Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 83.7%

Time: 22.6s

Alternatives: 11

Speedup: 8.8×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t_1 \cdot t_1\\ \frac{\frac{t_0 \cdot \left(\beta \cdot \alpha + t_0\right)}{t_2}}{t_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Initial Program: 16.2% 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, 0.1× speedup

Math

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

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 (+ alpha (fma i 2.0 beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (* (/ t_3 (fma t_4 t_4 -1.0)) (/ (fma i t_2 (* alpha beta)) (* t_4 t_4)))
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 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 = alpha + fma(i, 2.0, beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (t_3 / fma(t_4, t_4, -1.0)) * (fma(i, t_2, (alpha * beta)) / (t_4 * t_4));
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (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 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(t_3 / fma(t_4, t_4, -1.0)) * Float64(fma(i, t_2, Float64(alpha * beta)) / Float64(t_4 * t_4)));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - 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[(alpha + N[(i * 2.0 + 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[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$3 / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 43.0%

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

      1. associate-/l/ 39.1%

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

      2. times-frac 99.8%

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

   3. Simplified 99.7%

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

   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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 78.8%

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

4. Final simplification 85.9%

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

Alternative 2: 83.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := \alpha + \left(i + \beta\right)\\ t_5 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot t_4}{t_5} \cdot \frac{\mathsf{fma}\left(i, t_4, \alpha \cdot \beta\right)}{t_5}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (+ alpha (+ i beta)))
        (t_5 (+ beta (fma i 2.0 alpha))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) INFINITY)
     (/ (* (/ (* i t_4) t_5) (/ (fma i t_4 (* alpha beta)) t_5)) t_2)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = alpha + (i + beta);
	double t_5 = beta + fma(i, 2.0, alpha);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (((i * t_4) / t_5) * (fma(i, t_4, (alpha * beta)) / t_5)) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (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(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(alpha + Float64(i + beta))
	t_5 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(Float64(i * t_4) / t_5) * Float64(fma(i, t_4, Float64(alpha * beta)) / t_5)) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - 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[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(N[(i * t$95$4), $MachinePrecision] / t$95$5), $MachinePrecision] * N[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 43.0%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. +-commutative 99.7%

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

      10. *-commutative 99.7%

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

      11. fma-udef 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+l+ 99.7%

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

      14. *-commutative 99.7%

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

      15. +-commutative 99.7%

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

      16. *-commutative 99.7%

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

      17. associate-+r+ 99.7%

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

      18. +-commutative 99.7%

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

      19. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 78.8%

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

4. Final simplification 85.9%

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

Alternative 3: 80.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)\\ \mathbf{if}\;\frac{\frac{t_4}{t_1}}{t_2} \leq 0.1:\\ \;\;\;\;\frac{\frac{t_4}{\left(\alpha + \left(i + \beta\right)\right) \cdot \left(i \cdot 4\right) + {\left(\alpha + \beta\right)}^{2}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (* t_3 (+ t_3 (* alpha beta)))))
   (if (<= (/ (/ t_4 t_1) t_2) 0.1)
     (/
      (/ t_4 (+ (* (+ alpha (+ i beta)) (* i 4.0)) (pow (+ alpha beta) 2.0)))
      t_2)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = t_3 * (t_3 + (alpha * beta));
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.1) {
		tmp = (t_4 / (((alpha + (i + beta)) * (i * 4.0)) + pow((alpha + beta), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = t_1 + (-1.0d0)
    t_3 = i * (i + (alpha + beta))
    t_4 = t_3 * (t_3 + (alpha * beta))
    if (((t_4 / t_1) / t_2) <= 0.1d0) then
        tmp = (t_4 / (((alpha + (i + beta)) * (i * 4.0d0)) + ((alpha + beta) ** 2.0d0))) / t_2
    else
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / 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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = t_3 * (t_3 + (alpha * beta));
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.1) {
		tmp = (t_4 / (((alpha + (i + beta)) * (i * 4.0)) + Math.pow((alpha + beta), 2.0))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = t_1 + -1.0
	t_3 = i * (i + (alpha + beta))
	t_4 = t_3 * (t_3 + (alpha * beta))
	tmp = 0
	if ((t_4 / t_1) / t_2) <= 0.1:
		tmp = (t_4 / (((alpha + (i + beta)) * (i * 4.0)) + math.pow((alpha + beta), 2.0))) / t_2
	else:
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (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(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(t_3 * Float64(t_3 + Float64(alpha * beta)))
	tmp = 0.0
	if (Float64(Float64(t_4 / t_1) / t_2) <= 0.1)
		tmp = Float64(Float64(t_4 / Float64(Float64(Float64(alpha + Float64(i + beta)) * Float64(i * 4.0)) + (Float64(alpha + beta) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / 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 = t_1 + -1.0;
	t_3 = i * (i + (alpha + beta));
	t_4 = t_3 * (t_3 + (alpha * beta));
	tmp = 0.0;
	if (((t_4 / t_1) / t_2) <= 0.1)
		tmp = (t_4 / (((alpha + (i + beta)) * (i * 4.0)) + ((alpha + beta) ^ 2.0))) / t_2;
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / 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[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 0.1], N[(N[(t$95$4 / N[(N[(N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision] * N[(i * 4.0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(alpha + beta), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in i around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. distribute-lft-out 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. associate-+r+ 99.6%

