Octave 3.8, jcobi/4

Percentage Accurate: 16.2% → 83.3%

Time: 16.6s

Alternatives: 10

Speedup: 53.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 6.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{i \cdot t\_0}{t\_1}}{t\_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_0, \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ alpha beta))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= i 6.8e+147)
     (*
      (/ (/ (* i t_0) t_1) (+ t_1 1.0))
      (/ (/ (fma i t_0 (* alpha beta)) t_1) (+ t_1 -1.0)))
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = i + (alpha + beta);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (i <= 6.8e+147) {
		tmp = (((i * t_0) / t_1) / (t_1 + 1.0)) * ((fma(i, t_0, (alpha * beta)) / t_1) / (t_1 + -1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(i + Float64(alpha + beta))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (i <= 6.8e+147)
		tmp = Float64(Float64(Float64(Float64(i * t_0) / t_1) / Float64(t_1 + 1.0)) * Float64(Float64(fma(i, t_0, Float64(alpha * beta)) / t_1) / Float64(t_1 + -1.0)));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 6.8e+147], N[(N[(N[(N[(i * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]

Derivation

1. Split input into 2 regimes

2. if i < 6.8e147

   1. Initial program 36.7%

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

      1. associate-/l/ 31.5%

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

   3. Simplified 31.5%

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

   6. Applied egg-rr 86.7%

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

   if 6.8e147 < i

   1. Initial program 0.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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* 0.1%

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

      3. +-commutative 0.1%

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

      4. +-commutative 0.1%

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

      5. +-commutative 0.1%

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

      6. associate-+l+ 0.1%

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

      7. +-commutative 0.1%

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

      8. associate-*l* 0.1%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in i around inf 83.4%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #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 45.1%

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

      1. associate-/l/ 38.5%

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

   3. Simplified 38.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   8. Simplified 99.7%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #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%

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

   3. Taylor expanded in i around inf 3.4%

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

      1. associate--l+ 3.4%

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

      2. distribute-lft-out 3.4%

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

   5. Simplified 3.4%

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

   6. Taylor expanded in i around inf 75.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   7. Taylor expanded in beta around inf 73.7%

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

      1. associate-*r/ 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
   11. Step-by-step derivation

       1. distribute-lft-out 73.7%

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

   12. Simplified 73.7%

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

4. Final simplification 83.2%

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

Alternative 3: 79.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #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 45.1%

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

      1. associate-/l/ 38.5%

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

   3. Simplified 38.5%

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

   6. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. fma-define 99.6%

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

      3. +-commutative 99.6%

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

      4. fma-define 99.6%

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

      5. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   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%

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

   3. Taylor expanded in i around inf 3.4%

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

      1. associate--l+ 3.4%

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

      2. distribute-lft-out 3.4%

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

   5. Simplified 3.4%

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

   6. Taylor expanded in i around inf 75.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   7. Taylor expanded in beta around inf 73.7%

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

      1. associate-*r/ 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
   11. Step-by-step derivation

       1. distribute-lft-out 73.7%

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

   12. Simplified 73.7%

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

4. Final simplification 83.2%

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

Alternative 4: 76.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #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 45.1%

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

   3. Taylor expanded in alpha around 0 39.4%

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

      1. associate-/l* 90.4%

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

   5. Simplified 90.4%

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

   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%

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

   3. Taylor expanded in i around inf 3.4%

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

      1. associate--l+ 3.4%

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

      2. distribute-lft-out 3.4%

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

   5. Simplified 3.4%

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

   6. Taylor expanded in i around inf 75.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   7. Taylor expanded in beta around inf 73.7%

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

      1. associate-*r/ 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in i around 0 73.7%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
   11. Step-by-step derivation

       1. distribute-lft-out 73.7%

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

   12. Simplified 73.7%

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

4. Final simplification 79.8%

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

Alternative 5: 76.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #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 0.7%

