Average Error: 24.6 → 1.9
Time: 18.4s
Precision: binary64
Cost: 28996
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -1:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]
(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -1.0)
     (/
      (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha))))
      2.0)
     (/
      (pow
       (pow
        (+
         1.0
         (/
          (+ alpha beta)
          (*
           (/ (+ alpha (fma 2.0 i beta)) (- beta alpha))
           (+ alpha (+ beta (fma 2.0 i 2.0))))))
        3.0)
       0.3333333333333333)
      2.0))))
double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}
double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -1.0) {
		tmp = ((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha)))) / 2.0;
	} else {
		tmp = pow(pow((1.0 + ((alpha + beta) / (((alpha + fma(2.0, i, beta)) / (beta - alpha)) * (alpha + (beta + fma(2.0, i, 2.0)))))), 3.0), 0.3333333333333333) / 2.0;
	}
	return tmp;
}
function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0)
end
function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(beta / alpha)) + Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))) / 2.0);
	else
		tmp = Float64(((Float64(1.0 + Float64(Float64(alpha + beta) / Float64(Float64(Float64(alpha + fma(2.0, i, beta)) / Float64(beta - alpha)) * Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))))) ^ 3.0) ^ 0.3333333333333333) / 2.0);
	end
	return tmp
end
code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(1.0 + N[(N[(alpha + beta), $MachinePrecision] / N[(N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / 2.0), $MachinePrecision]]]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}
\begin{array}{l}
t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\
\mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -1:\\
\;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left({\left(1 + \frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes
  2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -1

    1. Initial program 63.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
    2. Simplified53.8

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (+.f64 alpha (+.f64 beta (fma.f64 2 i 2))) (/.f64 (fma.f64 2 i (+.f64 alpha beta)) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (+.f64 alpha (+.f64 beta (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) 2)))) (/.f64 (fma.f64 2 i (+.f64 alpha beta)) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) (/.f64 (fma.f64 2 i (+.f64 alpha beta)) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (fma.f64 2 i (+.f64 alpha beta)) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) (+.f64 alpha beta))) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 alpha beta) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 beta alpha)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 alpha beta) (/.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (-.f64 beta alpha))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) 1) 2): 87 points increase in error, 0 points decrease in error
    3. Taylor expanded in alpha around inf 6.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}}{2} \]
    4. Taylor expanded in beta around 0 6.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}}{2} \]

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

    1. Initial program 13.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
    2. Simplified0.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (/.f64 (+.f64 alpha beta) (/.f64 (+.f64 alpha (+.f64 beta (*.f64 2 i))) (-.f64 beta alpha))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (/.f64 (+.f64 alpha beta) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 beta alpha))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 87 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 87 points decrease in error
      (/.f64 (+.f64 (/.f64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 89 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 0 points increase in error, 2 points decrease in error
    3. Applied egg-rr0.6

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
  3. Recombined 2 regimes into one program.
  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -1:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(1 + \frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error1.9
Cost3524
\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -1:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
Alternative 2
Error2.2
Cost3268
\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.5:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \end{array} \]
Alternative 3
Error11.4
Cost1348
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+152}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]
Alternative 4
Error14.1
Cost836
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]
Alternative 5
Error16.0
Cost708
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + 2}{\alpha}}{2}\\ \end{array} \]
Alternative 6
Error14.5
Cost708
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]
Alternative 7
Error13.2
Cost708
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]
Alternative 8
Error18.0
Cost196
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+73}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 9
Error24.9
Cost64
\[0.5 \]

Error

Reproduce

herbie shell --seed 2022329 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))