Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 84.3%

Time: 20.2s

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: 16.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := 0.125 \cdot \frac{\alpha + \beta}{i}\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := \beta + \left(i + \alpha\right)\\ t_5 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;\frac{i \cdot t\_4}{\mathsf{fma}\left(t\_5, t\_5, -1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_4, \alpha \cdot \beta\right)}{t\_5}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + 0.0625\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* 0.125 (/ (+ alpha beta) i)))
        (t_3 (* i (+ i (+ alpha beta))))
        (t_4 (+ beta (+ i alpha)))
        (t_5 (fma i 2.0 (+ alpha beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (*
      (/ (* i t_4) (fma t_5 t_5 -1.0))
      (/ (/ (fma i t_4 (* alpha beta)) t_5) t_5))
     (- (+ t_2 0.0625) t_2))))

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 = 0.125 * ((alpha + beta) / i);
	double t_3 = i * (i + (alpha + beta));
	double t_4 = beta + (i + alpha);
	double t_5 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((i * t_4) / fma(t_5, t_5, -1.0)) * ((fma(i, t_4, (alpha * beta)) / t_5) / t_5);
	} else {
		tmp = (t_2 + 0.0625) - t_2;
	}
	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(0.125 * Float64(Float64(alpha + beta) / i))
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_4 = Float64(beta + Float64(i + alpha))
	t_5 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(i * t_4) / fma(t_5, t_5, -1.0)) * Float64(Float64(fma(i, t_4, Float64(alpha * beta)) / t_5) / t_5));
	else
		tmp = Float64(Float64(t_2 + 0.0625) - t_2);
	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[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * t$95$4), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

      1. associate-/l/ 45.2%

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

      2. 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 99.6%

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

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

   1. Initial program 0.0%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in i around inf 70.8%

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

      1. sub-neg 70.8%

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

      2. +-commutative 70.8%

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

      3. fma-define 70.8%

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

      4. distribute-lft-out 70.8%

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

      5. associate-*r/ 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. unsub-neg 70.8%

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

      2. fma-undefine 70.8%

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

      3. associate-/l* 70.8%

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

      4. associate-*r* 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-/l* 70.8%

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

      7. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 80.1%

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

Alternative 2: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := 0.125 \cdot \frac{\alpha + \beta}{i}\\ t_3 := i + \left(\alpha + \beta\right)\\ t_4 := i \cdot t\_3\\ t_5 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\frac{\frac{t\_4 \cdot \left(t\_4 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1} \leq \infty:\\ \;\;\;\;i \cdot \left(\frac{\mathsf{fma}\left(i, t\_3, \alpha \cdot \beta\right)}{\mathsf{fma}\left(t\_5, t\_5, -1\right)} \cdot \frac{t\_3}{t\_5 \cdot t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + 0.0625\right) - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* 0.125 (/ (+ alpha beta) i)))
        (t_3 (+ i (+ alpha beta)))
        (t_4 (* i t_3))
        (t_5 (+ alpha (fma i 2.0 beta))))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (*
      i
      (*
       (/ (fma i t_3 (* alpha beta)) (fma t_5 t_5 -1.0))
       (/ t_3 (* t_5 t_5))))
     (- (+ t_2 0.0625) t_2))))

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 = 0.125 * ((alpha + beta) / i);
	double t_3 = i + (alpha + beta);
	double t_4 = i * t_3;
	double t_5 = alpha + fma(i, 2.0, beta);
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = i * ((fma(i, t_3, (alpha * beta)) / fma(t_5, t_5, -1.0)) * (t_3 / (t_5 * t_5)));
	} else {
		tmp = (t_2 + 0.0625) - t_2;
	}
	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(0.125 * Float64(Float64(alpha + beta) / i))
	t_3 = Float64(i + Float64(alpha + beta))
	t_4 = Float64(i * t_3)
	t_5 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(i * Float64(Float64(fma(i, t_3, Float64(alpha * beta)) / fma(t_5, t_5, -1.0)) * Float64(t_3 / Float64(t_5 * t_5))));
	else
		tmp = Float64(Float64(t_2 + 0.0625) - t_2);
	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[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(i * N[(N[(N[(i * t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

