Octave 3.8, jcobi/4

Percentage Accurate: 17.1% → 84.5%

Time: 14.1s

Alternatives: 10

Speedup: 115.0×

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: 17.1% 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: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 + -1\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t\_3\\ t_5 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_4 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{i \cdot \left(t\_3 \cdot \frac{\mathsf{fma}\left(i, t\_3, \alpha \cdot \beta\right)}{t\_5 \cdot t\_5}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\beta}{i}, 0.0625\right) - \frac{\beta \cdot 0.125}{i}\\ \end{array} \end{array} \]

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 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3))
        (t_5 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/ (* i (* t_3 (/ (fma i t_3 (* alpha beta)) (* t_5 t_5)))) t_2)
     (- (fma 0.125 (/ beta i) 0.0625) (/ (* beta 0.125) i)))))

C

assert(alpha < beta && beta < i);
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 + (alpha + beta);
	double t_4 = i * t_3;
	double t_5 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (i * (t_3 * (fma(i, t_3, (alpha * beta)) / (t_5 * t_5)))) / t_2;
	} else {
		tmp = fma(0.125, (beta / i), 0.0625) - ((beta * 0.125) / i);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
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(alpha + beta))
	t_4 = Float64(i * t_3)
	t_5 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(i * Float64(t_3 * Float64(fma(i, t_3, Float64(alpha * beta)) / Float64(t_5 * t_5)))) / t_2);
	else
		tmp = Float64(fma(0.125, Float64(beta / i), 0.0625) - Float64(Float64(beta * 0.125) / i));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$3 * N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.125 * N[(beta / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(beta * 0.125), $MachinePrecision] / i), $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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 50.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)} - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      3. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \color{blue}{\left(\alpha + \beta\right)}\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      11. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \color{blue}{\frac{\frac{1}{8} \cdot \left(\alpha + \beta\right)}{i}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \color{blue}{\frac{\frac{1}{8} \cdot \left(\alpha + \beta\right)}{i}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \frac{\color{blue}{\frac{1}{8} \cdot \left(\alpha + \beta\right)}}{i} \]

      14. lower-+.f64 81.7

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

   8. Applied rewrites 81.7%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{\color{blue}{\frac{1}{8} \cdot \left(\alpha + \beta\right)}}{i} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.7%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{8}, \frac{\beta}{i}, \frac{1}{16}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]
   13. Step-by-step derivation

   14. Applied rewrites 81.7%

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

4. Final simplification 88.0%

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

Alternative 2: 84.5% accurate, 0.5× speedup

Math

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

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

C

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

Julia

alpha, beta, i = sort([alpha, beta, i])
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 = fma(i, 2.0, Float64(alpha + 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(fma(i, t_2, Float64(alpha * beta)) / fma(t_4, t_4, -1.0)) * Float64(t_3 / Float64(t_4 * t_4)));
	else
		tmp = Float64(fma(0.125, Float64(beta / i), 0.0625) - Float64(Float64(beta * 0.125) / i));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(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[(i * 2.0 + N[(alpha + 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[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(beta / i), $MachinePrecision] + 0.0625), $MachinePrecision] - N[(N[(beta * 0.125), $MachinePrecision] / i), $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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 48.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

   4. Applied rewrites 99.7%

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

   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   4. Step-by-step derivation

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)} - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      3. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\alpha + \beta\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. distribute-lft-in N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \color{blue}{\left(\alpha + \beta\right)}\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      11. associate-*r/ N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \color{blue}{\frac{\frac{1}{8} \cdot \left(\alpha + \beta\right)}{i}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \color{blue}{\frac{\frac{1}{8} \cdot \left(\alpha + \beta\right)}{i}} \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right) - \frac{\color{blue}{\frac{1}{8} \cdot \left(\alpha + \beta\right)}}{i} \]

      14. lower-+.f64 76.3

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{\color{blue}{\frac{1}{8} \cdot \left(\alpha + \beta\right)}}{i} \]
   10. Step-by-step derivation

   11. Applied rewrites 76.3%

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

   12. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{fma}\left(\frac{1}{8}, \frac{\beta}{i}, \frac{1}{16}\right) - \frac{1}{8} \cdot \color{blue}{\frac{\beta}{i}} \]
   13. Step-by-step derivation

   14. Applied rewrites 76.3%

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

4. Final simplification 84.5%

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

Reproduce

herbie shell --seed 2024227 
(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)))