Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 82.5%

Time: 22.8s

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: 16.3% 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: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := 0.25 \cdot \left(\left(\beta + \alpha\right) \cdot -0.5 + \left(\beta + \alpha\right) \cdot 0.25\right)\\ t_1 := \left(\beta + \alpha\right) \cdot -0.0625\\ \mathbf{if}\;\beta \leq 3.25 \cdot 10^{+146}:\\ \;\;\;\;0.0625 + \frac{t\_1 - \left(\frac{\left(\beta + \alpha\right) \cdot \left(t\_1 - t\_0\right)}{i} + \left(t\_0 + 0.015625 \cdot \frac{-1 + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}{i}\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.25 (+ (* (+ beta alpha) -0.5) (* (+ beta alpha) 0.25))))
        (t_1 (* (+ beta alpha) -0.0625)))
   (if (<= beta 3.25e+146)
     (+
      0.0625
      (/
       (-
        t_1
        (+
         (/ (* (+ beta alpha) (- t_1 t_0)) i)
         (+
          t_0
          (* 0.015625 (/ (+ -1.0 (* (+ beta alpha) (+ beta alpha))) i)))))
       i))
     (* (/ i beta) (/ i beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = 0.25 * (((beta + alpha) * -0.5) + ((beta + alpha) * 0.25));
	double t_1 = (beta + alpha) * -0.0625;
	double tmp;
	if (beta <= 3.25e+146) {
		tmp = 0.0625 + ((t_1 - ((((beta + alpha) * (t_1 - t_0)) / i) + (t_0 + (0.015625 * ((-1.0 + ((beta + alpha) * (beta + alpha))) / i))))) / i);
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.25d0 * (((beta + alpha) * (-0.5d0)) + ((beta + alpha) * 0.25d0))
    t_1 = (beta + alpha) * (-0.0625d0)
    if (beta <= 3.25d+146) then
        tmp = 0.0625d0 + ((t_1 - ((((beta + alpha) * (t_1 - t_0)) / i) + (t_0 + (0.015625d0 * (((-1.0d0) + ((beta + alpha) * (beta + alpha))) / i))))) / i)
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = 0.25 * (((beta + alpha) * -0.5) + ((beta + alpha) * 0.25));
	double t_1 = (beta + alpha) * -0.0625;
	double tmp;
	if (beta <= 3.25e+146) {
		tmp = 0.0625 + ((t_1 - ((((beta + alpha) * (t_1 - t_0)) / i) + (t_0 + (0.015625 * ((-1.0 + ((beta + alpha) * (beta + alpha))) / i))))) / i);
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = 0.25 * (((beta + alpha) * -0.5) + ((beta + alpha) * 0.25))
	t_1 = (beta + alpha) * -0.0625
	tmp = 0
	if beta <= 3.25e+146:
		tmp = 0.0625 + ((t_1 - ((((beta + alpha) * (t_1 - t_0)) / i) + (t_0 + (0.015625 * ((-1.0 + ((beta + alpha) * (beta + alpha))) / i))))) / i)
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(0.25 * Float64(Float64(Float64(beta + alpha) * -0.5) + Float64(Float64(beta + alpha) * 0.25)))
	t_1 = Float64(Float64(beta + alpha) * -0.0625)
	tmp = 0.0
	if (beta <= 3.25e+146)
		tmp = Float64(0.0625 + Float64(Float64(t_1 - Float64(Float64(Float64(Float64(beta + alpha) * Float64(t_1 - t_0)) / i) + Float64(t_0 + Float64(0.015625 * Float64(Float64(-1.0 + Float64(Float64(beta + alpha) * Float64(beta + alpha))) / i))))) / i));
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = 0.25 * (((beta + alpha) * -0.5) + ((beta + alpha) * 0.25));
	t_1 = (beta + alpha) * -0.0625;
	tmp = 0.0;
	if (beta <= 3.25e+146)
		tmp = 0.0625 + ((t_1 - ((((beta + alpha) * (t_1 - t_0)) / i) + (t_0 + (0.015625 * ((-1.0 + ((beta + alpha) * (beta + alpha))) / i))))) / i);
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.25 * N[(N[(N[(beta + alpha), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[(beta + alpha), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] * -0.0625), $MachinePrecision]}, If[LessEqual[beta, 3.25e+146], N[(0.0625 + N[(N[(t$95$1 - N[(N[(N[(N[(beta + alpha), $MachinePrecision] * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] + N[(t$95$0 + N[(0.015625 * N[(N[(-1.0 + N[(N[(beta + alpha), $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2499999999999998e146

   1. Initial program 20.5%

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

   3. Taylor expanded in i around inf 41.2%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv 41.2%

         \[\leadsto \frac{{i}^{2} \cdot \color{blue}{\left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) + \left(-0.25\right) \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. distribute-lft-out 41.2%

         \[\leadsto \frac{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right) + \left(-0.25\right) \cdot \frac{\alpha + \beta}{i}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. metadata-eval 41.2%

         \[\leadsto \frac{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \left(\alpha + \beta\right)}{i}\right) + \color{blue}{-0.25} \cdot \frac{\alpha + \beta}{i}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 41.2%

      \[\leadsto \frac{\color{blue}{{i}^{2} \cdot \left(\left(0.25 + 0.25 \cdot \frac{2 \cdot \left(\alpha + \beta\right)}{i}\right) + -0.25 \cdot \frac{\alpha + \beta}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   6. Taylor expanded in i around -inf 74.7%

      \[\leadsto \color{blue}{0.0625 + -1 \cdot \frac{\left(-1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)}{i} + \left(0.015625 \cdot \frac{{\left(\alpha + \beta\right)}^{2} - 1}{i} + 0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right)\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i}} \]
   7. Step-by-step derivation