   3. Simplified 99.3%

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

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

   1. Initial program 0.0%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in i around inf 70.8%

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

      1. sub-neg 70.8%

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

      2. +-commutative 70.8%

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

      3. fma-define 70.8%

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

      4. distribute-lft-out 70.8%

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

      5. associate-*r/ 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. unsub-neg 70.8%

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

      2. fma-undefine 70.8%

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

      3. associate-/l* 70.8%

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

      4. associate-*r* 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-/l* 70.8%

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

      7. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 80.0%

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

Alternative 3: 81.2% accurate, 0.1× speedup

Math

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

FPCore

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

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 = 0.125 * ((alpha + beta) / i);
	double t_3 = t_1 + -1.0;
	double t_4 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_3) <= ((double) INFINITY)) {
		tmp = (pow(i, 2.0) * (pow((i + beta), 2.0) / fma(i, ((i + beta) * 4.0), pow(beta, 2.0)))) / t_3;
	} else {
		tmp = (t_2 + 0.0625) - t_2;
	}
	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(0.125 * Float64(Float64(alpha + beta) / i))
	t_3 = Float64(t_1 + -1.0)
	t_4 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / t_3) <= Inf)
		tmp = Float64(Float64((i ^ 2.0) * Float64((Float64(i + beta) ^ 2.0) / fma(i, Float64(Float64(i + beta) * 4.0), (beta ^ 2.0)))) / t_3);
	else
		tmp = Float64(Float64(t_2 + 0.0625) - t_2);
	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[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$4 * N[(t$95$4 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], Infinity], N[(N[(N[Power[i, 2.0], $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[(i * N[(N[(i + beta), $MachinePrecision] * 4.0), $MachinePrecision] + N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(t$95$2 + 0.0625), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

   3. Taylor expanded in alpha around 0 46.1%

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

      1. associate-/l* 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in i around 0 91.2%

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

      1. fma-define 91.2%

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

      2. distribute-lft-out 91.2%

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

   8. Simplified 91.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in i around inf 70.8%

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

      1. sub-neg 70.8%

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

      2. +-commutative 70.8%

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

      3. fma-define 70.8%

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

      4. distribute-lft-out 70.8%

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

      5. associate-*r/ 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. unsub-neg 70.8%

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

      2. fma-undefine 70.8%

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

      3. associate-/l* 70.8%

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

      4. associate-*r* 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-/l* 70.8%

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

      7. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 77.3%

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

Alternative 4: 81.2% accurate, 0.1× speedup

Math

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

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 (* 0.125 (/ (+ alpha beta) i)))
        (t_4 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_4 (+ t_4 (* alpha beta))) t_1) t_2) INFINITY)
     (/
      (*
       (pow i 2.0)
       (/
        (pow (+ i beta) 2.0)
        (+ (pow beta 2.0) (* i (+ (* beta 4.0) (* i 4.0))))))
      t_2)
     (- (+ t_3 0.0625) t_3))))

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

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 + -1.0;
	double t_3 = 0.125 * ((alpha + beta) / i);
	double t_4 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow(i, 2.0) * (Math.pow((i + beta), 2.0) / (Math.pow(beta, 2.0) + (i * ((beta * 4.0) + (i * 4.0)))))) / t_2;
	} else {
		tmp = (t_3 + 0.0625) - t_3;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = t_1 + -1.0
	t_3 = 0.125 * ((alpha + beta) / i)
	t_4 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= math.inf:
		tmp = (math.pow(i, 2.0) * (math.pow((i + beta), 2.0) / (math.pow(beta, 2.0) + (i * ((beta * 4.0) + (i * 4.0)))))) / t_2
	else:
		tmp = (t_3 + 0.0625) - t_3
	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(0.125 * Float64(Float64(alpha + beta) / i))
	t_4 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_4 * Float64(t_4 + Float64(alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64((i ^ 2.0) * Float64((Float64(i + beta) ^ 2.0) / Float64((beta ^ 2.0) + Float64(i * Float64(Float64(beta * 4.0) + Float64(i * 4.0)))))) / t_2);
	else
		tmp = Float64(Float64(t_3 + 0.0625) - t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = t_1 + -1.0;
	t_3 = 0.125 * ((alpha + beta) / i);
	t_4 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_4 * (t_4 + (alpha * beta))) / t_1) / t_2) <= Inf)
		tmp = ((i ^ 2.0) * (((i + beta) ^ 2.0) / ((beta ^ 2.0) + (i * ((beta * 4.0) + (i * 4.0)))))) / t_2;
	else
		tmp = (t_3 + 0.0625) - t_3;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

