Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 84.2%

Time: 13.9s

Alternatives: 9

Speedup: 53.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 16.1% 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: 84.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i / N[(t$95$4 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * 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 43.6%

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

      1. times-frac 99.6%

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

      2. difference-of-sqr-1 99.6%

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

      3. times-frac 99.5%

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

   3. Applied egg-rr 99.6%

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

   if +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. Taylor expanded in i around inf 5.6%

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

      1. +-commutative 5.6%

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

      2. *-commutative 5.6%

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

      3. fma-udef 5.6%

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

      4. distribute-lft-out-- 5.6%

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

      5. distribute-lft-out 5.6%

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

      6. *-commutative 5.6%

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

      7. unpow2 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in i around inf 77.8%

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

4. Final simplification 84.3%

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

Alternative 2: 84.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + beta), $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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * 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 43.6%

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

      1. div-inv 43.6%

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

      2. *-commutative 43.6%

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

      3. associate-*l* 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

      9. pow2 99.5%

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

      10. pow-flip 99.3%

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

      11. +-commutative 99.3%

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

      12. *-commutative 99.3%

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

      13. fma-def 99.3%

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

   3. Applied egg-rr 99.3%

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

   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. Taylor expanded in i around inf 5.6%

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

      1. +-commutative 5.6%

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

      2. *-commutative 5.6%

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

      3. fma-udef 5.6%

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

      4. distribute-lft-out-- 5.6%

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

      5. distribute-lft-out 5.6%

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

      6. *-commutative 5.6%

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

      7. unpow2 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in i around inf 77.8%

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

4. Final simplification 84.2%

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

Alternative 3: 84.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + beta), $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[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[Power[N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * 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 43.6%

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

      1. associate-/l* 99.6%

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

      2. associate-/r/ 99.7%

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

      3. +-commutative 99.7%

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

      4. pow2 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. fma-def 99.7%

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

      10. +-commutative 99.7%

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

      11. *-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/l* 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   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. Taylor expanded in i around inf 5.6%

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

      1. +-commutative 5.6%

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

      2. *-commutative 5.6%

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

      3. fma-udef 5.6%

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

      4. distribute-lft-out-- 5.6%

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

      5. distribute-lft-out 5.6%

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

      6. *-commutative 5.6%

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

      7. unpow2 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in i around inf 77.8%

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

4. Final simplification 84.3%

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

Alternative 4: 80.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ alpha beta)))
        (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) 0.0625)
     (/
      (/
       t_4
       (+ (* (+ alpha beta) (+ (+ alpha beta) (* i 4.0))) (* 4.0 (* i i))))
      t_2)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* beta 2.0) (* alpha 2.0)) i)))
      (* (/ (+ alpha beta) i) 0.125)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	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) <= 0.0625) {
		tmp = (t_4 / (((alpha + beta) * ((alpha + beta) + (i * 4.0))) + (4.0 * (i * i)))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	}
	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 = (i * 2.0d0) + (alpha + beta)
    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) <= 0.0625d0) then
        tmp = (t_4 / (((alpha + beta) * ((alpha + beta) + (i * 4.0d0))) + (4.0d0 * (i * i)))) / t_2
    else
        tmp = (0.0625d0 + (0.0625d0 * (((beta * 2.0d0) + (alpha * 2.0d0)) / i))) - (((alpha + beta) / i) * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	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) <= 0.0625) {
		tmp = (t_4 / (((alpha + beta) * ((alpha + beta) + (i * 4.0))) + (4.0 * (i * i)))) / t_2;
	} else {
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (i * 2.0) + (alpha + beta)
	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) <= 0.0625:
		tmp = (t_4 / (((alpha + beta) * ((alpha + beta) + (i * 4.0))) + (4.0 * (i * i)))) / t_2
	else:
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(alpha + beta))
	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) <= 0.0625)
		tmp = Float64(Float64(t_4 / Float64(Float64(Float64(alpha + beta) * Float64(Float64(alpha + beta) + Float64(i * 4.0))) + Float64(4.0 * Float64(i * i)))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(beta * 2.0) + Float64(alpha * 2.0)) / i))) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (alpha + beta);
	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) <= 0.0625)
		tmp = (t_4 / (((alpha + beta) * ((alpha + beta) + (i * 4.0))) + (4.0 * (i * i)))) / t_2;
	else
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

