Octave 3.8, jcobi/4

Percentage Accurate: 16.1% → 83.8%

Time: 21.4s

Alternatives: 9

Speedup: 8.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.1% 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: 83.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$3 / N[(t$95$4 * t$95$4 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(N[(-1.0 - N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - -1.0), $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 52.3%

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

      1. associate-/l/ 46.0%

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

      2. times-frac 99.7%

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

   3. Simplified 99.7%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 81.2%

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

      1. expm1-log1p-u 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. expm1-udef 71.8%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)} - 1\right)} \]

      2. log1p-udef 71.8%

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

      3. rem-exp-log 81.2%

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

      4. +-commutative 81.2%

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

   9. Applied egg-rr 81.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(\left(1 + \frac{\beta + \alpha}{i}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

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

Alternative 2: 83.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(i * N[(t$95$4 * N[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] * N[Power[N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(N[(-1.0 - N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - -1.0), $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 52.3%

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

      1. expm1-log1p-u 49.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef 49.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 49.2%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\left(i \cdot \left(\left(\alpha + \left(\beta + i\right)\right) \cdot \mathsf{fma}\left(i, \alpha + \left(\beta + i\right), \beta \cdot \alpha\right)\right)\right) \cdot {\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{-2}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   5. Step-by-step derivation

      1. expm1-def 49.2%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(i \cdot \left(\left(\alpha + \left(\beta + i\right)\right) \cdot \mathsf{fma}\left(i, \alpha + \left(\beta + i\right), \beta \cdot \alpha\right)\right)\right) \cdot {\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{-2}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p 52.2%

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

      3. associate-*r* 66.3%

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

      4. associate-*l* 99.4%

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

      5. *-commutative 99.4%

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

   6. Simplified 99.4%

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

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

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 81.2%

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

      1. expm1-log1p-u 71.8%

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

   7. Applied egg-rr 71.8%

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

      1. expm1-udef 71.8%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)} - 1\right)} \]

      2. log1p-udef 71.8%

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

      3. rem-exp-log 81.2%

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

      4. +-commutative 81.2%

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

   9. Applied egg-rr 81.2%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(\left(1 + \frac{\beta + \alpha}{i}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

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

Alternative 3: 80.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = (t_2 * (t_2 + (alpha * beta))) / t_1
    if ((t_3 / (t_1 + (-1.0d0))) <= 0.1d0) then
        tmp = t_3 / (((4.0d0 * (i * (alpha + beta))) + ((4.0d0 * (i ** 2.0d0)) + ((alpha + beta) ** 2.0d0))) + (-1.0d0))
    else
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) + (0.125d0 * (((-1.0d0) - ((alpha + beta) / i)) - (-1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1)
	tmp = 0.0
	if (Float64(t_3 / Float64(t_1 + -1.0)) <= 0.1)
		tmp = Float64(t_3 / Float64(Float64(Float64(4.0 * Float64(i * Float64(alpha + beta))) + Float64(Float64(4.0 * (i ^ 2.0)) + (Float64(alpha + beta) ^ 2.0))) + -1.0));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) + Float64(0.125 * Float64(Float64(-1.0 - Float64(Float64(alpha + beta) / i)) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = (t_2 * (t_2 + (alpha * beta))) / t_1;
	tmp = 0.0;
	if ((t_3 / (t_1 + -1.0)) <= 0.1)
		tmp = t_3 / (((4.0 * (i * (alpha + beta))) + ((4.0 * (i ^ 2.0)) + ((alpha + beta) ^ 2.0))) + -1.0);
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) + (0.125 * ((-1.0 - ((alpha + beta) / i)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

      1. add-cbrt-cube 58.1%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} - 1} \]

      2. pow1/3 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{{\left(\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)\right)}^{0.3333333333333333}} - 1} \]

      3. pow3 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\color{blue}{\left({\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}^{3}\right)}}^{0.3333333333333333} - 1} \]

      4. pow2 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\color{blue}{\left({\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - 1} \]

      5. pow-pow 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\color{blue}{\left({\left(\left(\alpha + \beta\right) + 2 \cdot i\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - 1} \]

      6. +-commutative 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} - 1} \]

      7. *-commutative 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\left(\left(\beta + \alpha\right) + \color{blue}{i \cdot 2}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} - 1} \]

      8. associate-+r+ 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\color{blue}{\left(\beta + \left(\alpha + i \cdot 2\right)\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} - 1} \]

      9. +-commutative 54.5%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\left(\beta + \color{blue}{\left(i \cdot 2 + \alpha\right)}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} - 1} \]

      10. fma-def 54.5%

          \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\left(\beta + \color{blue}{\mathsf{fma}\left(i, 2, \alpha\right)}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} - 1} \]

      11. metadata-eval 54.5%

          \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{{\left({\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} - 1} \]

   4. Applied egg-rr 54.5%

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{{\left({\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{6}\right)}^{0.3333333333333333}} - 1} \]
   5. Step-by-step derivation

      1. unpow1/3 58.1%

         \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\sqrt[3]{{\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{6}}} - 1} \]

   6. Simplified 58.1%

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\sqrt[3]{{\left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}^{6}}} - 1} \]

   7. Taylor expanded in i around 0 99.7%

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

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

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 7.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in i around inf 81.8%

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

      1. expm1-log1p-u 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. expm1-udef 74.5%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)} - 1\right)} \]

      2. log1p-udef 74.5%

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

      3. rem-exp-log 81.8%

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

      4. +-commutative 81.8%

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

   9. Applied egg-rr 81.8%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(\left(1 + \frac{\beta + \alpha}{i}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

