Octave 3.8, jcobi/4

Percentage Accurate: 15.6% → 82.2%

Time: 11.4s

Alternatives: 8

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: 15.6% 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: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ \mathbf{if}\;i \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_1, i, \beta \cdot \alpha\right)}{t\_0}}{t\_0 - 1} \cdot \frac{\frac{i}{t\_0} \cdot t\_1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ (+ beta alpha) i)))
   (if (<= i 7.8e+99)
     (*
      (/ (/ (fma t_1 i (* beta alpha)) t_0) (- t_0 1.0))
      (/ (* (/ i t_0) t_1) (+ 1.0 t_0)))
     (-
      (fma (/ (* 2.0 (+ beta alpha)) i) 0.0625 0.0625)
      (* 0.125 (/ (+ beta alpha) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double tmp;
	if (i <= 7.8e+99) {
		tmp = ((fma(t_1, i, (beta * alpha)) / t_0) / (t_0 - 1.0)) * (((i / t_0) * t_1) / (1.0 + t_0));
	} else {
		tmp = fma(((2.0 * (beta + alpha)) / i), 0.0625, 0.0625) - (0.125 * ((beta + alpha) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	tmp = 0.0
	if (i <= 7.8e+99)
		tmp = Float64(Float64(Float64(fma(t_1, i, Float64(beta * alpha)) / t_0) / Float64(t_0 - 1.0)) * Float64(Float64(Float64(i / t_0) * t_1) / Float64(1.0 + t_0)));
	else
		tmp = Float64(fma(Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625, 0.0625) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[i, 7.8e+99], N[(N[(N[(N[(t$95$1 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * 0.0625 + 0.0625), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 7.79999999999999989e99

   1. Initial program 47.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 87.4%

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

   if 7.79999999999999989e99 < i

   1. Initial program 0.7%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 7.2

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

   5. Applied rewrites 7.2%

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

   7. Applied rewrites 7.1%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in i around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 88.8

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

   13. Applied rewrites 88.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 7.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\beta + \alpha\right) + i, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{\frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \left(\left(\beta + \alpha\right) + i\right)}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.82 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.82e-51)
   (-
    (fma (/ (* 2.0 (+ beta alpha)) i) 0.0625 0.0625)
    (* 0.125 (/ (+ beta alpha) i)))
   (/ (/ (+ alpha i) beta) (/ beta i))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.82e-51) {
		tmp = fma(((2.0 * (beta + alpha)) / i), 0.0625, 0.0625) - (0.125 * ((beta + alpha) / i));
	} else {
		tmp = ((alpha + i) / beta) / (beta / i);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.82e-51)
		tmp = Float64(fma(Float64(Float64(2.0 * Float64(beta + alpha)) / i), 0.0625, 0.0625) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(beta / i));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.82e-51], N[(N[(N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * 0.0625 + 0.0625), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.82000000000000013e-51

   1. Initial program 21.9%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 14.7

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

   5. Applied rewrites 14.7%

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

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 6.4%

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

   11. Taylor expanded in i around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. +-commutative N/A

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

       10. lower-+.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. +-commutative N/A

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

       15. lower-+.f64 91.2

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

   13. Applied rewrites 91.2%

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

   if 1.82000000000000013e-51 < alpha

   1. Initial program 6.3%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 14.3

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

   5. Applied rewrites 14.3%

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

   7. Applied rewrites 14.3%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.82 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2 \cdot \left(\beta + \alpha\right)}{i}, 0.0625, 0.0625\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.1e+129) 0.0625 (/ (/ (+ alpha i) beta) (/ beta i))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.1e+129) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / 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) :: tmp
    if (beta <= 3.1d+129) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.1e+129) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / i);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.1e+129:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) / (beta / i)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.1e+129)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.1e+129)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.1e+129], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1e129

   1. Initial program 18.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 83.9%

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

   if 3.1e129 < beta

   1. Initial program 0.1%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

Alternative 4: 77.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.1e+129) 0.0625 (* (/ i beta) (/ (+ alpha i) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.1e+129) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + 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 <= 3.1d+129) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + i) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.1e+129) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.1e+129:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.1e+129)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.1e+129)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + i) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.1e+129], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.1e129

   1. Initial program 18.7%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 83.9%

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

   if 3.1e129 < beta

   1. Initial program 0.1%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 54.5

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

   5. Applied rewrites 54.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+222}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.7e+222) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+222) {
		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 <= 4.7d+222) 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 <= 4.7e+222) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.7e+222:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.7e+222)
		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 <= 4.7e+222)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.7e+222], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.6999999999999999e222

   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. Applied rewrites 80.4%

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

   if 4.6999999999999999e222 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 73.8%

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

Alternative 6: 73.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 5.3 \cdot 10^{+228}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 5.3e+228) 0.0625 (* (/ alpha beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+228) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / 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 <= 5.3d+228) then
        tmp = 0.0625d0
    else
        tmp = (alpha / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 5.3e+228) {
		tmp = 0.0625;
	} else {
		tmp = (alpha / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 5.3e+228:
		tmp = 0.0625
	else:
		tmp = (alpha / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 5.3e+228)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(alpha / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 5.3e+228)
		tmp = 0.0625;
	else
		tmp = (alpha / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 5.3e+228], 0.0625, N[(N[(alpha / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.2999999999999999e228

   1. Initial program 16.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 79.5%

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

   if 5.2999999999999999e228 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in alpha around inf

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

   8. Applied rewrites 52.5%

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

Alternative 7: 71.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+271}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.8e+271) 0.0625 (* (/ i (* beta beta)) alpha)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.8e+271) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	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 <= 1.8d+271) then
        tmp = 0.0625d0
    else
        tmp = (i / (beta * beta)) * alpha
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.8e+271) {
		tmp = 0.0625;
	} else {
		tmp = (i / (beta * beta)) * alpha;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.8e+271:
		tmp = 0.0625
	else:
		tmp = (i / (beta * beta)) * alpha
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.8e+271)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(beta * beta)) * alpha);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.8e+271)
		tmp = 0.0625;
	else
		tmp = (i / (beta * beta)) * alpha;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.8e+271], 0.0625, N[(N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision] * alpha), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8000000000000002e271

   1. Initial program 15.9%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.9%

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

   if 1.8000000000000002e271 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 73.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+271}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \beta} \cdot \alpha\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.4% 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 15.5%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 75.9%

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

Reproduce

herbie shell --seed 2024273 
(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)))