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

      5. *-commutative 99.6%

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

      6. unpow2 99.6%

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

      7. distribute-rgt-in 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 0.10000000000000001 < (/.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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in i around inf 81.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\alpha + \left(i + \beta\right)\right) \cdot \left(i \cdot 4\right) + {\left(\alpha + \beta\right)}^{2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)\\ t_5 := \beta + i \cdot 2\\ \mathbf{if}\;\frac{\frac{t_4}{t_1}}{t_2} \leq 0.1:\\ \;\;\;\;\frac{\frac{t_4}{2 \cdot \left(i \cdot t_5\right) + \beta \cdot t_5}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ t_1 -1.0))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (* t_3 (+ t_3 (* alpha beta))))
        (t_5 (+ beta (* i 2.0))))
   (if (<= (/ (/ t_4 t_1) t_2) 0.1)
     (/ (/ t_4 (+ (* 2.0 (* i t_5)) (* beta t_5))) t_2)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = t_3 * (t_3 + (alpha * beta));
	double t_5 = beta + (i * 2.0);
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.1) {
		tmp = (t_4 / ((2.0 * (i * t_5)) + (beta * t_5))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = t_1 + (-1.0d0)
    t_3 = i * (i + (alpha + beta))
    t_4 = t_3 * (t_3 + (alpha * beta))
    t_5 = beta + (i * 2.0d0)
    if (((t_4 / t_1) / t_2) <= 0.1d0) then
        tmp = (t_4 / ((2.0d0 * (i * t_5)) + (beta * t_5))) / t_2
    else
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / 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 = t_1 + -1.0;
	double t_3 = i * (i + (alpha + beta));
	double t_4 = t_3 * (t_3 + (alpha * beta));
	double t_5 = beta + (i * 2.0);
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.1) {
		tmp = (t_4 / ((2.0 * (i * t_5)) + (beta * t_5))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = t_1 + -1.0
	t_3 = i * (i + (alpha + beta))
	t_4 = t_3 * (t_3 + (alpha * beta))
	t_5 = beta + (i * 2.0)
	tmp = 0
	if ((t_4 / t_1) / t_2) <= 0.1:
		tmp = (t_4 / ((2.0 * (i * t_5)) + (beta * t_5))) / t_2
	else:
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (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(t_1 + -1.0)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(t_3 * Float64(t_3 + Float64(alpha * beta)))
	t_5 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (Float64(Float64(t_4 / t_1) / t_2) <= 0.1)
		tmp = Float64(Float64(t_4 / Float64(Float64(2.0 * Float64(i * t_5)) + Float64(beta * t_5))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / 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 = t_1 + -1.0;
	t_3 = i * (i + (alpha + beta));
	t_4 = t_3 * (t_3 + (alpha * beta));
	t_5 = beta + (i * 2.0);
	tmp = 0.0;
	if (((t_4 / t_1) / t_2) <= 0.1)
		tmp = (t_4 / ((2.0 * (i * t_5)) + (beta * t_5))) / t_2;
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / 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[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 0.1], N[(N[(t$95$4 / N[(N[(2.0 * N[(i * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(beta * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-+r+ 99.6%