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

Alternative 4: 80.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) + Float64(0.125 * Float64(Float64(-1.0 - Float64(Float64(alpha + beta) / i)) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) + (0.125 * ((-1.0 - ((alpha + beta) / i)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(N[(-1.0 - N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

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

   1. Initial program 0.7%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 7.2%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in i around inf 81.8%

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

      1. expm1-log1p-u 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. expm1-udef 74.5%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)} - 1\right)} \]

      2. log1p-udef 74.5%

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

      3. rem-exp-log 81.8%

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

      4. +-commutative 81.8%

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

   9. Applied egg-rr 81.8%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(\left(1 + \frac{\beta + \alpha}{i}\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

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

Alternative 5: 77.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) + (0.125 * ((-1.0 - ((alpha + beta) / i)) - -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
    code = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) + (0.125d0 * (((-1.0d0) - ((alpha + beta) / i)) - (-1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.6%

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

   1. associate-/l/ 17.2%

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

   2. associate-*l* 17.2%

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

   3. times-frac 24.9%

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

3. Simplified 24.9%

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

5. Taylor expanded in i around inf 80.0%

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

   1. expm1-log1p-u 74.0%

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

7. Applied egg-rr 74.0%

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

   1. expm1-udef 74.0%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - 0.125 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)} - 1\right)} \]

   2. log1p-udef 74.0%

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

   3. rem-exp-log 80.0%

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

   4. +-commutative 80.0%

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

9. Applied egg-rr 80.0%

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

10. Final simplification 80.0%

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

Alternative 6: 74.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \left(0.0625 + t\_0\right) - t\_0 \end{array} \end{array} \]

FPCore

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

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_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
    t_0 = 0.125d0 * (beta / i)
    code = (0.0625d0 + t_0) - t_0
end function

Java

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

Python

def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	return (0.0625 + t_0) - t_0

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	return Float64(Float64(0.0625 + t_0) - t_0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = (0.0625 + t_0) - t_0;
end

Wolfram

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

Derivation

1. Initial program 19.6%

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

   1. associate-/l/ 17.2%

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

   2. associate-*l* 17.2%

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

   3. times-frac 24.9%

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

3. Simplified 24.9%

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

5. Taylor expanded in i around inf 80.0%

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

6. Taylor expanded in alpha around 0 77.1%

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

7. Taylor expanded in alpha around 0 78.1%

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

8. Final simplification 78.1%

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

Alternative 7: 74.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.5e+201) 0.0625 (/ i (* beta (/ beta i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+201) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.5d+201) then
        tmp = 0.0625d0
    else
        tmp = i / (beta * (beta / i))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.5e+201) {
		tmp = 0.0625;
	} else {
		tmp = i / (beta * (beta / i));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.5e+201:
		tmp = 0.0625
	else:
		tmp = i / (beta * (beta / i))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.5e+201)
		tmp = 0.0625;
	else
		tmp = Float64(i / Float64(beta * Float64(beta / i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.5e+201)
		tmp = 0.0625;
	else
		tmp = i / (beta * (beta / i));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 7.5e+201], 0.0625, N[(i / N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.5000000000000004e201

   1. Initial program 21.2%

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

      1. associate-/l/ 18.6%

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

      2. associate-*l* 18.6%

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

      3. times-frac 26.9%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in i around inf 79.3%

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

   if 7.5000000000000004e201 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in beta around inf 41.2%

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

      1. associate-/l* 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in alpha around 0 42.2%

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

      1. unpow2 42.2%

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

      2. *-un-lft-identity 42.2%

         \[\leadsto \frac{i}{\frac{\beta \cdot \beta}{\color{blue}{1 \cdot i}}} \]

      3. times-frac 52.5%

         \[\leadsto \frac{i}{\color{blue}{\frac{\beta}{1} \cdot \frac{\beta}{i}}} \]

   10. Applied egg-rr 52.5%

       \[\leadsto \frac{i}{\color{blue}{\frac{\beta}{1} \cdot \frac{\beta}{i}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+201}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 1.4e+264) 0.0625 0.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.4e+264) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.4d+264) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.4e+264) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.4e+264:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.4e+264)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.4e+264)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.4e+264], 0.0625, 0.0]

Derivation

1. Split input into 2 regimes

2. if beta < 1.39999999999999999e264

   1. Initial program 20.2%

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

      1. associate-/l/ 17.7%

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

      2. associate-*l* 17.7%

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

      3. times-frac 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in i around inf 76.4%

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

   if 1.39999999999999999e264 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in i around inf 73.6%

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

   6. Taylor expanded in i around 0 72.4%

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

      1. div-sub 72.4%

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

      2. distribute-lft-in 72.4%

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

      3. associate-*r* 72.4%

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

      4. metadata-eval 72.4%

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

      5. associate-*r/ 72.4%

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

      6. associate-*r/ 72.4%

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

      7. +-inverses 72.4%

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

   8. Simplified 72.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+264}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.5% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0

Derivation

1. Initial program 19.6%

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

   1. associate-/l/ 17.2%

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

   2. associate-*l* 17.2%

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

   3. times-frac 24.9%

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

3. Simplified 24.9%

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

5. Taylor expanded in i around inf 80.0%

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

6. Taylor expanded in i around 0 9.0%

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

   1. div-sub 9.0%

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

   2. distribute-lft-in 9.0%

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

   3. associate-*r* 9.0%

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

   4. metadata-eval 9.0%

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

   5. associate-*r/ 9.0%

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

   6. associate-*r/ 9.0%

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

   7. +-inverses 9.0%

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

8. Simplified 9.0%

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

9. Final simplification 9.0%

   \[\leadsto 0 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))