Octave 3.8, jcobi/4

Percentage Accurate: 15.8% → 84.4%

Time: 11.6s

Alternatives: 11

Speedup: 53.0×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 15.8% 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: 84.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(beta + Float64(i + alpha))
	t_4 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(i * t_3) / fma(t_4, t_4, -1.0)) * Float64(Float64(fma(i, t_3, Float64(alpha * beta)) / t_4) / t_4));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / 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[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$3), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.0%

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

      1. associate-/l/ 36.9%

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

      2. times-frac 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \]
   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in i around inf 72.6%

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

      1. log1p-expm1-u 63.4%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 63.3%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 63.3%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 72.6%

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

Alternative 2: 84.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(i + Float64(alpha + beta))
	t_4 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(i * Float64(Float64(fma(i, t_3, Float64(alpha * beta)) / fma(t_4, t_4, -1.0)) * Float64(t_3 / Float64(t_4 * t_4))));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / 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[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(i * N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.0%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{i \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} \]
   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in i around inf 72.6%

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

      1. log1p-expm1-u 63.4%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 63.3%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 63.3%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 72.6%

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

Alternative 3: 80.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(i * Float64(alpha + Float64(beta + i))) * Float64(Float64(i * Float64(beta + i)) / (fma(2.0, i, Float64(alpha + beta)) ^ 2.0))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / 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[(2.0 * i), $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[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(N[(i * N[(alpha + N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[Power[N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 44.0%

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

   3. Taylor expanded in alpha around 0 38.6%

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

      1. associate-/l* 90.0%

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

      2. associate-+l+ 90.0%

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

      3. pow2 90.0%

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

      4. +-commutative 90.0%

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

      5. fma-define 90.0%

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

   5. Applied egg-rr 90.0%

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

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

   1. Initial program 0.0%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in i around inf 72.6%

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

      1. log1p-expm1-u 63.4%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 63.3%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 63.3%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 63.3%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 72.6%

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

Alternative 4: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}\\ \mathbf{if}\;\frac{t\_3}{t\_1 - 1} \leq 0.1:\\ \;\;\;\;\frac{t\_3}{\left(i \cdot \left(4 \cdot i + 4 \cdot \left(\alpha + \beta\right)\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0625 \cdot i + 0.125 \cdot \left(\alpha + \beta\right)}{i} - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (/ (* t_2 (+ (* beta alpha) t_2)) t_1)))
   (if (<= (/ t_3 (- t_1 1.0)) 0.1)
     (/
      t_3
      (-
       (+
        (* i (+ (* 4.0 i) (* 4.0 (+ alpha beta))))
        (* (+ alpha beta) (+ alpha beta)))
       1.0))
     (-
      (/ (+ (* 0.0625 i) (* 0.125 (+ alpha beta))) i)
      (* 0.125 (/ (+ alpha beta) i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (t_2 * ((beta * alpha) + t_2)) / t_1;
	double tmp;
	if ((t_3 / (t_1 - 1.0)) <= 0.1) {
		tmp = t_3 / (((i * ((4.0 * i) + (4.0 * (alpha + beta)))) + ((alpha + beta) * (alpha + beta))) - 1.0);
	} else {
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / 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) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((alpha + beta) + i)
    t_3 = (t_2 * ((beta * alpha) + t_2)) / t_1
    if ((t_3 / (t_1 - 1.0d0)) <= 0.1d0) then
        tmp = t_3 / (((i * ((4.0d0 * i) + (4.0d0 * (alpha + beta)))) + ((alpha + beta) * (alpha + beta))) - 1.0d0)
    else
        tmp = (((0.0625d0 * i) + (0.125d0 * (alpha + beta))) / 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) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double t_3 = (t_2 * ((beta * alpha) + t_2)) / t_1;
	double tmp;
	if ((t_3 / (t_1 - 1.0)) <= 0.1) {
		tmp = t_3 / (((i * ((4.0 * i) + (4.0 * (alpha + beta)))) + ((alpha + beta) * (alpha + beta))) - 1.0);
	} else {
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	t_3 = (t_2 * ((beta * alpha) + t_2)) / t_1
	tmp = 0
	if (t_3 / (t_1 - 1.0)) <= 0.1:
		tmp = t_3 / (((i * ((4.0 * i) + (4.0 * (alpha + beta)))) + ((alpha + beta) * (alpha + beta))) - 1.0)
	else:
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1)
	tmp = 0.0
	if (Float64(t_3 / Float64(t_1 - 1.0)) <= 0.1)
		tmp = Float64(t_3 / Float64(Float64(Float64(i * Float64(Float64(4.0 * i) + Float64(4.0 * Float64(alpha + beta)))) + Float64(Float64(alpha + beta) * Float64(alpha + beta))) - 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / 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) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	t_3 = (t_2 * ((beta * alpha) + t_2)) / t_1;
	tmp = 0.0;
	if ((t_3 / (t_1 - 1.0)) <= 0.1)
		tmp = t_3 / (((i * ((4.0 * i) + (4.0 * (alpha + beta)))) + ((alpha + beta) * (alpha + beta))) - 1.0);
	else
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / 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[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$3 / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 0.1], N[(t$95$3 / N[(N[(N[(i * N[(N[(4.0 * i), $MachinePrecision] + N[(4.0 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in i around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.7%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in i around inf 75.0%

