Octave 3.8, jcobi/4

Average Error: 53.9 → 9.1

Time: 25.0s

Precision: binary64

Cost: 15820

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

\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := 1 + t_0\\ t_2 := \frac{i + \alpha}{t_1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ t_3 := \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{t_0 + -1}\\ t_4 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\left(\beta + i\right) + \alpha\right) \cdot \frac{i}{t_4}}{t_1} \cdot t_3\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+152}:\\ \;\;\;\;t_3 \cdot \frac{\frac{i}{\frac{t_4}{\beta + i}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* 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)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ beta alpha)))
        (t_1 (+ 1.0 t_0))
        (t_2 (* (/ (+ i alpha) t_1) (/ i (+ (+ beta alpha) (fma i 2.0 -1.0)))))
        (t_3
         (/
          (+ (* 0.5 (+ beta (+ i alpha))) (* -0.25 (+ beta alpha)))
          (+ t_0 -1.0)))
        (t_4 (+ beta (* i 2.0))))
   (if (<= beta 2.5e+111)
     (* (/ (* (+ (+ beta i) alpha) (/ i t_4)) t_1) t_3)
     (if (<= beta 3.1e+132)
       t_2
       (if (<= beta 1.2e+152) (* t_3 (/ (/ i (/ t_4 (+ beta i))) t_1)) t_2)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (beta + alpha));
	double t_1 = 1.0 + t_0;
	double t_2 = ((i + alpha) / t_1) * (i / ((beta + alpha) + fma(i, 2.0, -1.0)));
	double t_3 = ((0.5 * (beta + (i + alpha))) + (-0.25 * (beta + alpha))) / (t_0 + -1.0);
	double t_4 = beta + (i * 2.0);
	double tmp;
	if (beta <= 2.5e+111) {
		tmp = ((((beta + i) + alpha) * (i / t_4)) / t_1) * t_3;
	} else if (beta <= 3.1e+132) {
		tmp = t_2;
	} else if (beta <= 1.2e+152) {
		tmp = t_3 * ((i / (t_4 / (beta + i))) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(beta + alpha))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(Float64(Float64(i + alpha) / t_1) * Float64(i / Float64(Float64(beta + alpha) + fma(i, 2.0, -1.0))))
	t_3 = Float64(Float64(Float64(0.5 * Float64(beta + Float64(i + alpha))) + Float64(-0.25 * Float64(beta + alpha))) / Float64(t_0 + -1.0))
	t_4 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 2.5e+111)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + i) + alpha) * Float64(i / t_4)) / t_1) * t_3);
	elseif (beta <= 3.1e+132)
		tmp = t_2;
	elseif (beta <= 1.2e+152)
		tmp = Float64(t_3 * Float64(Float64(i / Float64(t_4 / Float64(beta + i))) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(i + alpha), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(i / N[(N[(beta + alpha), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(0.5 * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.5e+111], N[(N[(N[(N[(N[(beta + i), $MachinePrecision] + alpha), $MachinePrecision] * N[(i / t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[beta, 3.1e+132], t$95$2, If[LessEqual[beta, 1.2e+152], N[(t$95$3 * N[(N[(i / N[(t$95$4 / N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.4999999999999998e111

   1. Initial program 48.2

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

   2. Applied egg-rr 31.9

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

   3. Taylor expanded in i around inf 3.7

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\beta + \alpha\right) + 0.5 \cdot i\right) - 0.25 \cdot \left(\beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   4. Simplified 3.7

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (+.f64 (*.f64 1/2 (+.f64 (+.f64 alpha i) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 i alpha)) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite<= associate-+r+_binary64 (+.f64 i (+.f64 alpha beta)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 i (Rewrite<= +-commutative_binary64 (+.f64 beta alpha)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 beta alpha) i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (Rewrite<= *-commutative_binary64 (*.f64 -1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/4)) (+.f64 beta alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 3.9

