Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 96.8%

Time: 11.1s

Alternatives: 13

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: 16.1% 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: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{\left(\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(\beta + i\right)\right) \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \left(\alpha + \left(\beta + i\right)\right)}{\mathsf{fma}\left(2, i, 1 + \left(\alpha + \beta\right)\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - 1\right)} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (/
  (*
   (* (/ i (fma 2.0 i beta)) (+ beta i))
   (/
    (* (/ i (fma 2.0 i (+ alpha beta))) (+ alpha (+ beta i)))
    (fma 2.0 i (+ 1.0 (+ alpha beta)))))
  (fma 2.0 i (- (+ alpha beta) 1.0))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (((i / fma(2.0, i, beta)) * (beta + i)) * (((i / fma(2.0, i, (alpha + beta))) * (alpha + (beta + i))) / fma(2.0, i, (1.0 + (alpha + beta))))) / fma(2.0, i, ((alpha + beta) - 1.0));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i / fma(2.0, i, beta)) * Float64(beta + i)) * Float64(Float64(Float64(i / fma(2.0, i, Float64(alpha + beta))) * Float64(alpha + Float64(beta + i))) / fma(2.0, i, Float64(1.0 + Float64(alpha + beta))))) / fma(2.0, i, Float64(Float64(alpha + beta) - 1.0)))
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(N[(N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(beta + i), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 39.8%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 37.4

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

7. Applied rewrites 37.4%

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

9. Applied rewrites 87.8%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lower-/.f64 N/A

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

11. Applied rewrites 87.8%

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

Alternative 2: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{\left(\left(i + \beta\right) + \alpha\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 1\right)} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - 1\right)} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (*
  (/
   (* (+ (+ i beta) alpha) (/ i (fma 2.0 i (+ alpha beta))))
   (fma 2.0 i (+ (+ alpha beta) 1.0)))
  (/
   (* (/ i (fma 2.0 i beta)) (+ i beta))
   (fma 2.0 i (- (+ alpha beta) 1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return ((((i + beta) + alpha) * (i / fma(2.0, i, (alpha + beta)))) / fma(2.0, i, ((alpha + beta) + 1.0))) * (((i / fma(2.0, i, beta)) * (i + beta)) / fma(2.0, i, ((alpha + beta) - 1.0)));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(i + beta) + alpha) * Float64(i / fma(2.0, i, Float64(alpha + beta)))) / fma(2.0, i, Float64(Float64(alpha + beta) + 1.0))) * Float64(Float64(Float64(i / fma(2.0, i, beta)) * Float64(i + beta)) / fma(2.0, i, Float64(Float64(alpha + beta) - 1.0))))
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(i + beta), $MachinePrecision] + alpha), $MachinePrecision] * N[(i / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 39.8%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 37.4

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

7. Applied rewrites 37.4%

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

9. Applied rewrites 87.8%

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

Alternative 3: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta\right)} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 1\right)} \cdot \left(\frac{\left(i + \beta\right) + \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) - 1\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (*
  (/ (* (/ i (fma 2.0 i beta)) (+ i beta)) (fma 2.0 i (+ (+ alpha beta) 1.0)))
  (*
   (/ (+ (+ i beta) alpha) (fma 2.0 i (- (+ alpha beta) 1.0)))
   (/ i (fma 2.0 i (+ alpha beta))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (((i / fma(2.0, i, beta)) * (i + beta)) / fma(2.0, i, ((alpha + beta) + 1.0))) * ((((i + beta) + alpha) / fma(2.0, i, ((alpha + beta) - 1.0))) * (i / fma(2.0, i, (alpha + beta))));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i / fma(2.0, i, beta)) * Float64(i + beta)) / fma(2.0, i, Float64(Float64(alpha + beta) + 1.0))) * Float64(Float64(Float64(Float64(i + beta) + alpha) / fma(2.0, i, Float64(Float64(alpha + beta) - 1.0))) * Float64(i / fma(2.0, i, Float64(alpha + beta)))))
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(N[(N[(i / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(i + beta), $MachinePrecision] + alpha), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 39.8%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 37.4

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

7. Applied rewrites 37.4%

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

9. Applied rewrites 87.8%

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

Alternative 4: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \left(\frac{\frac{\beta + i}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) + 1} \cdot i\right) \cdot \frac{\frac{i}{t\_0} \cdot \left(\left(\beta + \alpha\right) + i\right)}{t\_0 - 1} \end{array} \end{array} \]

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 (fma 2.0 i (+ beta alpha))))
   (*
    (* (/ (/ (+ beta i) (fma 2.0 i beta)) (+ (fma 2.0 i beta) 1.0)) i)
    (/ (* (/ i t_0) (+ (+ beta alpha) i)) (- t_0 1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	return ((((beta + i) / fma(2.0, i, beta)) / (fma(2.0, i, beta) + 1.0)) * i) * (((i / t_0) * ((beta + alpha) + i)) / (t_0 - 1.0));
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	return Float64(Float64(Float64(Float64(Float64(beta + i) / fma(2.0, i, beta)) / Float64(fma(2.0, i, beta) + 1.0)) * i) * Float64(Float64(Float64(i / t_0) * Float64(Float64(beta + alpha) + i)) / Float64(t_0 - 1.0)))
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[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(beta + i), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.8%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 39.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. difference-of-sqr-1 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

