Octave 3.8, jcobi/4

Percentage Accurate: 16.4% → 84.8%

Time: 23.8s

Alternatives: 9

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + 1\right)}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1e+202)
   0.0625
   (/
    (/ (+ i alpha) (+ (+ beta alpha) (fma i 2.0 -1.0)))
    (/ (+ (fma i 2.0 alpha) (+ beta 1.0)) i))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1e+202) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / ((beta + alpha) + fma(i, 2.0, -1.0))) / ((fma(i, 2.0, alpha) + (beta + 1.0)) / i);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1e+202)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))) / Float64(Float64(fma(i, 2.0, alpha) + Float64(beta + 1.0)) / i));
	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, 1e+202], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * 2.0 + alpha), $MachinePrecision] + N[(beta + 1.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.999999999999999e201

   1. Initial program 19.9%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in i around inf 83.4%

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

   if 9.999999999999999e201 < 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. Taylor expanded in beta around -inf 34.5%

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

      1. associate-*r* 34.5%

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

      2. neg-mul-1 34.5%

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

      3. distribute-lft-out 34.5%

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

   4. Simplified 34.5%

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

      1. difference-of-sqr-1 34.5%

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

      2. associate-+r+ 34.5%

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

      3. *-commutative 34.5%

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

      4. fma-udef 34.5%

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

      5. sub-neg 34.5%

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

      6. metadata-eval 34.5%

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

      7. associate-+r+ 34.5%

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

      8. *-commutative 34.5%

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

      9. fma-udef 34.5%

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

      10. associate-*r* 34.5%

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

      11. times-frac 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. remove-double-neg 86.6%

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

      2. fma-udef 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. +-commutative 86.6%

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

      6. fma-udef 86.6%

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

      7. +-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. clear-num 86.5%

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

      3. frac-times 53.1%

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

      4. *-lft-identity 53.1%

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

      5. *-commutative 53.1%

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

      6. associate-+r+ 53.1%

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

      7. +-commutative 53.1%

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

      8. associate-+l+ 53.1%

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

      9. associate-+r+ 53.1%

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

      10. +-commutative 53.1%

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

      11. +-commutative 53.1%

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

   10. Applied egg-rr 53.1%

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

       1. +-commutative 53.1%

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

       2. associate-/r* 86.8%

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

       3. +-commutative 86.8%

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

       4. associate-+r+ 86.8%

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

   12. Simplified 86.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha\right) + \left(\beta + 1\right)}{i}}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)} \cdot \frac{i + \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.2e+201)
   0.0625
   (*
    (/ i (+ 1.0 (+ beta (fma i 2.0 alpha))))
    (/ (+ i alpha) (+ alpha (+ beta (fma i 2.0 -1.0)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = (i / (1.0 + (beta + fma(i, 2.0, alpha)))) * ((i + alpha) / (alpha + (beta + fma(i, 2.0, -1.0))));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(1.0 + Float64(beta + fma(i, 2.0, alpha)))) * Float64(Float64(i + alpha) / Float64(alpha + Float64(beta + fma(i, 2.0, -1.0)))));
	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, 9.2e+201], 0.0625, N[(N[(i / N[(1.0 + N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2000000000000004e201

   1. Initial program 19.9%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in i around inf 83.4%

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

   if 9.2000000000000004e201 < 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. Taylor expanded in beta around -inf 34.5%

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

      1. associate-*r* 34.5%

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

      2. neg-mul-1 34.5%

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

      3. distribute-lft-out 34.5%

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

   4. Simplified 34.5%

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

      1. difference-of-sqr-1 34.5%

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

      2. associate-+r+ 34.5%

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

      3. *-commutative 34.5%

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

      4. fma-udef 34.5%

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

      5. sub-neg 34.5%

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

      6. metadata-eval 34.5%

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

      7. associate-+r+ 34.5%

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

      8. *-commutative 34.5%

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

      9. fma-udef 34.5%

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

      10. associate-*r* 34.5%

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

      11. times-frac 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. remove-double-neg 86.6%

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

      2. fma-udef 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. +-commutative 86.6%

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

      6. fma-udef 86.6%

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

      7. +-commutative 86.6%

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

   8. Simplified 86.6%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)} \cdot \frac{i + \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, -1\right)\right)}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.2e+201)
   0.0625
   (/ (* i (/ (+ i alpha) beta)) (+ beta (fma i 2.0 alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((i + alpha) / beta)) / (beta + fma(i, 2.0, alpha));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) / Float64(beta + fma(i, 2.0, alpha)));
	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, 9.2e+201], 0.0625, N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2000000000000004e201

   1. Initial program 19.9%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in i around inf 83.4%

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

   if 9.2000000000000004e201 < 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 17.7%