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\alpha + \left(i + \beta\right)} \cdot \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right)}\right)} - 1\right)} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   6. Simplified 3.7

      \[\leadsto \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\left(\beta + i\right) + \alpha\right)}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (/.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 beta alpha))) (+.f64 (+.f64 beta i) alpha)) (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 i (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta)))) (+.f64 (+.f64 beta i) alpha)) (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (Rewrite<= +-commutative_binary64 (+.f64 alpha (+.f64 beta i)))) (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 i beta)))) (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta))))) (+.f64 1 (fma.f64 i 2 (+.f64 beta alpha)))): 3 points increase in error, 1 points decrease in error
      (/.f64 (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta)))) (+.f64 1 (fma.f64 i 2 (Rewrite=> +-commutative_binary64 (+.f64 alpha beta))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 i (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta)))) (Rewrite<= +-commutative_binary64 (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 i (*.f64 (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta))) (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)))): 2 points increase in error, 1 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 i (*.f64 (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta))) (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1)))))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 i (*.f64 (/.f64 (fma.f64 i 2 (+.f64 alpha beta)) (+.f64 alpha (+.f64 i beta))) (+.f64 (fma.f64 i 2 (+.f64 alpha beta)) 1))))) 1)): 47 points increase in error, 44 points decrease in error

   7. Taylor expanded in alpha around 0 3.8

      \[\leadsto \frac{\color{blue}{\frac{i}{\beta + 2 \cdot i}} \cdot \left(\left(\beta + i\right) + \alpha\right)}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   if 2.4999999999999998e111 < beta < 3.0999999999999998e132 or 1.2e152 < beta

   1. Initial program 63.4

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

   2. Taylor expanded in beta around inf 48.8

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

   3. Applied egg-rr 16.8

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

   if 3.0999999999999998e132 < beta < 1.2e152

   1. Initial program 63.1

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

   2. Applied egg-rr 32.8

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

   3. Taylor expanded in i around inf 27.9

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{\left(0.5 \cdot \left(\beta + \alpha\right) + 0.5 \cdot i\right) - 0.25 \cdot \left(\beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   4. Simplified 27.9

      \[\leadsto \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (+.f64 (*.f64 1/2 (+.f64 (+.f64 alpha i) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 i alpha)) beta)) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite<= associate-+r+_binary64 (+.f64 i (+.f64 alpha beta)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (+.f64 i (Rewrite<= +-commutative_binary64 (+.f64 beta alpha)))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 beta alpha) i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i))) (*.f64 (+.f64 beta alpha) -1/4)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (Rewrite<= *-commutative_binary64 (*.f64 -1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 (Rewrite<= metadata-eval (neg.f64 1/4)) (+.f64 beta alpha))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (*.f64 1/2 (+.f64 beta alpha)) (*.f64 1/2 i)) (*.f64 1/4 (+.f64 beta alpha)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in alpha around 0 27.9

      \[\leadsto \frac{\frac{i}{\color{blue}{\frac{\beta + 2 \cdot i}{\beta + i}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   6. Simplified 27.9

      \[\leadsto \frac{\frac{i}{\color{blue}{\frac{\beta + i \cdot 2}{\beta + i}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{0.5 \cdot \left(\left(\alpha + i\right) + \beta\right) + \left(\beta + \alpha\right) \cdot -0.25}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof
      (/.f64 (+.f64 beta (*.f64 i 2)) (+.f64 beta i)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))) (+.f64 beta i)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(\left(\beta + i\right) + \alpha\right) \cdot \frac{i}{\beta + i \cdot 2}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{i + \alpha}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	21444

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := 1 + t_0\\ t_2 := t_0 + -1\\ \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\left(\beta + i\right) + \alpha\right) \cdot \frac{i}{\beta + i \cdot 2}}{t_1} \cdot \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \left(\beta + \alpha\right)}}}{t_1} \cdot \frac{i + \alpha}{t_2}\\ \end{array} \]

Alternative 2
Error	9.1
Cost	15820

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := 1 + t_0\\ t_2 := \frac{i + \alpha}{t_1} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ t_3 := \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{t_0 + -1} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{t_1}\\ \mathbf{if}\;\beta \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	9.1
Cost	14540

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i + \alpha}{1 + t_0} \cdot \frac{i}{\left(\beta + \alpha\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ t_2 := \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{t_0 + -1} \cdot \left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(1 + i \cdot 2\right)}\right)\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\beta \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	9.6
Cost	9420

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \left(\beta + \left(i + \alpha\right)\right) + -0.25 \cdot \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \left(\frac{i}{\beta + i \cdot 2} \cdot \frac{\beta + i}{\beta + \left(1 + i \cdot 2\right)}\right)\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2.1 \cdot 10^{+159}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	11.3
Cost	844

\[\begin{array}{l} t_0 := \frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.8 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+159}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	11.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	11.2
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 2 \cdot 10^{+160}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	11.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.1 \cdot 10^{+110}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+159}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 10
Error	18.7
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022328 
(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)))