Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 83.1%

Time: 14.7s

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: 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: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 1.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{i \cdot t\_0}{t\_1}}{t\_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_0, \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ alpha beta))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= i 1.05e+133)
     (*
      (/ (/ (* i t_0) t_1) (+ t_1 1.0))
      (/ (/ (fma i t_0 (* alpha beta)) t_1) (+ t_1 -1.0)))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = i + (alpha + beta);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (i <= 1.05e+133) {
		tmp = (((i * t_0) / t_1) / (t_1 + 1.0)) * ((fma(i, t_0, (alpha * beta)) / t_1) / (t_1 + -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(i + Float64(alpha + beta))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (i <= 1.05e+133)
		tmp = Float64(Float64(Float64(Float64(i * t_0) / t_1) / Float64(t_1 + 1.0)) * Float64(Float64(fma(i, t_0, Float64(alpha * beta)) / t_1) / Float64(t_1 + -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 1.05e+133], N[(N[(N[(N[(i * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 1.05e133

   1. Initial program 35.5%

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

   4. Applied egg-rr 86.8%

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

   if 1.05e133 < 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 i around inf

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

   5. Simplified 89.3%

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

Alternative 2: 78.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \left(i \cdot \frac{1}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.4e+158)
   0.0625
   (*
    (/ (+ i alpha) (+ alpha (+ 1.0 (fma i 2.0 beta))))
    (* i (/ 1.0 (+ alpha (+ -1.0 (fma i 2.0 beta))))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.4e+158) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (1.0 + fma(i, 2.0, beta)))) * (i * (1.0 / (alpha + (-1.0 + fma(i, 2.0, beta)))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.4e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(1.0 + fma(i, 2.0, beta)))) * Float64(i * Float64(1.0 / Float64(alpha + Float64(-1.0 + fma(i, 2.0, beta))))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.4e+158], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i * N[(1.0 / N[(alpha + N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.40000000000000021e158

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 80.5%

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

   if 7.40000000000000021e158 < 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

      \[\leadsto \frac{\color{blue}{i \cdot \left(\alpha + i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 14.2

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

   5. Simplified 14.2%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

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

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 75.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 76.1

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

   9. Applied egg-rr 76.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \left(i \cdot \frac{1}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.4e+158)
   0.0625
   (*
    (/ (+ i alpha) (+ alpha (+ 1.0 (fma i 2.0 beta))))
    (/ i (+ alpha (+ -1.0 (fma i 2.0 beta)))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.4e+158) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (1.0 + fma(i, 2.0, beta)))) * (i / (alpha + (-1.0 + fma(i, 2.0, beta))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.4e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(1.0 + fma(i, 2.0, beta)))) * Float64(i / Float64(alpha + Float64(-1.0 + fma(i, 2.0, beta)))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.4e+158], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(alpha + N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.40000000000000021e158

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 80.5%

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

   if 7.40000000000000021e158 < 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

      \[\leadsto \frac{\color{blue}{i \cdot \left(\alpha + i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 14.2

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

   5. Simplified 14.2%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

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

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 75.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i}{-1 + \mathsf{fma}\left(i, 2, \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.4e+158)
   0.0625
   (* (/ i (+ 1.0 (fma i 2.0 beta))) (/ i (+ -1.0 (fma i 2.0 beta))))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.4e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / (1.0 + fma(i, 2.0, beta))) * (i / (-1.0 + fma(i, 2.0, beta)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.4e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(1.0 + fma(i, 2.0, beta))) * Float64(i / Float64(-1.0 + fma(i, 2.0, beta))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.4e+158], 0.0625, N[(N[(i / N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.40000000000000021e158

   1. Initial program 16.6%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 80.5%

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

   if 7.40000000000000021e158 < 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

      \[\leadsto \frac{\color{blue}{i \cdot \left(\alpha + i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 14.2

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

   5. Simplified 14.2%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 14.2

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

   8. Simplified 14.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f64 N/A

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

      12. lower-/.f64 69.5

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

   10. Applied egg-rr 69.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i}{-1 + \mathsf{fma}\left(i, 2, \beta\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+183)
   0.0625
   (* (/ (+ i alpha) (+ alpha (+ 1.0 (fma i 2.0 beta)))) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (1.0 + fma(i, 2.0, beta)))) * (i / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(1.0 + fma(i, 2.0, beta)))) * Float64(i / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+183], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000046e183

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 79.4%

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

   if 7.20000000000000046e183 < 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

      \[\leadsto \frac{\color{blue}{i \cdot \left(\alpha + i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 17.3