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

   4. Taylor expanded in beta around inf 17.9%

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

      1. associate-/r* 29.1%

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

      2. fma-udef 29.1%

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

      3. *-commutative 29.1%

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

      4. +-commutative 29.1%

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

      5. associate-*r/ 29.1%

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

   6. Applied egg-rr 86.1%

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

   7. Taylor expanded in i around 0 86.1%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}\\ \end{array} \]

Alternative 4: 84.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + \alpha}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.2e+201) 0.0625 (* (/ (+ i alpha) beta) (/ i (+ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / (beta + alpha));
	}
	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 <= 9.2d+201) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / (beta + alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / (beta + alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.2e+201:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / (beta + alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / Float64(beta + alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / (beta + alpha));
	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, 9.2e+201], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2000000000000004e201

   1. Initial program 19.9%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in i around inf 83.4%

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

   if 9.2000000000000004e201 < 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 17.7%

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

   4. Taylor expanded in beta around inf 17.9%

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

   5. Taylor expanded in i around 0 86.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + \alpha}\\ \end{array} \]

Alternative 5: 84.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.2e+201) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	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 <= 9.2d+201) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+201) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.2e+201:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.2e+201)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	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, 9.2e+201], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.2000000000000004e201

   1. Initial program 19.9%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in i around inf 83.4%

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

   if 9.2000000000000004e201 < 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 17.7%

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

   4. Taylor expanded in beta around inf 17.9%

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

   5. Taylor expanded in beta around inf 85.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6: 74.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.3e+223) 0.0625 (* i (/ i (* beta beta)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.3e+223) {
		tmp = 0.0625;
	} else {
		tmp = i * (i / (beta * beta));
	}
	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 <= 3.3d+223) then
        tmp = 0.0625d0
    else
        tmp = i * (i / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.3e+223) {
		tmp = 0.0625;
	} else {
		tmp = i * (i / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.3e+223:
		tmp = 0.0625
	else:
		tmp = i * (i / (beta * beta))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.3e+223)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.3e+223)
		tmp = 0.0625;
	else
		tmp = i * (i / (beta * beta));
	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, 3.3e+223], 0.0625, N[(i * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.3e223

   1. Initial program 19.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in i around inf 81.1%

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

   if 3.3e223 < 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 18.2%

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

   4. Taylor expanded in beta around inf 39.8%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

   6. Simplified 42.6%

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

   7. Taylor expanded in i around inf 40.1%

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

      1. unpow2 40.1%

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

      2. associate-*r/ 42.6%

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

      3. unpow2 42.6%

         \[\leadsto i \cdot \frac{i}{\color{blue}{\beta \cdot \beta}} \]

   9. Simplified 42.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \]

Alternative 7: 75.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{\alpha}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.25e+223) 0.0625 (/ i (* beta (/ beta alpha)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+223) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / alpha));
	}
	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 <= 2.25d+223) then
        tmp = 0.0625d0
    else
        tmp = i / (beta * (beta / alpha))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.25e+223) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / alpha));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.25e+223:
		tmp = 0.0625
	else:
		tmp = i / (beta * (beta / alpha))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.25e+223)
		tmp = 0.0625;
	else
		tmp = Float64(i / Float64(beta * Float64(beta / alpha)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.25e+223)
		tmp = 0.0625;
	else
		tmp = i / (beta * (beta / alpha));
	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, 2.25e+223], 0.0625, N[(i / N[(beta * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.25e223

   1. Initial program 19.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in i around inf 81.1%

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

   if 2.25e223 < 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 18.2%

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

   4. Taylor expanded in beta around inf 39.8%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

   6. Simplified 42.6%

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

      1. associate-/l* 59.2%

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

      2. associate-/r/ 59.1%

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

      3. +-commutative 59.1%

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

   8. Applied egg-rr 59.1%

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

   9. Taylor expanded in i around 0 43.3%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{\alpha}}\\ \end{array} \]

Alternative 8: 76.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{+221}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.35e+221) 0.0625 (/ i (* beta (/ beta i)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.35e+221) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / 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 <= 2.35d+221) then
        tmp = 0.0625d0
    else
        tmp = i / (beta * (beta / 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 <= 2.35e+221) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / i));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.35e+221:
		tmp = 0.0625
	else:
		tmp = i / (beta * (beta / i))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.35e+221)
		tmp = 0.0625;
	else
		tmp = Float64(i / Float64(beta * Float64(beta / i)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.35e+221)
		tmp = 0.0625;
	else
		tmp = i / (beta * (beta / 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, 2.35e+221], 0.0625, N[(i / N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.35000000000000003e221

   1. Initial program 19.3%

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

   3. Simplified 40.3%

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

   4. Taylor expanded in i around inf 81.1%

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

   if 2.35000000000000003e221 < 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 18.2%

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

   4. Taylor expanded in beta around inf 39.8%

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

      1. associate-/l* 42.6%

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

      2. unpow2 42.6%

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

   6. Simplified 42.6%

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

      1. associate-/l* 59.2%

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

      2. associate-/r/ 59.1%

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

      3. +-commutative 59.1%

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

   8. Applied egg-rr 59.1%

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

   9. Taylor expanded in i around inf 58.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.35 \cdot 10^{+221}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \end{array} \]

Alternative 9: 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 17.7%

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

3. Simplified 38.4%

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

4. Taylor expanded in i around inf 74.9%

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

5. Final simplification 74.9%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023297 
(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)))