   3. Taylor expanded in alpha around 0 46.1%

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

      1. associate-/l* 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in i around 0 91.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in i around inf 70.8%

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

      1. sub-neg 70.8%

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

      2. +-commutative 70.8%

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

      3. fma-define 70.8%

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

      4. distribute-lft-out 70.8%

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

      5. associate-*r/ 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. unsub-neg 70.8%

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

      2. fma-undefine 70.8%

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

      3. associate-/l* 70.8%

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

      4. associate-*r* 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-/l* 70.8%

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

      7. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 77.3%

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

Alternative 5: 81.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * (i + (alpha + beta));
	double t_1 = (alpha + beta) + (i * 2.0);
	double t_2 = t_1 * t_1;
	double t_3 = t_2 + -1.0;
	double t_4 = 0.125 * ((alpha + beta) / i);
	double tmp;
	if ((((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow(i, 2.0) * (Math.pow((i + beta), 2.0) / Math.pow((beta + (i * 2.0)), 2.0))) / t_3;
	} else {
		tmp = (t_4 + 0.0625) - t_4;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = i * (i + (alpha + beta))
	t_1 = (alpha + beta) + (i * 2.0)
	t_2 = t_1 * t_1
	t_3 = t_2 + -1.0
	t_4 = 0.125 * ((alpha + beta) / i)
	tmp = 0
	if (((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= math.inf:
		tmp = (math.pow(i, 2.0) * (math.pow((i + beta), 2.0) / math.pow((beta + (i * 2.0)), 2.0))) / t_3
	else:
		tmp = (t_4 + 0.0625) - t_4
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = i * (i + (alpha + beta));
	t_1 = (alpha + beta) + (i * 2.0);
	t_2 = t_1 * t_1;
	t_3 = t_2 + -1.0;
	t_4 = 0.125 * ((alpha + beta) / i);
	tmp = 0.0;
	if ((((t_0 * (t_0 + (alpha * beta))) / t_2) / t_3) <= Inf)
		tmp = ((i ^ 2.0) * (((i + beta) ^ 2.0) / ((beta + (i * 2.0)) ^ 2.0))) / t_3;
	else
		tmp = (t_4 + 0.0625) - t_4;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

   3. Taylor expanded in alpha around 0 46.1%

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

      1. associate-/l* 91.2%

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

   5. Simplified 91.2%

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

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

   1. Initial program 0.0%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in i around inf 70.8%

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

      1. sub-neg 70.8%

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

      2. +-commutative 70.8%

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

      3. fma-define 70.8%

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

      4. distribute-lft-out 70.8%

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

      5. associate-*r/ 70.8%

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

   7. Applied egg-rr 70.8%

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

      1. unsub-neg 70.8%

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

      2. fma-undefine 70.8%

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

      3. associate-/l* 70.8%

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

      4. associate-*r* 70.8%

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

      5. metadata-eval 70.8%

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

      6. associate-/l* 70.8%

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

      7. associate-/l* 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 77.3%

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

Alternative 6: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t\_2 \cdot \left(t\_2 + \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ t_4 := 0.125 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{if}\;t\_3 \leq 0.1:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(t\_4 + 0.0625\right) - t\_4\\ \end{array} \end{array} \]

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)))
        (t_4 (* 0.125 (/ (+ alpha beta) i))))
   (if (<= t_3 0.1) t_3 (- (+ t_4 0.0625) t_4))))