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

      1. log1p-expm1-u 67.7%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 67.7%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 67.7%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 67.7%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 75.0%

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

Alternative 5: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0625 \cdot i + 0.125 \cdot \left(\alpha + \beta\right)}{i} - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i)))
        (t_3 (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0))))
   (if (<= t_3 0.1)
     t_3
     (-
      (/ (+ (* 0.0625 i) (* 0.125 (+ alpha beta))) i)
      (* 0.125 (/ (+ alpha beta) i))))))

C

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

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	t_3 = ((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / 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) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	t_3 = ((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / 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[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $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[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 99.4%

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

   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 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 0.7%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in i around inf 75.0%

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

      1. log1p-expm1-u 67.7%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 67.7%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 67.7%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 67.7%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 67.7%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 75.0%

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

Alternative 6: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;i \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{t\_0 \cdot t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + 0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= i 4.5e+32)
     (/ (* i (+ alpha i)) (- (* t_0 t_0) 1.0))
     (/
      (-
       (+ (* 0.0625 i) (* 0.0625 (+ (* 2.0 alpha) (* 2.0 beta))))
       (* 0.125 (+ alpha beta)))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (i <= 4.5e+32) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else {
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (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) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (i <= 4.5d+32) then
        tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0d0)
    else
        tmp = (((0.0625d0 * i) + (0.0625d0 * ((2.0d0 * alpha) + (2.0d0 * beta)))) - (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) + (2.0 * i);
	double tmp;
	if (i <= 4.5e+32) {
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	} else {
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (0.125 * (alpha + beta))) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if i <= 4.5e+32:
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0)
	else:
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (0.125 * (alpha + beta))) / i
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (i <= 4.5e+32)
		tmp = Float64(Float64(i * Float64(alpha + i)) / Float64(Float64(t_0 * t_0) - 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.0625 * Float64(Float64(2.0 * alpha) + Float64(2.0 * beta)))) - Float64(0.125 * Float64(alpha + beta))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (i <= 4.5e+32)
		tmp = (i * (alpha + i)) / ((t_0 * t_0) - 1.0);
	else
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (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[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 4.5e+32], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.0625 * N[(N[(2.0 * alpha), $MachinePrecision] + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if i < 4.5000000000000003e32

   1. Initial program 61.4%

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

   3. Taylor expanded in beta around inf 51.8%

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

   if 4.5000000000000003e32 < i

   1. Initial program 10.8%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in i around inf 77.0%

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

   6. Taylor expanded in i around 0 77.0%

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

Alternative 7: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 3.55 \cdot 10^{+31}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + 0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 3.55e+31)
   (* i (/ (+ alpha i) (* beta beta)))
   (/
    (-
     (+ (* 0.0625 i) (* 0.0625 (+ (* 2.0 alpha) (* 2.0 beta))))
     (* 0.125 (+ alpha beta)))
    i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.55e+31) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (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) :: tmp
    if (i <= 3.55d+31) then
        tmp = i * ((alpha + i) / (beta * beta))
    else
        tmp = (((0.0625d0 * i) + (0.0625d0 * ((2.0d0 * alpha) + (2.0d0 * beta)))) - (0.125d0 * (alpha + beta))) / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 3.55e+31) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (0.125 * (alpha + beta))) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 3.55e+31:
		tmp = i * ((alpha + i) / (beta * beta))
	else:
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (0.125 * (alpha + beta))) / i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 3.55e+31)
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.0625 * Float64(Float64(2.0 * alpha) + Float64(2.0 * beta)))) - Float64(0.125 * Float64(alpha + beta))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 3.55e+31)
		tmp = i * ((alpha + i) / (beta * beta));
	else
		tmp = (((0.0625 * i) + (0.0625 * ((2.0 * alpha) + (2.0 * beta)))) - (0.125 * (alpha + beta))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 3.55e+31], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.0625 * N[(N[(2.0 * alpha), $MachinePrecision] + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 3.5499999999999998e31