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

   5. Simplified 17.3%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

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

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

   7. Applied egg-rr 82.9%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 81.2

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

   10. Simplified 81.2%

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

Alternative 6: 77.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \left(i \cdot \frac{1}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+183) 0.0625 (* (/ (+ i alpha) beta) (* i (/ 1.0 beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i * (1.0 / 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 <= 7.2d+183) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i * (1.0d0 / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i * (1.0 / beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.2e+183:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i * (1.0 / beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i * Float64(1.0 / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i * (1.0 / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+183], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000046e183

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 79.4%

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

   if 7.20000000000000046e183 < 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 17.3

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

   5. Simplified 17.3%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. *-commutative 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. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

      1. clear-num N/A

         \[\leadsto \frac{i + \alpha}{\beta} \cdot \color{blue}{\frac{1}{\frac{\beta}{i}}} \]

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 81.1

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

   9. Applied egg-rr 81.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \left(i \cdot \frac{1}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+183) 0.0625 (* (/ i beta) (/ (+ i alpha) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / 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 <= 7.2d+183) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.2e+183:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+183], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000046e183

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 79.4%

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

   if 7.20000000000000046e183 < 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 17.3

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

   5. Simplified 17.3%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. *-commutative 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. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+183}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+183) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / 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 <= 7.2d+183) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+183) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.2e+183:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.2e+183)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+183], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.20000000000000046e183

   1. Initial program 16.3%

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

   3. Taylor expanded in i around inf

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

   5. Simplified 79.4%

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

   if 7.20000000000000046e183 < 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 17.3

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

   5. Simplified 17.3%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. *-commutative 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. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 80.9

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

   7. Applied egg-rr 80.9%

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

   8. Taylor expanded in i around inf

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

      1. lower-/.f64 72.9

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

   10. Simplified 72.9%

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

Alternative 9: 67.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.25e+42) (/ (* i (+ i alpha)) (* beta beta)) 0.0625))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.25e+42) {
		tmp = (i * (i + alpha)) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	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 (i <= 1.25d+42) then
        tmp = (i * (i + alpha)) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.25e+42) {
		tmp = (i * (i + alpha)) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.25e+42:
		tmp = (i * (i + alpha)) / (beta * beta)
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.25e+42)
		tmp = Float64(Float64(i * Float64(i + alpha)) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.25e+42)
		tmp = (i * (i + alpha)) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.25e+42], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1.25000000000000002e42

   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

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   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 16.0

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

   5. Simplified 16.0%

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

   if 1.25000000000000002e42 < 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 i around inf

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

   5. Simplified 80.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.25e+42) (* i (/ (+ i alpha) (* beta beta))) 0.0625))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.25e+42) {
		tmp = i * ((i + alpha) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	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 (i <= 1.25d+42) then
        tmp = i * ((i + alpha) / (beta * beta))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.25e+42) {
		tmp = i * ((i + alpha) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.25e+42:
		tmp = i * ((i + alpha) / (beta * beta))
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.25e+42)
		tmp = Float64(i * Float64(Float64(i + alpha) / Float64(beta * beta)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.25e+42)
		tmp = i * ((i + alpha) / (beta * beta));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.25e+42], N[(i * N[(N[(i + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]

Derivation

1. Split input into 2 regimes

2. if i < 1.25000000000000002e42

   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

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   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 16.0

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

   5. Simplified 16.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 16.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 16.1

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

   7. Applied egg-rr 16.1%

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

   if 1.25000000000000002e42 < 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 i around inf

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

   5. Simplified 80.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.25 \cdot 10^{+42}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+243) 0.0625 (* i (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+243) {
		tmp = 0.0625;
	} else {
		tmp = i * (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 <= 5.3d+243) then
        tmp = 0.0625d0
    else
        tmp = i * (i / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+243) {
		tmp = 0.0625;
	} else {
		tmp = i * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+243:
		tmp = 0.0625
	else:
		tmp = i * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+243)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+243)
		tmp = 0.0625;
	else
		tmp = i * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+243], 0.0625, N[(i * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2999999999999997e243

   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 i around inf

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

   5. Simplified 76.6%

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

   if 5.2999999999999997e243 < 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

      \[\leadsto \frac{\color{blue}{i \cdot \left(\alpha + i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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 14.1

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

   5. Simplified 14.1%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. unpow2 N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 14.1

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

   8. Simplified 14.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 17.2

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

   10. Applied egg-rr 17.2%

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

   11. Taylor expanded in beta around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 17.2

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

   13. Simplified 17.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+243) 0.0625 (* alpha (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+243) {
		tmp = 0.0625;
	} 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 <= 5.3d+243) then
        tmp = 0.0625d0
    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 <= 5.3e+243) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+243:
		tmp = 0.0625
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+243)
		tmp = 0.0625;
	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 <= 5.3e+243)
		tmp = 0.0625;
	else
		tmp = alpha * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+243], 0.0625, N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2999999999999997e243

   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 i around inf

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

   5. Simplified 76.6%

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

   if 5.2999999999999997e243 < 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 13: 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)))