Octave 3.8, jcobi/4

Percentage Accurate: 16.0% → 84.4%

Time: 16.8s

Alternatives: 7

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t_2\\ t_4 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(t_4, t_4, -1\right)} \cdot \frac{\mathsf{fma}\left(i, t_2, \alpha \cdot \beta\right)}{\frac{t_4 \cdot t_4}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (+ alpha (fma i 2.0 beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (*
      (/ i (fma t_4 t_4 -1.0))
      (/ (fma i t_2 (* alpha beta)) (/ (* t_4 t_4) t_2)))
     (- (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* 0.0625 (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = alpha + fma(i, 2.0, beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (i / fma(t_4, t_4, -1.0)) * (fma(i, t_2, (alpha * beta)) / ((t_4 * t_4) / t_2));
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(i / fma(t_4, t_4, -1.0)) * Float64(fma(i, t_2, Float64(alpha * beta)) / Float64(Float64(t_4 * t_4) / t_2)));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) - Float64(0.0625 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(i / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$4 * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 35.6%

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

      1. associate-/l/ 28.8%

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

      2. associate-*l* 28.7%

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

      3. times-frac 48.0%

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

   3. Simplified 99.5%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in i around inf 15.0%

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

      1. cancel-sign-sub-inv 15.0%

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

      2. distribute-lft-out 15.0%

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

      3. distribute-lft-out 15.0%

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

      4. +-commutative 15.0%

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

      5. metadata-eval 15.0%

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

      6. +-commutative 15.0%

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

   6. Simplified 15.0%

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

   7. Taylor expanded in i around inf 71.9%

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

   8. Taylor expanded in alpha around 0 65.0%

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

      1. distribute-rgt-out 65.0%

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

      2. metadata-eval 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in alpha around 0 66.5%

       \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\frac{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}{i + \left(\alpha + \beta\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := i \cdot \left(\alpha + \left(i + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{t_4 \cdot \left(\mathsf{fma}\left(\alpha, \beta, t_4\right) \cdot {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{-2}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (* i (+ alpha (+ i beta)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (* t_4 (* (fma alpha beta t_4) (pow (fma i 2.0 (+ alpha beta)) -2.0)))
      t_2)
     (- (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* 0.0625 (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = i * (alpha + (i + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (t_4 * (fma(alpha, beta, t_4) * pow(fma(i, 2.0, (alpha + beta)), -2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(i * Float64(alpha + Float64(i + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(t_4 * Float64(fma(alpha, beta, t_4) * (fma(i, 2.0, Float64(alpha + beta)) ^ -2.0))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) - Float64(0.0625 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(t$95$4 * N[(N[(alpha * beta + t$95$4), $MachinePrecision] * N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 35.6%

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

      1. expm1-log1p-u 33.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 33.3%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Applied egg-rr 33.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(\left(i \cdot \left(\alpha + \left(\beta + i\right)\right)\right) \cdot \mathsf{fma}\left(\alpha, \beta, i \cdot \left(\alpha + \left(\beta + i\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{-2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   4. Step-by-step derivation

      1. expm1-def 33.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(i \cdot \left(\alpha + \left(\beta + i\right)\right)\right) \cdot \mathsf{fma}\left(\alpha, \beta, i \cdot \left(\alpha + \left(\beta + i\right)\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{-2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 35.5%

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

      3. associate-*l* 99.3%

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

      4. +-commutative 99.3%

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

      5. +-commutative 99.3%

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

      6. +-commutative 99.3%

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

   5. Simplified 99.3%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in i around inf 15.0%

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

      1. cancel-sign-sub-inv 15.0%

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

      2. distribute-lft-out 15.0%

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

      3. distribute-lft-out 15.0%

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

      4. +-commutative 15.0%

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

      5. metadata-eval 15.0%

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

      6. +-commutative 15.0%

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

   6. Simplified 15.0%

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

   7. Taylor expanded in i around inf 71.9%

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

   8. Taylor expanded in alpha around 0 65.0%

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

      1. distribute-rgt-out 65.0%

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

      2. metadata-eval 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in alpha around 0 66.5%

       \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\left(i \cdot \left(\alpha + \left(i + \beta\right)\right)\right) \cdot \left(\mathsf{fma}\left(\alpha, \beta, i \cdot \left(\alpha + \left(i + \beta\right)\right)\right) \cdot {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{-2}\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 3: 81.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.1)
     t_3
     (- (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* 0.0625 (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.25d0 * ((beta * 0.25d0) / i))) - (0.0625d0 * (beta / i))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) - Float64(0.0625 * Float64(beta / i)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (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_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

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

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 4.8%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in i around inf 30.2%