   1. Initial program 64.5%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in beta around inf 45.1%

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

      1. unpow2 45.1%

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

   7. Applied egg-rr 45.1%

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

   if 3.5499999999999998e31 < i

   1. Initial program 10.8%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in i around inf 76.7%

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

   6. Taylor expanded in i around 0 76.7%

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

Alternative 8: 73.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.7 \cdot 10^{+32}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.0625 \cdot i + 0.125 \cdot \left(\alpha + \beta\right)}{i} - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.7e+32)
   (* i (/ (+ alpha i) (* beta beta)))
   (-
    (/ (+ (* 0.0625 i) (* 0.125 (+ alpha beta))) i)
    (* 0.125 (/ (+ alpha beta) i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.7e+32) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / 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) :: tmp
    if (i <= 1.7d+32) then
        tmp = i * ((alpha + i) / (beta * beta))
    else
        tmp = (((0.0625d0 * i) + (0.125d0 * (alpha + beta))) / i) - (0.125d0 * ((alpha + beta) / i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.7e+32) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 1.7e+32:
		tmp = i * ((alpha + i) / (beta * beta))
	else:
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 1.7e+32)
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * Float64(alpha + beta))) / i) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 1.7e+32)
		tmp = i * ((alpha + i) / (beta * beta));
	else
		tmp = (((0.0625 * i) + (0.125 * (alpha + beta))) / i) - (0.125 * ((alpha + beta) / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 1.7e+32], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.69999999999999989e32

   1. Initial program 61.4%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in beta around inf 43.2%

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

      1. unpow2 43.2%

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

   7. Applied egg-rr 43.2%

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

   if 1.69999999999999989e32 < i

   1. Initial program 10.8%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in i around inf 77.0%

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

      1. log1p-expm1-u 72.4%

         \[\leadsto \left(0.0625 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      2. log1p-undefine 72.4%

         \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      3. associate-*r/ 72.4%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{0.0625 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i}}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

      4. distribute-lft-out 72.4%

         \[\leadsto \left(0.0625 + \log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \color{blue}{\left(2 \cdot \left(\alpha + \beta\right)\right)}}{i}\right)\right)\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied egg-rr 72.4%

      \[\leadsto \left(0.0625 + \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.0625 \cdot \left(2 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in i around 0 77.0%

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

Alternative 9: 70.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 5.2 \cdot 10^{+31}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.0625 \cdot i + 0.125 \cdot \beta\right) - 0.125 \cdot \left(\alpha + \beta\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 5.2e+31)
   (* i (/ (+ alpha i) (* beta beta)))
   (/ (- (+ (* 0.0625 i) (* 0.125 beta)) (* 0.125 (+ alpha beta))) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 5.2e+31) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.125 * beta)) - (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) :: tmp
    if (i <= 5.2d+31) then
        tmp = i * ((alpha + i) / (beta * beta))
    else
        tmp = (((0.0625d0 * i) + (0.125d0 * beta)) - (0.125d0 * (alpha + beta))) / i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 5.2e+31) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = (((0.0625 * i) + (0.125 * beta)) - (0.125 * (alpha + beta))) / i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if i <= 5.2e+31:
		tmp = i * ((alpha + i) / (beta * beta))
	else:
		tmp = (((0.0625 * i) + (0.125 * beta)) - (0.125 * (alpha + beta))) / i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 5.2e+31)
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 * i) + Float64(0.125 * beta)) - Float64(0.125 * Float64(alpha + beta))) / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (i <= 5.2e+31)
		tmp = i * ((alpha + i) / (beta * beta));
	else
		tmp = (((0.0625 * i) + (0.125 * beta)) - (0.125 * (alpha + beta))) / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 5.2e+31], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0625 * i), $MachinePrecision] + N[(0.125 * beta), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 5.2e31

   1. Initial program 64.5%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in beta around inf 45.1%

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

      1. unpow2 45.1%

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

   7. Applied egg-rr 45.1%

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

   if 5.2e31 < i

   1. Initial program 10.8%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in i around inf 76.7%

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

   6. Taylor expanded in i around 0 76.7%

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

   7. Taylor expanded in alpha around 0 73.3%

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

Alternative 10: 72.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.5e+229) 0.0625 (* i (/ (+ alpha i) (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.5e+229) {
		tmp = 0.0625;
	} else {
		tmp = i * ((alpha + i) / (beta * beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.5d+229) then
        tmp = 0.0625d0
    else
        tmp = i * ((alpha + i) / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.5e+229) {
		tmp = 0.0625;
	} else {
		tmp = i * ((alpha + i) / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.5e+229:
		tmp = 0.0625
	else:
		tmp = i * ((alpha + i) / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.5e+229)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.5e+229)
		tmp = 0.0625;
	else
		tmp = i * ((alpha + i) / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.5e+229], 0.0625, N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.50000000000000023e229

   1. Initial program 16.7%

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

   3. Simplified 39.0%

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

   5. Taylor expanded in i around inf 76.7%

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

   if 4.50000000000000023e229 < beta

   1. Initial program 0.0%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in beta around inf 26.2%

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

      1. unpow2 26.2%

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

   7. Applied egg-rr 26.2%

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

Alternative 11: 70.8% 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 15.0%

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

3. Simplified 35.6%

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

5. Taylor expanded in i around inf 69.1%

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

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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)))