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

      11. +-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in alpha around 0 94.6%

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

   if 0.10000000000000001 < (/.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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in i around inf 81.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{2 \cdot \left(i \cdot \left(\beta + i \cdot 2\right)\right) + \beta \cdot \left(\beta + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ t_1 -1.0))))
   (if (<= t_3 0.1)
     t_3
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 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 * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / 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 t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / 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))
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (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(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / 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));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / 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]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.5%

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

   if 0.10000000000000001 < (/.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 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in i around inf 81.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.0625 \cdot \left(\left(\left(\alpha + \beta\right) \cdot 2\right) \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.0625 (* (* (+ alpha beta) 2.0) (/ 1.0 i))))
  (* 0.125 (/ (+ alpha beta) i))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * ((alpha + beta) / i));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.0625d0 * (((alpha + beta) * 2.0d0) * (1.0d0 / i)))) - (0.125d0 * ((alpha + beta) / i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * ((alpha + beta) / i));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * ((alpha + beta) / i))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha + beta) * 2.0) * Float64(1.0 / i)))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * ((alpha + beta) / i));
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha + beta), $MachinePrecision] * 2.0), $MachinePrecision] * N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 13.3%

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

   2. associate-*l* 13.2%

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

   3. times-frac 22.1%

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

3. Simplified 22.1%

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

5. Taylor expanded in i around inf 81.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
6. Step-by-step derivation

   1. div-inv 80.3%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(2 \cdot \alpha + 2 \cdot \beta\right) \cdot \frac{1}{i}\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   2. distribute-lft-out 80.3%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)} \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied egg-rr 80.3%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(2 \cdot \left(\alpha + \beta\right)\right) \cdot \frac{1}{i}\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

8. Final simplification 80.3%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(\left(\alpha + \beta\right) \cdot 2\right) \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
9. Add Preprocessing

Alternative 7: 76.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
  (* 0.125 (/ (+ alpha beta) i))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
end

Wolfram

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

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 13.3%

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

   2. associate-*l* 13.2%

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

   3. times-frac 22.1%

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

3. Simplified 22.1%

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

5. Taylor expanded in i around inf 81.8%

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

6. Final simplification 81.8%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]
7. Add Preprocessing

Alternative 8: 71.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) + 0.125 \cdot \frac{-1}{\frac{i}{\alpha + \beta}} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (+
  (+ 0.0625 (* 0.0625 (* 2.0 (/ beta i))))
  (* 0.125 (/ -1.0 (/ i (+ alpha beta))))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (2.0 * (beta / i)))) + (0.125 * (-1.0 / (i / (alpha + beta))));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.0625d0 * (2.0d0 * (beta / i)))) + (0.125d0 * ((-1.0d0) / (i / (alpha + beta))))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (2.0 * (beta / i)))) + (0.125 * (-1.0 / (i / (alpha + beta))));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (2.0 * (beta / i)))) + (0.125 * (-1.0 / (i / (alpha + beta))))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(2.0 * Float64(beta / i)))) + Float64(0.125 * Float64(-1.0 / Float64(i / Float64(alpha + beta)))))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (2.0 * (beta / i)))) + (0.125 * (-1.0 / (i / (alpha + beta))));
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.0625 * N[(2.0 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(-1.0 / N[(i / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 13.3%

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

   2. associate-*l* 13.2%

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

   3. times-frac 22.1%

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

3. Simplified 22.1%

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

5. Taylor expanded in i around inf 81.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
6. Step-by-step derivation

   1. clear-num 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   2. inv-pow 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]

7. Applied egg-rr 79.6%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

9. Simplified 79.6%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

10. Taylor expanded in alpha around 0 76.2%

    \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(2 \cdot \frac{\beta}{i}\right)}\right) - 0.125 \cdot \frac{1}{\frac{i}{\alpha + \beta}} \]

11. Final simplification 76.2%

    \[\leadsto \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) + 0.125 \cdot \frac{-1}{\frac{i}{\alpha + \beta}} \]
12. Add Preprocessing

Alternative 9: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.0625 \cdot \left(\left(\left(\alpha + \beta\right) \cdot 2\right) \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\beta}{i} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.0625 (* (* (+ alpha beta) 2.0) (/ 1.0 i))))
  (* 0.125 (/ beta i))))

