Octave 3.8, jcobi/4

Percentage Accurate: 15.5% → 83.7%

Time: 15.3s

Alternatives: 12

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: 15.5% 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: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{i \cdot t\_0}{t\_1}}{t\_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_0, \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ alpha beta))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= i 5.8e+137)
     (*
      (/ (/ (* i t_0) t_1) (+ t_1 1.0))
      (/ (/ (fma i t_0 (* alpha beta)) t_1) (+ t_1 -1.0)))
     (*
      (/ t_0 (+ alpha (+ 1.0 (fma i 2.0 beta))))
      (/ (* i 0.25) (+ alpha (+ -1.0 (fma i 2.0 beta))))))))

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 <= 5.8e+137) {
		tmp = (((i * t_0) / t_1) / (t_1 + 1.0)) * ((fma(i, t_0, (alpha * beta)) / t_1) / (t_1 + -1.0));
	} else {
		tmp = (t_0 / (alpha + (1.0 + fma(i, 2.0, beta)))) * ((i * 0.25) / (alpha + (-1.0 + fma(i, 2.0, beta))));
	}
	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 <= 5.8e+137)
		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 = Float64(Float64(t_0 / Float64(alpha + Float64(1.0 + fma(i, 2.0, beta)))) * Float64(Float64(i * 0.25) / Float64(alpha + Float64(-1.0 + fma(i, 2.0, beta)))));
	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, 5.8e+137], 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], N[(N[(t$95$0 / N[(alpha + N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / N[(alpha + N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 5.79999999999999969e137

   1. Initial program 41.0%

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

      1. lift-/.f64 N/A

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

      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 85.8%

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

   if 5.79999999999999969e137 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 8.7%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 8.7%

      \[\leadsto \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot \color{blue}{0.25}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

   9. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + 1\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \left(\alpha + \beta\right)}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t\_2\\ t_4 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{-1 + t\_1} \leq \infty:\\ \;\;\;\;\frac{t\_2 \cdot \left(i \cdot \frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{t\_4 \cdot t\_4}\right)}{\mathsf{fma}\left(i, t\_2 \cdot 4, \mathsf{fma}\left(\alpha + \beta, \alpha + \beta, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\beta}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ -1.0 t_1)) INFINITY)
     (/
      (* t_2 (* i (/ (fma i t_2 (* alpha beta)) (* t_4 t_4))))
      (fma i (* t_2 4.0) (fma (+ alpha beta) (+ alpha beta) -1.0)))
     (+ (fma 0.125 (/ beta i) 0.0625) (* -0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= ((double) INFINITY)) {
		tmp = (t_2 * (i * (fma(i, t_2, (alpha * beta)) / (t_4 * t_4)))) / fma(i, (t_2 * 4.0), fma((alpha + beta), (alpha + beta), -1.0));
	} else {
		tmp = fma(0.125, (beta / i), 0.0625) + (-0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= Inf)
		tmp = Float64(Float64(t_2 * Float64(i * Float64(fma(i, t_2, Float64(alpha * beta)) / Float64(t_4 * t_4)))) / fma(i, Float64(t_2 * 4.0), fma(Float64(alpha + beta), Float64(alpha + beta), -1.0)));
	else
		tmp = Float64(fma(0.125, Float64(beta / i), 0.0625) + Float64(-0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$2 * N[(i * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * N[(t$95$2 * 4.0), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * N[(alpha + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(beta / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 53.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in i around 0

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. sub-neg N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 0.2%

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in i around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 73.3

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 66.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq \infty:\\ \;\;\;\;\frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\mathsf{fma}\left(i, \left(i + \left(\alpha + \beta\right)\right) \cdot 4, \mathsf{fma}\left(\alpha + \beta, \alpha + \beta, -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\beta}{i}, 0.0625\right) + -0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ -1.0 t_1)) INFINITY)
     (* (/ (fma i t_2 (* alpha beta)) (fma t_4 t_4 -1.0)) (/ t_3 (* t_4 t_4)))
     (+ (fma 0.125 (/ beta i) 0.0625) (* -0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= ((double) INFINITY)) {
		tmp = (fma(i, t_2, (alpha * beta)) / fma(t_4, t_4, -1.0)) * (t_3 / (t_4 * t_4));
	} else {
		tmp = fma(0.125, (beta / i), 0.0625) + (-0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= Inf)
		tmp = Float64(Float64(fma(i, t_2, Float64(alpha * beta)) / fma(t_4, t_4, -1.0)) * Float64(t_3 / Float64(t_4 * t_4)));
	else
		tmp = Float64(fma(0.125, Float64(beta / i), 0.0625) + Float64(-0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(beta / i), $MachinePrecision] + 0.0625), $MachinePrecision] + N[(-0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 53.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

   4. Applied rewrites 99.6%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 0.2%

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in i around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 73.3

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

   10. Applied rewrites 73.3%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 66.9%

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

4. Final simplification 79.7%

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

Alternative 4: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{-1 + t\_1} \leq 10^{-7}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ -1.0 t_1)) 1e-7)
     (/ (* i (+ i alpha)) (* beta beta))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = (i * (i + alpha)) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / ((-1.0d0) + t_1)) <= 1d-7) then
        tmp = (i * (i + alpha)) / (beta * beta)
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = (i * (i + alpha)) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7:
		tmp = (i * (i + alpha)) / (beta * beta)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= 1e-7)
		tmp = Float64(Float64(i * Float64(i + alpha)) / Float64(beta * beta));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7)
		tmp = (i * (i + alpha)) / (beta * beta);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(i * N[(i + alpha), $MachinePrecision]), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 9.9999999999999995e-8