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

      1. cancel-sign-sub-inv 30.2%

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

      2. distribute-lft-out 30.2%

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

      3. distribute-lft-out 30.2%

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

      4. +-commutative 30.2%

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

      5. metadata-eval 30.2%

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

      6. +-commutative 30.2%

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

   6. Simplified 30.2%

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

   7. Taylor expanded in i around inf 74.5%

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

   8. Taylor expanded in alpha around 0 69.1%

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

      1. distribute-rgt-out 69.1%

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

      2. metadata-eval 69.1%

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

   10. Simplified 69.1%

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

   11. Taylor expanded in alpha around 0 70.3%

       \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \color{blue}{\frac{\beta}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 4: 78.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

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 + ((beta / i) * 0.125d0)) - (0.125d0 * ((alpha + beta) / i))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 10.7%

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

   1. associate-/l/ 8.7%

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

   2. associate-*l* 8.6%

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

   3. times-frac 14.5%

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

3. Simplified 34.7%

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

4. Taylor expanded in i around inf 75.0%

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

5. Taylor expanded in alpha around 0 70.1%

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

6. Final simplification 70.1%

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

Alternative 5: 78.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (- (+ 0.0625 (* 0.25 (/ (* beta 0.25) i))) (* 0.0625 (/ beta i))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
}

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 + (0.25d0 * ((beta * 0.25d0) / i))) - (0.0625d0 * (beta / i))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	return (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.25 * Float64(Float64(beta * 0.25) / i))) - Float64(0.0625 * Float64(beta / i)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.25 * ((beta * 0.25) / i))) - (0.0625 * (beta / i));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.25 * N[(N[(beta * 0.25), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.7%

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

   1. associate-/l/ 8.7%

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

   2. associate-*l* 8.6%

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

   3. times-frac 14.5%

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

3. Simplified 34.7%

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

4. Taylor expanded in i around inf 35.5%

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

   1. cancel-sign-sub-inv 35.5%

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

   2. distribute-lft-out 35.5%

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

   3. distribute-lft-out 35.5%

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

   4. +-commutative 35.5%

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

   5. metadata-eval 35.5%

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

   6. +-commutative 35.5%

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

6. Simplified 35.5%

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

7. Taylor expanded in i around inf 75.0%

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

8. Taylor expanded in alpha around 0 70.1%

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

   1. distribute-rgt-out 70.1%

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

   2. metadata-eval 70.1%

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

10. Simplified 70.1%

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

11. Taylor expanded in alpha around 0 71.2%

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

12. Final simplification 71.2%

    \[\leadsto \left(0.0625 + 0.25 \cdot \frac{\beta \cdot 0.25}{i}\right) - 0.0625 \cdot \frac{\beta}{i} \]

Alternative 6: 74.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.12e+229) 0.0625 0.0))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.12e+229) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.12d+229) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.12e+229) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.12e+229:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.12e+229)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.12e+229)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.12e+229], 0.0625, 0.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.12e229

   1. Initial program 11.7%

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

      1. associate-/l/ 9.5%

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

      2. associate-*l* 9.4%

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

      3. times-frac 15.8%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in i around inf 71.8%

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

   if 1.12e229 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in i around inf 48.4%

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

      1. clear-num 43.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

      2. inv-pow 43.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]

   6. Applied egg-rr 43.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 43.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   8. Simplified 43.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   9. Taylor expanded in i around 0 36.5%

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

       1. div-sub 36.5%

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

       2. +-commutative 36.5%

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

       3. distribute-lft-in 36.5%

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

       4. associate-*r* 36.5%

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

       5. metadata-eval 36.5%

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

       6. *-commutative 36.5%

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

       7. associate-*l/ 36.5%

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

       8. associate-*r/ 36.5%

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

       9. *-commutative 36.5%

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

       10. +-commutative 36.5%

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

       11. +-inverses 36.5%

           \[\leadsto \color{blue}{0} \]

   11. Simplified 36.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 9.8% accurate, 53.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ 0 \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0)

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	return 0.0;
}

Fortran

NOTE: alpha and beta should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	return 0.0;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	return 0.0

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	return 0.0
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 10.7%

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

   1. associate-/l/ 8.7%

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

   2. associate-*l* 8.6%

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

   3. times-frac 14.5%

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

3. Simplified 34.7%

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

4. Taylor expanded in i around inf 75.0%

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

   1. clear-num 73.5%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   2. inv-pow 73.5%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]

6. Applied egg-rr 73.5%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 73.5%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

8. Simplified 73.5%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

9. Taylor expanded in i around 0 11.5%

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

    1. div-sub 11.5%

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

    2. +-commutative 11.5%

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

    3. distribute-lft-in 11.5%

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

    4. associate-*r* 11.5%

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

    5. metadata-eval 11.5%

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

    6. *-commutative 11.5%

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

    7. associate-*l/ 11.5%

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

    8. associate-*r/ 11.5%

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

    9. *-commutative 11.5%

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

    10. +-commutative 11.5%

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

    11. +-inverses 11.5%

        \[\leadsto \color{blue}{0} \]

11. Simplified 11.5%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 11.5%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023314 
(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)))