6. Applied rewrites 44.4%

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

7. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 34.2

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

9. Applied rewrites 34.2%

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

11. Applied rewrites 83.6%

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

Alternative 5: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{t\_0 + 1} \cdot \frac{\frac{i}{t\_0} \cdot \left(\left(\beta + \alpha\right) + i\right)}{t\_0 - 1}\\ \end{array} \end{array} \]

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 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 5.7e+85)
     0.0625
     (*
      (/ (+ alpha i) (+ t_0 1.0))
      (/ (* (/ i t_0) (+ (+ beta alpha) i)) (- t_0 1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 5.7e+85) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / (t_0 + 1.0)) * (((i / t_0) * ((beta + alpha) + i)) / (t_0 - 1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 5.7e+85)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(t_0 + 1.0)) * Float64(Float64(Float64(i / t_0) * Float64(Float64(beta + alpha) + i)) / Float64(t_0 - 1.0)));
	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[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.7e+85], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.7000000000000002e85

   1. Initial program 22.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.6%

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

   if 5.7000000000000002e85 < beta

   1. Initial program 3.9%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 28.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-sqr-1 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 33.6%

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

   7. Taylor expanded in beta around inf

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

      1. lower-+.f64 65.3

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

   9. Applied rewrites 65.3%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.7 \cdot 10^{+85}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\left(\beta + \alpha\right) + i\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{i}{t\_0} \cdot \left(\left(\beta + \alpha\right) + i\right)}{t\_0 - 1}\\ \end{array} \end{array} \]

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 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 6e+88)
     0.0625
     (*
      (/ (+ alpha i) beta)
      (/ (* (/ i t_0) (+ (+ beta alpha) i)) (- t_0 1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (((i / t_0) * ((beta + alpha) + i)) / (t_0 - 1.0));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(Float64(Float64(i / t_0) * Float64(Float64(beta + alpha) + i)) / Float64(t_0 - 1.0)));
	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[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6e+88], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000011e88

   1. Initial program 22.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.8%

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

   if 6.00000000000000011e88 < beta

   1. Initial program 4.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing
   3. 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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 29.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. difference-of-sqr-1 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 63.6

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

   9. Applied rewrites 63.6%

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

Alternative 7: 83.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+88) 0.0625 (/ (/ (+ alpha i) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6d+88) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+88:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+88], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000011e88

   1. Initial program 22.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.8%

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

   if 6.00000000000000011e88 < beta

   1. Initial program 4.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 63.1%

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

Alternative 8: 83.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+88) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6d+88) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+88:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+88], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000011e88

   1. Initial program 22.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.8%

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

   if 6.00000000000000011e88 < beta

   1. Initial program 4.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 9: 80.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+88}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+88) 0.0625 (* (/ i beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6d+88) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+88) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+88:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+88)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+88], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.00000000000000011e88

   1. Initial program 22.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.8%

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

   if 6.00000000000000011e88 < beta

   1. Initial program 4.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 59.4%

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

Alternative 10: 71.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.1e+30) (* i (/ (+ alpha i) (* beta beta))) 0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.1e+30) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.1d+30) then
        tmp = i * ((alpha + i) / (beta * beta))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.1e+30) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if i <= 2.1e+30:
		tmp = i * ((alpha + i) / (beta * beta))
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.1e+30)
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.1e+30)
		tmp = i * ((alpha + i) / (beta * beta));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 2.1e+30], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 2.1e30

   1. Initial program 58.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 49.8%

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

   if 2.1e30 < i

   1. Initial program 13.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;i \leq 2.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 2.1e+30) (/ (* i i) (* beta beta)) 0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.1e+30) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (i <= 2.1d+30) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 2.1e+30) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if i <= 2.1e+30:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 2.1e+30)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 2.1e+30)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[i, 2.1e+30], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 2.1e30

   1. Initial program 58.2%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 50.1%

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

   if 2.1e30 < i

   1. Initial program 13.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+213}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+213) 0.0625 (* alpha (/ i (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+213) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2d+213) then
        tmp = 0.0625d0
    else
        tmp = alpha * (i / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+213) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+213:
		tmp = 0.0625
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+213)
		tmp = 0.0625;
	else
		tmp = Float64(alpha * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+213)
		tmp = 0.0625;
	else
		tmp = alpha * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+213], 0.0625, N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999997e213

   1. Initial program 20.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 74.9%

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

   if 1.99999999999999997e213 < 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. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 44.5%

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

Alternative 13: 70.7% accurate, 115.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 69.7%

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

Reproduce

herbie shell --seed 2024303 
(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)))