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

   3. Taylor expanded in i around inf 22.7%

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

      1. associate--l+ 22.7%

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

      2. distribute-lft-out 22.7%

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

   5. Simplified 22.7%

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

   6. Taylor expanded in i around inf 77.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   7. Taylor expanded in beta around inf 75.4%

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

      1. associate-*r/ 75.4%

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

   9. Simplified 75.4%

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

   10. Taylor expanded in i around 0 75.4%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
   11. Step-by-step derivation

       1. distribute-lft-out 75.4%

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

   12. Simplified 75.4%

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

4. Final simplification 79.2%

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

Alternative 6: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{0.0625 \cdot \left(i + \beta\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+194)
   (- (/ (* 0.0625 (+ i beta)) i) (* 0.0625 (/ (+ alpha beta) i)))
   (pow (/ i beta) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+194) {
		tmp = ((0.0625 * (i + beta)) / i) - (0.0625 * ((alpha + beta) / i));
	} else {
		tmp = pow((i / beta), 2.0);
	}
	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 <= 3.8d+194) then
        tmp = ((0.0625d0 * (i + beta)) / i) - (0.0625d0 * ((alpha + beta) / i))
    else
        tmp = (i / beta) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+194) {
		tmp = ((0.0625 * (i + beta)) / i) - (0.0625 * ((alpha + beta) / i));
	} else {
		tmp = Math.pow((i / beta), 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.8e+194:
		tmp = ((0.0625 * (i + beta)) / i) - (0.0625 * ((alpha + beta) / i))
	else:
		tmp = math.pow((i / beta), 2.0)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.8e+194)
		tmp = Float64(Float64(Float64(0.0625 * Float64(i + beta)) / i) - Float64(0.0625 * Float64(Float64(alpha + beta) / i)));
	else
		tmp = Float64(i / beta) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.8e+194)
		tmp = ((0.0625 * (i + beta)) / i) - (0.0625 * ((alpha + beta) / i));
	else
		tmp = (i / beta) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.8e+194], N[(N[(N[(0.0625 * N[(i + beta), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] - N[(0.0625 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(i / beta), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999999e194

   1. Initial program 18.9%

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

   3. Taylor expanded in i around inf 34.7%

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

      1. associate--l+ 34.7%

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

      2. distribute-lft-out 34.7%

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

   5. Simplified 34.7%

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

   6. Taylor expanded in i around inf 81.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   7. Taylor expanded in beta around inf 80.0%

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

      1. associate-*r/ 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in i around 0 80.0%

       \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
   11. Step-by-step derivation

       1. distribute-lft-out 80.0%

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

   12. Simplified 80.0%

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

   if 3.7999999999999999e194 < 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* 12.8%

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

      3. +-commutative 12.8%

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

      4. +-commutative 12.8%

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

      5. +-commutative 12.8%

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

      6. associate-+l+ 12.8%

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

      7. +-commutative 12.8%

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

      8. associate-*l* 12.8%

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

   3. Simplified 12.8%

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

   5. Taylor expanded in beta around inf 19.9%

      \[\leadsto i \cdot \color{blue}{\frac{\left(\alpha + \left(i + \left(\frac{i \cdot \left(\alpha + i\right)}{\beta} + \frac{{\left(\alpha + i\right)}^{2}}{\beta}\right)\right)\right) - \frac{\left(\alpha + i\right) \cdot \left(4 \cdot \alpha + 8 \cdot i\right)}{\beta}}{{\beta}^{2}}} \]

   6. Taylor expanded in alpha around 0 23.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\left(i + 2 \cdot \frac{{i}^{2}}{\beta}\right) - 8 \cdot \frac{{i}^{2}}{\beta}\right)}{{\beta}^{2}}} \]

   7. Taylor expanded in i around 0 23.7%

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

      1. unpow2 23.7%

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

      2. unpow2 23.7%

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

      3. times-frac 58.3%

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

      4. unpow1 58.3%

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

      5. pow-plus 58.3%

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

      6. metadata-eval 58.3%

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

   9. Simplified 58.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+194}:\\ \;\;\;\;\frac{0.0625 \cdot \left(i + \beta\right)}{i} - 0.0625 \cdot \frac{\alpha + \beta}{i}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{i}{\beta}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