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 t_4 = 0.125 * ((alpha + beta) / i);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (t_4 + 0.0625) - t_4;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + (-1.0d0))
    t_4 = 0.125d0 * ((alpha + beta) / i)
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (t_4 + 0.0625d0) - t_4
    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 t_4 = 0.125 * ((alpha + beta) / i);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (t_4 + 0.0625) - t_4;
	}
	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)
	t_4 = 0.125 * ((alpha + beta) / i)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (t_4 + 0.0625) - t_4
	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))
	t_4 = Float64(0.125 * Float64(Float64(alpha + beta) / i))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(t_4 + 0.0625) - t_4);
	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);
	t_4 = 0.125 * ((alpha + beta) / i);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (t_4 + 0.0625) - t_4;
	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]}, Block[{t$95$4 = N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(t$95$4 + 0.0625), $MachinePrecision] - t$95$4), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

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

   1. Initial program 0.6%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in i around inf 74.6%

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

      1. sub-neg 74.6%

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

      2. +-commutative 74.6%

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

      3. fma-define 74.6%

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

      4. distribute-lft-out 74.6%

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

      5. associate-*r/ 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. unsub-neg 74.6%

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

      2. fma-undefine 74.6%

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

      3. associate-/l* 74.6%

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

      4. associate-*r* 74.6%

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

      5. metadata-eval 74.6%

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

      6. associate-/l* 74.6%

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

      7. associate-/l* 74.6%

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

   9. Simplified 74.6%

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

4. Final simplification 78.6%

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

Alternative 7: 75.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.6 \cdot 10^{+165}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{i + \alpha}{\beta} \cdot \frac{1}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.6e+165) 0.0625 (* i (* (/ (+ i alpha) beta) (/ 1.0 beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.6e+165) {
		tmp = 0.0625;
	} else {
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 9.6d+165) then
        tmp = 0.0625d0
    else
        tmp = i * (((i + alpha) / beta) * (1.0d0 / beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.6e+165) {
		tmp = 0.0625;
	} else {
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.6e+165:
		tmp = 0.0625
	else:
		tmp = i * (((i + alpha) / beta) * (1.0 / beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.6e+165)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(Float64(i + alpha) / beta) * Float64(1.0 / beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.6e+165)
		tmp = 0.0625;
	else
		tmp = i * (((i + alpha) / beta) * (1.0 / beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 9.6e+165], 0.0625, N[(i * N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.60000000000000002e165

   1. Initial program 20.0%

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

   3. Simplified 40.4%

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

   5. Taylor expanded in i around inf 79.0%

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

   if 9.60000000000000002e165 < beta

   1. Initial program 0.0%

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

   3. Simplified 8.9%

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

   5. Taylor expanded in beta around inf 19.6%

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

   6. Taylor expanded in beta around inf 27.9%

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

   7. Taylor expanded in beta around inf 48.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.6 \cdot 10^{+165}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \left(\frac{i + \alpha}{\beta} \cdot \frac{1}{\beta}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+228}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 9.2e+228) 0.0625 0.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.2e+228) {
		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 <= 9.2d+228) 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 <= 9.2e+228) {
		tmp = 0.0625;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.2e+228:
		tmp = 0.0625
	else:
		tmp = 0.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.2e+228)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.2e+228)
		tmp = 0.0625;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 9.2e+228], 0.0625, 0.0]

Derivation

1. Split input into 2 regimes

2. if beta < 9.20000000000000052e228

   1. Initial program 18.4%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in i around inf 75.7%

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

   if 9.20000000000000052e228 < beta

   1. Initial program 0.0%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in i around inf 42.0%

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

   6. Taylor expanded in i around 0 31.0%

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

      1. div-sub 31.0%

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

      2. distribute-lft-in 31.0%

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

      3. associate-*r* 31.2%

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

      4. metadata-eval 31.2%

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

      5. +-inverses 31.2%

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

   8. Simplified 31.2%

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

Alternative 9: 9.9% 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 16.4%

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

3. Simplified 34.7%

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

5. Taylor expanded in i around inf 75.1%

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

6. Taylor expanded in i around 0 9.4%

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

   1. div-sub 9.4%

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

   2. distribute-lft-in 9.4%

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

   3. associate-*r* 9.4%

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

   4. metadata-eval 9.4%

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

   5. +-inverses 9.4%

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

8. Simplified 9.4%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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)))