      1. unpow2 74.7%

         \[\leadsto 0.0625 + -1 \cdot \frac{\left(-1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)}{i} + \left(0.015625 \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)} - 1}{i} + 0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right)\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

      2. +-commutative 74.7%

         \[\leadsto 0.0625 + -1 \cdot \frac{\left(-1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)}{i} + \left(0.015625 \cdot \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \left(\alpha + \beta\right) - 1}{i} + 0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right)\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

      3. +-commutative 74.7%

         \[\leadsto 0.0625 + -1 \cdot \frac{\left(-1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)}{i} + \left(0.015625 \cdot \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\beta + \alpha\right)} - 1}{i} + 0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right)\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   8. Applied egg-rr 74.7%

      \[\leadsto 0.0625 + -1 \cdot \frac{\left(-1 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)\right)}{i} + \left(0.015625 \cdot \frac{\color{blue}{\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)} - 1}{i} + 0.25 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) + 0.25 \cdot \left(\alpha + \beta\right)\right)\right)\right) - -0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   if 3.2499999999999998e146 < 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 12.1%

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

   5. Taylor expanded in alpha around 0 12.3%

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

   6. Taylor expanded in beta around inf 21.0%

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

      1. associate-*r/ 19.4%

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

      2. unpow2 19.4%

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

      3. frac-times 76.7%

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

      4. add-exp-log 72.0%

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

      5. pow2 72.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{i}{\beta}\right)}^{2}\right)}} \]

   8. Applied egg-rr 72.0%

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

      1. rem-exp-log 76.7%

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

      2. unpow2 76.7%

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

   10. Applied egg-rr 76.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.25 \cdot 10^{+146}:\\ \;\;\;\;0.0625 + \frac{\left(\beta + \alpha\right) \cdot -0.0625 - \left(\frac{\left(\beta + \alpha\right) \cdot \left(\left(\beta + \alpha\right) \cdot -0.0625 - 0.25 \cdot \left(\left(\beta + \alpha\right) \cdot -0.5 + \left(\beta + \alpha\right) \cdot 0.25\right)\right)}{i} + \left(0.25 \cdot \left(\left(\beta + \alpha\right) \cdot -0.5 + \left(\beta + \alpha\right) \cdot 0.25\right) + 0.015625 \cdot \frac{-1 + \left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)}{i}\right)\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+147}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.8e+147) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+147) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.8d+147) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.8e+147) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.8e+147:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.8e+147)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.8e+147)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.8e+147], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.8000000000000001e147

   1. Initial program 20.5%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in i around inf 78.6%

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

   if 2.8000000000000001e147 < 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 12.1%

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

   5. Taylor expanded in alpha around 0 12.3%

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

   6. Taylor expanded in beta around inf 21.0%

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

      1. associate-*r/ 19.4%

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

      2. unpow2 19.4%

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

      3. frac-times 76.7%

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

      4. add-exp-log 72.0%

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

      5. pow2 72.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{i}{\beta}\right)}^{2}\right)}} \]

   8. Applied egg-rr 72.0%

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

      1. rem-exp-log 76.7%

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

      2. unpow2 76.7%

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

   10. Applied egg-rr 76.7%

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

Alternative 3: 76.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.25 \cdot 10^{+146}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.25e+146) 0.0625 (* i (/ (/ i beta) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.25e+146) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / 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 <= 3.25d+146) then
        tmp = 0.0625d0
    else
        tmp = i * ((i / 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 <= 3.25e+146) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.25e+146:
		tmp = 0.0625
	else:
		tmp = i * ((i / beta) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.25e+146)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(i / 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 <= 3.25e+146)
		tmp = 0.0625;
	else
		tmp = i * ((i / 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, 3.25e+146], 0.0625, N[(i * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.2499999999999998e146

   1. Initial program 20.5%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in i around inf 78.6%

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

   if 3.2499999999999998e146 < 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 12.1%

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

   5. Taylor expanded in alpha around 0 12.3%

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

   6. Taylor expanded in beta around inf 21.0%

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

      1. add-sqr-sqrt 21.0%

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

      2. sqrt-div 21.0%

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

      3. sqrt-pow1 21.0%

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

      4. metadata-eval 21.0%

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

      5. pow1 21.0%

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

      6. sqrt-div 21.0%

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

      7. sqrt-pow1 55.4%

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

      8. metadata-eval 55.4%

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

      9. pow1 55.4%

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

   8. Applied egg-rr 55.4%

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

      1. associate-*r/ 55.4%

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

      2. associate-*l/ 55.3%

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

      3. rem-square-sqrt 55.5%

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

   10. Simplified 55.5%

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

Alternative 4: 70.8% 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 16.6%

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

3. Simplified 40.8%

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

5. Taylor expanded in i around inf 66.8%

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

Reproduce

herbie shell --seed 2024141 
(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)))