3. Taylor expanded in i around inf 32.0%

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

   1. associate--l+ 32.0%

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

   2. distribute-lft-out 32.0%

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

5. Simplified 32.0%

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

6. Taylor expanded in i around inf 77.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

7. Taylor expanded in beta around inf 75.7%

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

   1. associate-*r/ 75.7%

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

9. Simplified 75.7%

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

10. Taylor expanded in i around 0 75.7%

    \[\leadsto \color{blue}{\frac{0.0625 \cdot \beta + 0.0625 \cdot i}{i}} - 0.0625 \cdot \frac{\alpha + \beta}{i} \]
11. Step-by-step derivation

    1. distribute-lft-out 75.7%

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

12. Simplified 75.7%

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

13. Final simplification 75.7%

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

Alternative 8: 74.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

3. Taylor expanded in i around inf 32.0%

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

   1. associate--l+ 32.0%

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

   2. distribute-lft-out 32.0%

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

5. Simplified 32.0%

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

6. Taylor expanded in i around inf 77.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

7. Taylor expanded in beta around inf 75.7%

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

   1. associate-*r/ 75.7%

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

9. Simplified 75.7%

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

10. Taylor expanded in alpha around 0 76.5%

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

11. Final simplification 76.5%

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

Alternative 9: 71.0% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.6%

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

   1. associate-/l/ 14.1%

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

   2. associate-/l* 16.9%

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

   3. +-commutative 16.9%

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

   4. +-commutative 16.9%

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

   5. +-commutative 16.9%

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

   6. associate-+l+ 16.9%

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

   7. +-commutative 16.9%

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

   8. associate-*l* 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in i around inf 73.7%

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

Alternative 10: 9.8% 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 16.6%

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

3. Taylor expanded in i around inf 32.0%

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

   1. associate--l+ 32.0%

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

   2. distribute-lft-out 32.0%

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

5. Simplified 32.0%

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

6. Taylor expanded in i around inf 77.2%

   \[\leadsto \color{blue}{\left(0.0625 + 0.25 \cdot \frac{0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)}{i}\right) - 0.0625 \cdot \frac{\alpha + \beta}{i}} \]

7. Taylor expanded in i around 0 7.0%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right) - 0.0625 \cdot \left(\alpha + \beta\right)}{i}} \]
8. Step-by-step derivation

   1. div-sub 7.0%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \left(0.5 \cdot \left(\alpha + \beta\right) - 0.25 \cdot \left(\alpha + \beta\right)\right)}{i} - \frac{0.0625 \cdot \left(\alpha + \beta\right)}{i}} \]

   2. distribute-rgt-out-- 7.0%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\left(\alpha + \beta\right) \cdot \left(0.5 - 0.25\right)\right)}}{i} - \frac{0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   3. metadata-eval 7.0%

      \[\leadsto \frac{0.25 \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{0.25}\right)}{i} - \frac{0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   4. *-commutative 7.0%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\left(0.25 \cdot \left(\alpha + \beta\right)\right)}}{i} - \frac{0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   5. associate-*r* 7.0%

      \[\leadsto \frac{\color{blue}{\left(0.25 \cdot 0.25\right) \cdot \left(\alpha + \beta\right)}}{i} - \frac{0.0625 \cdot \left(\alpha + \beta\right)}{i} \]

   6. metadata-eval 7.0%

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

   7. associate-*r/ 7.0%

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

   8. associate-*r/ 7.0%

      \[\leadsto 0.0625 \cdot \frac{\alpha + \beta}{i} - \color{blue}{0.0625 \cdot \frac{\alpha + \beta}{i}} \]

   9. +-inverses 7.0%

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

9. Simplified 7.0%

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

Reproduce

herbie shell --seed 2024160 
(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)))