0))) -1.0))
     (- (+ 0.0625 t_0) t_0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (i <= 2.7e+25) {
		tmp = (i * (alpha + i)) / ((beta * ((beta + alpha) + (i * 2.0))) + -1.0);
	} else {
		tmp = (0.0625 + t_0) - t_0;
	}
	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) :: tmp
    t_0 = 0.125d0 * (beta / i)
    if (i <= 2.7d+25) then
        tmp = (i * (alpha + i)) / ((beta * ((beta + alpha) + (i * 2.0d0))) + (-1.0d0))
    else
        tmp = (0.0625d0 + t_0) - t_0
    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 = 0.125 * (beta / i);
	double tmp;
	if (i <= 2.7e+25) {
		tmp = (i * (alpha + i)) / ((beta * ((beta + alpha) + (i * 2.0))) + -1.0);
	} else {
		tmp = (0.0625 + t_0) - t_0;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	tmp = 0
	if i <= 2.7e+25:
		tmp = (i * (alpha + i)) / ((beta * ((beta + alpha) + (i * 2.0))) + -1.0)
	else:
		tmp = (0.0625 + t_0) - t_0
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (i <= 2.7e+25)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(beta * Float64(Float64(beta + alpha) + Float64(i * 2.0))) + -1.0));
	else
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = 0.0;
	if (i <= 2.7e+25)
		tmp = (i * (alpha + i)) / ((beta * ((beta + alpha) + (i * 2.0))) + -1.0);
	else
		tmp = (0.0625 + t_0) - t_0;
	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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.7e+25], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(beta * N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.7e25

   1. Initial program 44.1%

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

   3. Taylor expanded in beta around inf 39.4%

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

   4. Taylor expanded in beta around inf 49.6%

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

   5. Taylor expanded in beta around inf 40.6%

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

   if 2.7e25 < i

   1. Initial program 13.7%

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

   3. Simplified 39.9%

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

   5. Taylor expanded in i around inf 77.6%

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

   6. Taylor expanded in alpha around 0 74.0%

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

      1. associate-*r/ 74.0%

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

   8. Simplified 74.0%

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

   9. Taylor expanded in i around inf 74.0%

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

   10. Taylor expanded in alpha around 0 74.8%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.7 \cdot 10^{+25}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta + \alpha} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.5e+204) 0.0625 (* (/ alpha (+ beta alpha)) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+204) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / (beta + alpha)) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 8.5d+204) then
        tmp = 0.0625d0
    else
        tmp = (alpha / (beta + alpha)) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+204) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / (beta + alpha)) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+204:
		tmp = 0.0625
	else:
		tmp = (alpha / (beta + alpha)) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+204)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha / Float64(beta + alpha)) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+204)
		tmp = 0.0625;
	else
		tmp = (alpha / (beta + alpha)) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+204], 0.0625, N[(N[(alpha / N[(beta + alpha), $MachinePrecision]), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.5e204

   1. Initial program 18.2%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in i around inf 76.1%

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

   if 8.5e204 < beta

   1. Initial program 0.0%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in i around 0 21.2%

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

      1. times-frac 21.8%

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

      2. sub-neg 21.8%

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

      3. metadata-eval 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in beta around inf 38.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta + \alpha} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \left(0.0625 + t\_0\right) - t\_0 \end{array} \end{array} \]

FPCore

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

Java

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

Julia

alpha, beta, i = sort([alpha, beta, i])
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, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = (0.0625 + t_0) - t_0;
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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]

Derivation

1. Initial program 15.9%

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

3. Simplified 42.6%

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

5. Taylor expanded in i around inf 74.9%

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

6. Taylor expanded in alpha around 0 71.1%

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

   1. associate-*r/ 71.1%

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

8. Simplified 71.1%

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

9. Taylor expanded in i around inf 71.1%

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

10. Taylor expanded in alpha around 0 71.8%

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

11. Final simplification 71.8%

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

Alternative 8: 70.9% 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 15.9%

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

3. Simplified 42.6%

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

5. Taylor expanded in i around inf 68.5%

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

6. Final simplification 68.5%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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)))