Average Error: 53.5 → 12.8
Time: 6.4s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := \left(i + \alpha\right) + \beta\\ \mathbf{if}\;i \leq 2.3 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(i, t_1, \alpha \cdot \beta\right)}{t_0}}{{t_0}^{2} + -1} \cdot \left(t_1 \cdot \frac{i}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
(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 (+ alpha (fma i 2.0 beta))) (t_1 (+ (+ i alpha) beta)))
   (if (<= i 2.3e+123)
     (expm1
      (log1p
       (*
        (/ (/ (fma i t_1 (* alpha beta)) t_0) (+ (pow t_0 2.0) -1.0))
        (* t_1 (/ i t_0)))))
     0.0625)))
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 = alpha + fma(i, 2.0, beta);
	double t_1 = (i + alpha) + beta;
	double tmp;
	if (i <= 2.3e+123) {
		tmp = expm1(log1p((((fma(i, t_1, (alpha * beta)) / t_0) / (pow(t_0, 2.0) + -1.0)) * (t_1 * (i / t_0)))));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

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 = Float64(alpha + fma(i, 2.0, beta))
	t_1 = Float64(Float64(i + alpha) + beta)
	tmp = 0.0
	if (i <= 2.3e+123)
		tmp = expm1(log1p(Float64(Float64(Float64(fma(i, t_1, Float64(alpha * beta)) / t_0) / Float64((t_0 ^ 2.0) + -1.0)) * Float64(t_1 * Float64(i / t_0)))));
	else
		tmp = 0.0625;
	end
	return tmp
end

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[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i + alpha), $MachinePrecision] + beta), $MachinePrecision]}, If[LessEqual[i, 2.3e+123], N[(Exp[N[Log[1 + N[(N[(N[(N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], 0.0625]]]

\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 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\
t_1 := \left(i + \alpha\right) + \beta\\
\mathbf{if}\;i \leq 2.3 \cdot 10^{+123}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(i, t_1, \alpha \cdot \beta\right)}{t_0}}{{t_0}^{2} + -1} \cdot \left(t_1 \cdot \frac{i}{t_0}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;0.0625\\


\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 2.2999999999999999e123

    1. Initial program 38.6

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

    2. Simplified26.0

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

    3. Applied egg-rr14.5

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

    if 2.2999999999999999e123 < i

    1. Initial program 64.0

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

    2. Simplified63.8

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

    3. Taylor expanded in i around inf 11.6

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.3 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\mathsf{fma}\left(i, \left(i + \alpha\right) + \beta, \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1} \cdot \left(\left(\left(i + \alpha\right) + \beta\right) \cdot \frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))