   1. Initial program 99.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. 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 65.3

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

   5. Applied rewrites 65.3%

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

   if 9.9999999999999995e-8 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 72.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq 10^{-7}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{-1 + t\_1} \leq 10^{-7}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ -1.0 t_1)) 1e-7)
     (* i (/ (+ i alpha) (* beta beta)))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = i * ((i + alpha) / (beta * beta));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / ((-1.0d0) + t_1)) <= 1d-7) then
        tmp = i * ((i + alpha) / (beta * beta))
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = i * ((i + alpha) / (beta * beta));
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7:
		tmp = i * ((i + alpha) / (beta * beta))
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= 1e-7)
		tmp = Float64(i * Float64(Float64(i + alpha) / Float64(beta * beta)));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7)
		tmp = i * ((i + alpha) / (beta * beta));
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-7], N[(i * N[(N[(i + alpha), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 9.9999999999999995e-8

   1. Initial program 99.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. 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 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 65.1%

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

   if 9.9999999999999995e-8 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 72.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq 10^{-7}:\\ \;\;\;\;i \cdot \frac{i + \alpha}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{-1 + t\_1} \leq 10^{-7}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ -1.0 t_1)) 1e-7)
     (/ (* i i) (* beta beta))
     (+ 0.0625 (/ 0.015625 (* i i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / ((-1.0d0) + t_1)) <= 1d-7) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625 + (0.015625 / (i * i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625 + (0.015625 / (i * i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1)) <= 1e-7)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)) <= 1e-7)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625 + (0.015625 / (i * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1e-7], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 9.9999999999999995e-8

   1. Initial program 99.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. 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 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 60.2%

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

   if 9.9999999999999995e-8 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 15.2%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 32.6%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 72.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq 10^{-7}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{i + \left(\alpha + \beta\right)}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.6e+162)
   (*
    (/ (+ i (+ alpha beta)) (+ alpha (+ 1.0 (fma i 2.0 beta))))
    (/ (* i 0.25) (+ alpha (+ -1.0 (fma i 2.0 beta)))))
   (* (/ (+ i alpha) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.6e+162) {
		tmp = ((i + (alpha + beta)) / (alpha + (1.0 + fma(i, 2.0, beta)))) * ((i * 0.25) / (alpha + (-1.0 + fma(i, 2.0, beta))));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.6e+162)
		tmp = Float64(Float64(Float64(i + Float64(alpha + beta)) / Float64(alpha + Float64(1.0 + fma(i, 2.0, beta)))) * Float64(Float64(i * 0.25) / Float64(alpha + Float64(-1.0 + fma(i, 2.0, beta)))));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.6e+162], N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 0.25), $MachinePrecision] / N[(alpha + N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.6000000000000004e162

   1. Initial program 24.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in i around inf

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

   7. Applied rewrites 38.1%

      \[\leadsto \frac{\left(i + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot \color{blue}{0.25}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

   9. Applied rewrites 76.3%

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

   if 8.6000000000000004e162 < 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 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 78.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{i + \left(\alpha + \beta\right)}{\alpha + \left(1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot \frac{i \cdot 0.25}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+160}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.25e+160)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (* (/ (+ i alpha) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.25e+160) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + 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 <= 1.25d+160) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + 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 <= 1.25e+160) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.25e+160:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.25e+160)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.25e+160)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.25e+160], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.25e160

   1. Initial program 24.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 35.4%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 75.3%

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

   if 1.25e160 < 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 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 78.8%

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

Alternative 9: 74.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+162}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.6e+162)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (/ (* i i) beta) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.6e+162) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} 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 <= 8.6d+162) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    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 <= 8.6e+162) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i * i) / beta) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.6e+162:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i * i) / beta) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.6e+162)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i * i) / beta) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.6e+162)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i * i) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.6e+162], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 8.6000000000000004e162

   1. Initial program 24.1%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 35.4%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 75.3%

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

   if 8.6000000000000004e162 < 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 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 78.7%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 53.0%

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

Alternative 10: 73.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+233}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.7e+233)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* (+ alpha beta) 0.0) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((alpha + beta) * 0.0) / 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 <= 4.7d+233) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((alpha + beta) * 0.0d0) / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+233) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((alpha + beta) * 0.0) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.7e+233:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((alpha + beta) * 0.0) / i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.7e+233)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(alpha + beta) * 0.0) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.7e+233)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((alpha + beta) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.7e+233], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000003e233

   1. Initial program 22.5%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 15.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in i around inf

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

   11. Applied rewrites 71.8%

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

   if 4.70000000000000003e233 < 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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in i around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 54.5

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in i around 0

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

   13. Applied rewrites 40.7%

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot 0}{\color{blue}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+233}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.7e+233) 0.0625 (/ (* (+ alpha beta) 0.0) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+233) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + beta) * 0.0) / 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 <= 4.7d+233) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + beta) * 0.0d0) / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.7e+233) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + beta) * 0.0) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.7e+233:
		tmp = 0.0625
	else:
		tmp = ((alpha + beta) * 0.0) / i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.7e+233)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + beta) * 0.0) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.7e+233)
		tmp = 0.0625;
	else
		tmp = ((alpha + beta) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.7e+233], 0.0625, N[(N[(N[(alpha + beta), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.70000000000000003e233

   1. Initial program 22.5%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 71.7%

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

   if 4.70000000000000003e233 < 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 alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in i around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 54.5

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in i around 0

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

   13. Applied rewrites 40.7%

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot 0}{\color{blue}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.4% 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 20.8%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 67.6%

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

Reproduce

herbie shell --seed 2024225 
(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)))