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

   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. Taylor expanded in i around 0 99.5%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. associate-*r* 99.6%

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

      5. distribute-rgt-out 99.6%

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

      6. *-commutative 99.6%

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

      7. *-commutative 99.6%

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

      8. unpow2 99.6%

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

   4. Simplified 99.6%

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

   if 0.0625 < (/.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 2.3%

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

   2. Taylor expanded in i around inf 80.0%

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

4. Final simplification 82.2%

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

Alternative 5: 80.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ alpha beta)))
        (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 0.0625)
     t_3
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* beta 2.0) (* alpha 2.0)) i)))
      (* (/ (+ alpha beta) i) 0.125)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	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 <= 0.0625) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	}
	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 = (i * 2.0d0) + (alpha + beta)
    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 <= 0.0625d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * (((beta * 2.0d0) + (alpha * 2.0d0)) / i))) - (((alpha + beta) / i) * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	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 <= 0.0625) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (i * 2.0) + (alpha + beta)
	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 <= 0.0625:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125)
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(alpha + beta))
	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 <= 0.0625)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(beta * 2.0) + Float64(alpha * 2.0)) / i))) - Float64(Float64(Float64(alpha + beta) / i) * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (alpha + beta);
	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 <= 0.0625)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) - (((alpha + beta) / i) * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + beta), $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, 0.0625], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * 0.125), $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)) < 0.0625

   1. Initial program 99.5%

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

   if 0.0625 < (/.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 2.3%

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

   2. Taylor expanded in i around inf 80.0%

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

4. Final simplification 82.1%

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

Alternative 6: 77.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

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 * (((beta * 2.0d0) + (alpha * 2.0d0)) / i))) - (((alpha + beta) / i) * 0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.0%

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

2. Taylor expanded in i around inf 78.8%

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

3. Final simplification 78.8%

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

Alternative 7: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

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 + (0.125d0 * (beta / i))) - (((alpha + beta) / i) * 0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 13.0%

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

2. Taylor expanded in i around inf 78.8%

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

3. Taylor expanded in beta around inf 74.2%

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

4. Taylor expanded in i around 0 74.2%

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

5. Final simplification 74.2%

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

Alternative 8: 72.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+238}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{\alpha}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.5e+238) 0.0625 (/ i (/ (* beta beta) alpha))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.5e+238) {
		tmp = 0.0625;
	} else {
		tmp = i / ((beta * beta) / alpha);
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.5d+238) then
        tmp = 0.0625d0
    else
        tmp = i / ((beta * beta) / alpha)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.5e+238) {
		tmp = 0.0625;
	} else {
		tmp = i / ((beta * beta) / alpha);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.5e+238:
		tmp = 0.0625
	else:
		tmp = i / ((beta * beta) / alpha)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.5e+238)
		tmp = 0.0625;
	else
		tmp = Float64(i / Float64(Float64(beta * beta) / alpha));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.5e+238)
		tmp = 0.0625;
	else
		tmp = i / ((beta * beta) / alpha);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.5e+238], 0.0625, N[(i / N[(N[(beta * beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.50000000000000003e238

   1. Initial program 13.9%

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

   2. Taylor expanded in i around inf 76.2%

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

   if 3.50000000000000003e238 < 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. Taylor expanded in beta around inf 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-/l* 33.3%

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

      3. unpow2 33.3%

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

   4. Simplified 33.3%

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

   5. Taylor expanded in i around 0 33.3%

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

      1. unpow2 33.3%

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

   7. Simplified 33.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+238}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{\alpha}}\\ \end{array} \]

Alternative 9: 70.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 13.0%

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

2. Taylor expanded in i around inf 72.1%

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

3. Final simplification 72.1%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023174 
(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)))