Average Error: 54.4 → 1.8
Time: 27.5s
Precision: binary64
Cost: 21572
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.077031944364273 \cdot 10^{+205}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{\beta + i \cdot 2}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
(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
 (if (<= alpha 4.077031944364273e+205)
   (/
    (* i (/ i (/ (+ beta (fma i 2.0 -1.0)) (/ (+ i beta) (+ beta (* i 2.0))))))
    (* (/ (fma i 2.0 beta) (+ i beta)) (+ (fma i 2.0 beta) (+ alpha 1.0))))
   (* (/ (+ alpha i) beta) (/ i beta))))
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 tmp;
	if (alpha <= 4.077031944364273e+205) {
		tmp = (i * (i / ((beta + fma(i, 2.0, -1.0)) / ((i + beta) / (beta + (i * 2.0)))))) / ((fma(i, 2.0, beta) / (i + beta)) * (fma(i, 2.0, beta) + (alpha + 1.0)));
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	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)
	tmp = 0.0
	if (alpha <= 4.077031944364273e+205)
		tmp = Float64(Float64(i * Float64(i / Float64(Float64(beta + fma(i, 2.0, -1.0)) / Float64(Float64(i + beta) / Float64(beta + Float64(i * 2.0)))))) / Float64(Float64(fma(i, 2.0, beta) / Float64(i + beta)) * Float64(fma(i, 2.0, beta) + Float64(alpha + 1.0))));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	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_] := If[LessEqual[alpha, 4.077031944364273e+205], N[(N[(i * N[(i / N[(N[(beta + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 2.0 + beta), $MachinePrecision] + N[(alpha + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $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}
\mathbf{if}\;\alpha \leq 4.077031944364273 \cdot 10^{+205}:\\
\;\;\;\;\frac{i \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{\beta + i \cdot 2}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.07703194436427327e205

    1. Initial program 54.2

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

    2. Taylor expanded in alpha around 0 54.3

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

    3. Simplified42.1

      \[\leadsto \frac{\color{blue}{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      Proof
      (/.f64 (*.f64 i i) (/.f64 (pow.f64 (+.f64 beta (*.f64 i 2)) 2) (pow.f64 (+.f64 beta i) 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) (/.f64 (pow.f64 (+.f64 beta (*.f64 i 2)) 2) (pow.f64 (+.f64 beta i) 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 i 2) (/.f64 (pow.f64 (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))) 2) (pow.f64 (+.f64 beta i) 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta i) 2)) (pow.f64 (+.f64 beta (*.f64 2 i)) 2))): 53 points increase in error, 3 points decrease in error

    4. Applied egg-rr1.5

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

    5. Applied egg-rr1.5

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

    6. Taylor expanded in alpha around 0 36.6

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

    7. Simplified1.5

      \[\leadsto \frac{\frac{i}{\color{blue}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{\beta + i \cdot 2}}}} \cdot i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(1 + \alpha\right)\right)} \]
      Proof
      (/.f64 (+.f64 beta (fma.f64 i 2 -1)) (/.f64 (+.f64 i beta) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 beta (fma.f64 i 2 (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 (+.f64 i beta) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 beta (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 i 2) 1))) (/.f64 (+.f64 i beta) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 beta (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) 1)) (/.f64 (+.f64 i beta) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 beta (*.f64 2 i)) 1)) (/.f64 (+.f64 i beta) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta i)) (+.f64 beta (*.f64 i 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (/.f64 (+.f64 beta i) (+.f64 beta (Rewrite<= *-commutative_binary64 (*.f64 2 i))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 (+.f64 beta (*.f64 2 i)) 1) (+.f64 beta (*.f64 2 i))) (+.f64 beta i))): 168 points increase in error, 4 points decrease in error

    if 4.07703194436427327e205 < alpha

    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. Taylor expanded in beta around inf 60.9

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

    3. Simplified17.3

      \[\leadsto \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]
      Proof
      (*.f64 (/.f64 (+.f64 i alpha) beta) (/.f64 i beta)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 i alpha) i) (*.f64 beta beta))): 94 points increase in error, 18 points decrease in error
      (/.f64 (*.f64 (+.f64 i alpha) i) (Rewrite<= unpow2_binary64 (pow.f64 beta 2))): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.077031944364273 \cdot 10^{+205}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{\frac{i + \beta}{\beta + i \cdot 2}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error1.7
Cost15300
\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.077031944364273 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + 1\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 2
Error10.2
Cost15176
\[\begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 8.946413973199257 \cdot 10^{+116}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.660564394231339 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(i + \beta\right)}^{2}}}}{-1 + t_0 \cdot t_0}\\ \mathbf{elif}\;\beta \leq 7.789084165389078 \cdot 10^{+191}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 3
Error10.3
Cost14724
\[\begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ t_1 := \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}\\ \mathbf{if}\;i \leq 9.73910914178214 \cdot 10^{+142}:\\ \;\;\;\;\frac{{t_1}^{2}}{-1 + t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \]
Alternative 4
Error9.9
Cost1476
\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.789084165389078 \cdot 10^{+191}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \alpha \cdot 2}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 5
Error9.9
Cost1092
\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.789084165389078 \cdot 10^{+191}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\alpha + \beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 6
Error10.0
Cost708
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4354504510993 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + i\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
Alternative 7
Error9.9
Cost708
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4354504510993 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 8
Error15.8
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.1111165729742165 \cdot 10^{+209}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha}}\\ \end{array} \]
Alternative 9
Error11.2
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4354504510993 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
Alternative 10
Error11.2
Cost580
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.4354504510993 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]
Alternative 11
Error16.6
Cost196
\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.8317915168290554 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 12
Error57.8
Cost64
\[0 \]

Error

Reproduce

herbie shell --seed 2022318 
(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)))