Average Error: 54.1 → 10.3
Time: 7.8s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t_2\\ t_4 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{t_3}{t_4} \cdot \frac{\mathsf{fma}\left(i, t_2, \alpha \cdot \beta\right)}{t_4}}{-1 + {t_4}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \end{array} \]
(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 beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (+ alpha (fma i 2.0 beta))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (/
      (* (/ t_3 t_4) (/ (fma i t_2 (* alpha beta)) t_4))
      (+ -1.0 (pow t_4 2.0)))
     (+
      (+ 0.0625 (* 0.0625 (/ (+ (* beta 2.0) (* alpha 2.0)) i)))
      (* (/ (+ alpha beta) i) -0.125)))))
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 + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = alpha + fma(i, 2.0, beta);
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((t_3 / t_4) * (fma(i, t_2, (alpha * beta)) / t_4)) / (-1.0 + pow(t_4, 2.0));
	} else {
		tmp = (0.0625 + (0.0625 * (((beta * 2.0) + (alpha * 2.0)) / i))) + (((alpha + beta) / i) * -0.125);
	}
	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(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(t_3 / t_4) * Float64(fma(i, t_2, Float64(alpha * beta)) / t_4)) / Float64(-1.0 + (t_4 ^ 2.0)));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(beta * 2.0) + Float64(alpha * 2.0)) / i))) + Float64(Float64(Float64(alpha + beta) / i) * -0.125));
	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[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] * N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(beta * 2.0), $MachinePrecision] + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]]]]]]]

\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 := \left(\alpha + \beta\right) + i \cdot 2\\
t_1 := t_0 \cdot t_0\\
t_2 := i + \left(\alpha + \beta\right)\\
t_3 := i \cdot t_2\\
t_4 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\
\mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\
\;\;\;\;\frac{\frac{t_3}{t_4} \cdot \frac{\mathsf{fma}\left(i, t_2, \alpha \cdot \beta\right)}{t_4}}{-1 + {t_4}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 35.2

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

    2. Simplified26.9

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

    3. Applied egg-rr0.2

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

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

    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. Simplified60.5

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

    3. Taylor expanded in i around inf 15.6

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

  4. Final simplification10.3

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

Reproduce

herbie shell --seed 2022210 
(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)))