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * (beta / i));
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.0625d0 * (((alpha + beta) * 2.0d0) * (1.0d0 / i)))) - (0.125d0 * (beta / i))
end function

Java

public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * (beta / i));
}

Python

def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * (beta / i))

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha + beta) * 2.0) * Float64(1.0 / i)))) - Float64(0.125 * Float64(beta / i)))
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (((alpha + beta) * 2.0) * (1.0 / i)))) - (0.125 * (beta / i));
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha + beta), $MachinePrecision] * 2.0), $MachinePrecision] * N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 13.3%

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

   2. associate-*l* 13.2%

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

   3. times-frac 22.1%

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

3. Simplified 22.1%

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

5. Taylor expanded in i around inf 81.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
6. Step-by-step derivation

   1. div-inv 80.3%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(2 \cdot \alpha + 2 \cdot \beta\right) \cdot \frac{1}{i}\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   2. distribute-lft-out 80.3%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)} \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied egg-rr 80.3%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(\left(2 \cdot \left(\alpha + \beta\right)\right) \cdot \frac{1}{i}\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

8. Taylor expanded in alpha around 0 76.9%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(2 \cdot \left(\alpha + \beta\right)\right) \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]

9. Final simplification 76.9%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \left(\left(\left(\alpha + \beta\right) \cdot 2\right) \cdot \frac{1}{i}\right)\right) - 0.125 \cdot \frac{\beta}{i} \]
10. Add Preprocessing

Alternative 10: 72.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 8.5e+224) 0.0625 0.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+224) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	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 <= 8.5d+224) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.5e+224) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.5e+224:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.5e+224)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.5e+224)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 8.5e+224], 0.0625, 0.0]

Derivation

1. Split input into 2 regimes

2. if beta < 8.50000000000000046e224

   1. Initial program 15.8%

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

      1. associate-/l/ 14.4%

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

      2. associate-*l* 14.4%

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

      3. times-frac 24.0%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in i around inf 80.2%

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

   if 8.50000000000000046e224 < 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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 48.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
   6. Step-by-step derivation

      1. clear-num 28.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

      2. inv-pow 28.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]

   7. Applied egg-rr 28.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 28.2%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   9. Simplified 28.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   10. Taylor expanded in i around 0 33.6%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
   11. Step-by-step derivation

       1. div-sub 33.6%

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

       2. distribute-lft-in 33.6%

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

       3. associate-*r/ 33.6%

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

       4. associate-*r/ 38.6%

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

       5. associate-*r* 38.6%

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

       6. metadata-eval 38.6%

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

       7. associate-*r/ 38.6%

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

       8. +-inverses 38.6%

          \[\leadsto \color{blue}{0} \]

   12. Simplified 38.6%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.5 \cdot 10^{+224}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.0% accurate, 53.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0)

C

double code(double alpha, double beta, double i) {
	return 0.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0;
}

Python

def code(alpha, beta, i):
	return 0.0

Julia

function code(alpha, beta, i)
	return 0.0
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0;
end

Wolfram

code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 14.6%

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

   1. associate-/l/ 13.3%

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

   2. associate-*l* 13.2%

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

   3. times-frac 22.1%

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

3. Simplified 22.1%

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

5. Taylor expanded in i around inf 81.8%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]
6. Step-by-step derivation

   1. clear-num 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

   2. inv-pow 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]

7. Applied egg-rr 79.6%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{{\left(\frac{i}{\alpha + \beta}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 79.6%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

9. Simplified 79.6%

   \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}} \]

10. Taylor expanded in i around 0 9.9%

    \[\leadsto \color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}} \]
11. Step-by-step derivation

    1. div-sub 9.9%

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

    2. distribute-lft-in 9.9%

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

    3. associate-*r/ 9.9%

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

    4. associate-*r/ 10.3%

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

    5. associate-*r* 10.3%

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

    6. metadata-eval 10.3%

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

    7. associate-*r/ 10.3%

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

    8. +-inverses 10.3%

       \[\leadsto \color{blue}{0} \]

12. Simplified 10.3%

    \[\leadsto \color{blue}{0} \]

13. Final simplification 10.3%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))