Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 77.6%

Time: 11.9s

Alternatives: 5

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: 15.7% 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: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ \mathbf{if}\;i \leq 7.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{t\_0 \cdot t\_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0))))
   (if (<= i 7.5e+132)
     (/ (* 0.25 (* i i)) (+ (* t_0 t_0) -1.0))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (i <= 7.5e+132) {
		tmp = (0.25 * (i * i)) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

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) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    if (i <= 7.5d+132) then
        tmp = (0.25d0 * (i * i)) / ((t_0 * t_0) + (-1.0d0))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (i <= 7.5e+132) {
		tmp = (0.25 * (i * i)) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	tmp = 0
	if i <= 7.5e+132:
		tmp = (0.25 * (i * i)) / ((t_0 * t_0) + -1.0)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	tmp = 0.0
	if (i <= 7.5e+132)
		tmp = Float64(Float64(0.25 * Float64(i * i)) / Float64(Float64(t_0 * t_0) + -1.0));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	tmp = 0.0;
	if (i <= 7.5e+132)
		tmp = (0.25 * (i * i)) / ((t_0 * t_0) + -1.0);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 7.5e+132], N[(N[(0.25 * N[(i * i), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 7.50000000000000017e132

   1. Initial program 35.8%

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

   3. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. lower-*.f64 75.9

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

   5. Simplified 75.9%

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

   if 7.50000000000000017e132 < i

   1. Initial program 0.3%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 0.0%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{4 \cdot {i}^{2} - 1} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{{i}^{2} \cdot 4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{{i}^{2} \cdot 4 + \color{blue}{-1}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{\mathsf{fma}\left({i}^{2}, 4, -1\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

      11. lower-*.f64 10.2

          \[\leadsto \frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

   8. Simplified 10.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(i \cdot i, 4, -1\right)}} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. lower-*.f64 88.8

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   11. Simplified 88.8%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{0.25 \cdot \left(i \cdot i\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ \mathbf{if}\;i \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{t\_0 \cdot t\_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0))))
   (if (<= i 2.4e+42)
     (/ (* i (+ i alpha)) (+ (* t_0 t_0) -1.0))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (i <= 2.4e+42) {
		tmp = (i * (i + alpha)) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

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) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    if (i <= 2.4d+42) then
        tmp = (i * (i + alpha)) / ((t_0 * t_0) + (-1.0d0))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double tmp;
	if (i <= 2.4e+42) {
		tmp = (i * (i + alpha)) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	tmp = 0
	if i <= 2.4e+42:
		tmp = (i * (i + alpha)) / ((t_0 * t_0) + -1.0)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	tmp = 0.0
	if (i <= 2.4e+42)
		tmp = Float64(Float64(i * Float64(i + alpha)) / Float64(Float64(t_0 * t_0) + -1.0));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	tmp = 0.0;
	if (i <= 2.4e+42)
		tmp = (i * (i + alpha)) / ((t_0 * t_0) + -1.0);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.4e+42], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.3999999999999999e42

   1. Initial program 61.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 40.7

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

   5. Simplified 40.7%

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

   if 2.3999999999999999e42 < i

   1. Initial program 8.7%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 7.3%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{4 \cdot {i}^{2} - 1} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{{i}^{2} \cdot 4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{{i}^{2} \cdot 4 + \color{blue}{-1}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{\mathsf{fma}\left({i}^{2}, 4, -1\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

      11. lower-*.f64 28.3

          \[\leadsto \frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

   8. Simplified 28.3%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(i \cdot i, 4, -1\right)}} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. lower-*.f64 80.6

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   11. Simplified 80.6%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+242}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2e+242)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* alpha (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+242) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

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) :: tmp
    if (beta <= 2d+242) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = alpha * (i / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2e+242) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 2e+242:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2e+242)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(alpha * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2e+242)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = alpha * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 2e+242], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.0000000000000001e242

   1. Initial program 15.5%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 12.0%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{4 \cdot {i}^{2} - 1} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{{i}^{2} \cdot 4} + \left(\mathsf{neg}\left(1\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{{i}^{2} \cdot 4 + \color{blue}{-1}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{\mathsf{fma}\left({i}^{2}, 4, -1\right)}} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

      11. lower-*.f64 30.4

          \[\leadsto \frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

   8. Simplified 30.4%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(i \cdot i, 4, -1\right)}} \]

   9. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

       6. lower-*.f64 77.0

          \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

   11. Simplified 77.0%

       \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]

   if 2.0000000000000001e242 < 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. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 14.1

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

   5. Simplified 14.1%

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

   6. Taylor expanded in i around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 17.2

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

   8. Simplified 17.2%

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

Alternative 4: 70.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ 0.0625 + \frac{0.015625}{i \cdot i} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (+ 0.0625 (/ 0.015625 (* i i))))

C

double code(double alpha, double beta, double i) {
	return 0.0625 + (0.015625 / (i * i));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0 + (0.015625d0 / (i * i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625 + (0.015625 / (i * i));
}

Python

def code(alpha, beta, i):
	return 0.0625 + (0.015625 / (i * i))

Julia

function code(alpha, beta, i)
	return Float64(0.0625 + Float64(0.015625 / Float64(i * i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625 + (0.015625 / (i * i));
end

Wolfram

code[alpha_, beta_, i_] := N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. sub-neg N/A

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

   19. unpow2 N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 N/A

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

5. Simplified 11.7%

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

6. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{i}^{2}}{4 \cdot {i}^{2} - 1}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot {i}^{2}}{4 \cdot {i}^{2} - 1}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot {i}^{2}}}{4 \cdot {i}^{2} - 1} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(i \cdot i\right)}}{4 \cdot {i}^{2} - 1} \]

   6. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{4 \cdot {i}^{2} + \left(\mathsf{neg}\left(1\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{{i}^{2} \cdot 4} + \left(\mathsf{neg}\left(1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{{i}^{2} \cdot 4 + \color{blue}{-1}} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\color{blue}{\mathsf{fma}\left({i}^{2}, 4, -1\right)}} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

   11. lower-*.f64 29.5

       \[\leadsto \frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(\color{blue}{i \cdot i}, 4, -1\right)} \]

8. Simplified 29.5%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \left(i \cdot i\right)}{\mathsf{fma}\left(i \cdot i, 4, -1\right)}} \]

9. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
10. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]

    2. associate-*r/ N/A

       \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} \]

    3. metadata-eval N/A

       \[\leadsto \frac{1}{16} + \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{1}{16} + \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} \]

    5. unpow2 N/A

       \[\leadsto \frac{1}{16} + \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} \]

    6. lower-*.f64 75.2

       \[\leadsto 0.0625 + \frac{0.015625}{\color{blue}{i \cdot i}} \]

11. Simplified 75.2%

    \[\leadsto \color{blue}{0.0625 + \frac{0.015625}{i \cdot i}} \]
12. Add Preprocessing

Alternative 5: 70.2% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

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

Fortran

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

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

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 15.0%

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

3. Taylor expanded in i around inf

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

5. Simplified 74.7%

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

Reproduce

herbie shell --seed 2024215 
(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)))