Octave 3.8, jcobi/4

Percentage Accurate: 15.5% → 85.1%

Time: 21.1s

Alternatives: 4

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: 15.5% 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: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{i + \alpha}}{\beta}\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(i \cdot t\_0\right)\\ \end{array} \end{array} \]

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 (/ (sqrt (+ i alpha)) beta)))
   (if (<= beta 1.8e+124) 0.0625 (* t_0 (* i t_0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = sqrt((i + alpha)) / beta;
	double tmp;
	if (beta <= 1.8e+124) {
		tmp = 0.0625;
	} else {
		tmp = t_0 * (i * 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 = sqrt((i + alpha)) / beta
    if (beta <= 1.8d+124) then
        tmp = 0.0625d0
    else
        tmp = t_0 * (i * 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 = Math.sqrt((i + alpha)) / beta;
	double tmp;
	if (beta <= 1.8e+124) {
		tmp = 0.0625;
	} else {
		tmp = t_0 * (i * t_0);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = math.sqrt((i + alpha)) / beta
	tmp = 0
	if beta <= 1.8e+124:
		tmp = 0.0625
	else:
		tmp = t_0 * (i * t_0)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(sqrt(Float64(i + alpha)) / beta)
	tmp = 0.0
	if (beta <= 1.8e+124)
		tmp = 0.0625;
	else
		tmp = Float64(t_0 * Float64(i * t_0));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = sqrt((i + alpha)) / beta;
	tmp = 0.0;
	if (beta <= 1.8e+124)
		tmp = 0.0625;
	else
		tmp = t_0 * (i * 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[(N[Sqrt[N[(i + alpha), $MachinePrecision]], $MachinePrecision] / beta), $MachinePrecision]}, If[LessEqual[beta, 1.8e+124], 0.0625, N[(t$95$0 * N[(i * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.79999999999999993e124

   1. Initial program 18.8%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in i around inf 81.6%

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

   if 1.79999999999999993e124 < beta

   1. Initial program 4.2%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in beta around inf 23.4%

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

   6. Taylor expanded in beta around inf 53.4%

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

      1. add-exp-log 50.5%

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

      2. un-div-inv 50.5%

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

      3. +-commutative 50.5%

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

   8. Applied egg-rr 50.5%

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

      1. rem-exp-log 53.4%

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

      2. *-commutative 53.4%

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

      3. add-sqr-sqrt 53.3%

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

      4. associate-*l* 53.3%

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

      5. associate-/l/ 23.4%

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

      6. sqrt-div 23.4%

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

      7. sqrt-unprod 53.3%

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

      8. add-sqr-sqrt 53.5%

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

      9. associate-/l/ 23.4%

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

      10. sqrt-div 23.4%

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

      11. sqrt-unprod 72.7%

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

      12. add-sqr-sqrt 72.9%

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

   10. Applied egg-rr 72.9%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{i + \alpha}}{\beta} \cdot \left(i \cdot \frac{\sqrt{i + \alpha}}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{i + \alpha}{\beta}\right) \cdot \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.8e+124) 0.0625 (* (* i (/ (+ i alpha) beta)) (/ 1.0 beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.8e+124) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((i + alpha) / beta)) * (1.0 / 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 <= 1.8d+124) then
        tmp = 0.0625d0
    else
        tmp = (i * ((i + alpha) / beta)) * (1.0d0 / 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 <= 1.8e+124) {
		tmp = 0.0625;
	} else {
		tmp = (i * ((i + alpha) / beta)) * (1.0 / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.8e+124:
		tmp = 0.0625
	else:
		tmp = (i * ((i + alpha) / beta)) * (1.0 / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.8e+124)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) * Float64(1.0 / 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 <= 1.8e+124)
		tmp = 0.0625;
	else
		tmp = (i * ((i + alpha) / beta)) * (1.0 / 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, 1.8e+124], 0.0625, N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.79999999999999993e124

   1. Initial program 18.8%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in i around inf 81.6%

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

   if 1.79999999999999993e124 < beta

   1. Initial program 4.2%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in beta around inf 23.4%

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

   6. Taylor expanded in beta around inf 53.4%

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

      1. add-exp-log 50.5%

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

      2. un-div-inv 50.5%

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

      3. +-commutative 50.5%

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

   8. Applied egg-rr 50.5%

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

      1. rem-exp-log 53.4%

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

      2. associate-*r/ 72.8%

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

      3. div-inv 72.8%

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

   10. Applied egg-rr 72.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{i + \alpha}{\beta}\right) \cdot \frac{1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.9e+124) 0.0625 (* i (/ (/ (+ i alpha) beta) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+124) {
		tmp = 0.0625;
	} else {
		tmp = i * (((i + alpha) / beta) / 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 <= 2.9d+124) then
        tmp = 0.0625d0
    else
        tmp = i * (((i + alpha) / beta) / 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 <= 2.9e+124) {
		tmp = 0.0625;
	} else {
		tmp = i * (((i + alpha) / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.9e+124:
		tmp = 0.0625
	else:
		tmp = i * (((i + alpha) / beta) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.9e+124)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(Float64(i + alpha) / beta) / 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 <= 2.9e+124)
		tmp = 0.0625;
	else
		tmp = i * (((i + alpha) / beta) / 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, 2.9e+124], 0.0625, N[(i * N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.90000000000000021e124

   1. Initial program 18.8%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in i around inf 81.6%

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

   if 2.90000000000000021e124 < beta

   1. Initial program 4.2%

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

   3. Simplified 13.1%

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

   5. Taylor expanded in beta around inf 23.4%

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

   6. Taylor expanded in beta around inf 53.4%

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

      1. un-div-inv 53.4%

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

      2. +-commutative 53.4%

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

   8. Applied egg-rr 53.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+124}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.1%

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

3. Simplified 35.2%

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

5. Taylor expanded in i around inf 70.9%

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

6. Final simplification 70.9%

   \[\leadsto 0.0625 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))