Octave 3.8, jcobi/4

Percentage Accurate: 15.5% → 94.4%

Time: 17.4s

Alternatives: 15

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 15.5% 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: 94.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. accelerator-lowering-fma.f64 44.4

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

6. Applied egg-rr 44.4%

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

   1. div-inv N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. *-lowering-*.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. +-commutative N/A

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

   15. accelerator-lowering-fma.f64 44.3

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

8. Applied egg-rr 44.3%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-+r+ N/A

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

   8. associate-*l/ N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

10. Applied egg-rr 92.5%

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

11. Final simplification 92.5%

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

Alternative 2: 94.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. accelerator-lowering-fma.f64 44.4

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

6. Applied egg-rr 44.4%

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

   1. div-inv N/A

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

   2. +-commutative N/A

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

   3. associate-+r+ N/A

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

   4. *-lowering-*.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. associate-+r+ N/A

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

   12. +-commutative N/A

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

   13. +-lowering-+.f64 N/A

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

   14. +-commutative N/A

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

   15. accelerator-lowering-fma.f64 44.3

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

8. Applied egg-rr 44.3%

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

   1. *-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. +-lowering-+.f64 N/A

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

   16. accelerator-lowering-fma.f64 92.5

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

10. Applied egg-rr 92.5%

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

11. Final simplification 92.5%

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

Alternative 3: 42.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \frac{i \cdot \frac{i + \left(\beta + \alpha\right)}{t\_0}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + 1\right)} \cdot \frac{\frac{\mathsf{fma}\left(i, \alpha + \left(i + \beta\right), \beta \cdot \alpha\right)}{t\_0}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. accelerator-lowering-fma.f64 44.4

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

6. Applied egg-rr 44.4%

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

7. Final simplification 44.4%

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

Alternative 4: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. accelerator-lowering-fma.f64 44.4

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

6. Applied egg-rr 44.4%

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

8. Applied egg-rr 44.0%

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

9. Final simplification 44.0%

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

Alternative 5: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. clear-num N/A

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

   2. clear-num N/A

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

   3. frac-times N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

6. Applied egg-rr 44.0%

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

7. Final simplification 44.0%

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

Alternative 6: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. frac-times N/A

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

   4. clear-num N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

6. Applied egg-rr 43.4%

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

7. Final simplification 43.4%

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

Alternative 7: 37.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

4. Applied egg-rr 39.3%

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

5. Final simplification 39.3%

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

Alternative 8: 37.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

4. Applied egg-rr 16.9%

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

6. Applied egg-rr 39.3%

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

7. Final simplification 39.3%

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

Alternative 9: 37.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 39.3%

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

5. Final simplification 39.3%

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

Alternative 10: 37.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 26.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

6. Applied egg-rr 39.2%

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

7. Final simplification 39.2%

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

Alternative 11: 37.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. frac-times N/A

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

   2. frac-times N/A

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

   3. associate-+r+ N/A

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

   4. associate-+r+ N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. difference-of-sqr--1 N/A

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

   8. associate-/r* N/A

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

6. Applied egg-rr 39.2%

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

7. Final simplification 39.2%

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

Alternative 12: 17.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathsf{fma}\left(i + \beta, \alpha, i \cdot \left(i + \beta\right)\right) \cdot \frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\mathsf{fma}\left(t\_0, t\_0, -1\right) \cdot \left(t\_0 \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 18.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-in N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 N/A

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

   11. +-lowering-+.f64 18.6

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

6. Applied egg-rr 18.6%

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

7. Final simplification 18.6%

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

Alternative 13: 17.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. associate-/l/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 18.6%

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

5. Final simplification 18.6%

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

Alternative 14: 15.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. accelerator-lowering-fma.f64 44.4

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

6. Applied egg-rr 44.4%

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

8. Applied egg-rr 16.1%

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

Alternative 15: 15.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. times-frac N/A

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

   2. difference-of-sqr-1 N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

4. Applied egg-rr 44.3%

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

   1. frac-times N/A

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

   2. frac-times N/A

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

   3. associate-+r+ N/A

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

   4. associate-+r+ N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. difference-of-sqr--1 N/A

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

   8. associate-/r* N/A

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

6. Applied egg-rr 16.1%

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

7. Final simplification 16.1%

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

Reproduce

herbie shell --seed 2024208 
(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)))