Octave 3.8, jcobi/4

Average Error: 15.1% → 96.8%

Time: 40.7s

Precision: binary64

Cost: 27712.00

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

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

Math

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

↓

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

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
 (/
  (*
   (/ (/ i (fma i 2.0 beta)) (/ (+ 1.0 (fma i 2.0 (+ beta alpha))) (+ i beta)))
   (/ i (+ (fma i 2.0 beta) -1.0)))
  (/ (fma i 2.0 beta) (+ i beta))))

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) {
	return (((i / fma(i, 2.0, beta)) / ((1.0 + fma(i, 2.0, (beta + alpha))) / (i + beta))) * (i / (fma(i, 2.0, beta) + -1.0))) / (fma(i, 2.0, beta) / (i + beta));
}

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)
	return Float64(Float64(Float64(Float64(i / fma(i, 2.0, beta)) / Float64(Float64(1.0 + fma(i, 2.0, Float64(beta + alpha))) / Float64(i + beta))) * Float64(i / Float64(fma(i, 2.0, beta) + -1.0))) / Float64(fma(i, 2.0, beta) / Float64(i + beta)))
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_] := N[(N[(N[(N[(i / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.1

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

2. Taylor expanded in alpha around 0 14.9

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

3. Simplified 34.1

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

   [Start] 14.9	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 34.1	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 34.1	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [=>] 34.1	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 96.8

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

5. Taylor expanded in alpha around 0 37.4

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

6. Simplified 96.8

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

   [Start] 37.4	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \left(\beta + i\right)}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
   +-commutative [<=] 37.4	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \color{blue}{\left(i + \beta\right)}}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
   times-frac [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \color{blue}{\left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right)} \]
   sub-neg [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\color{blue}{\left(\beta + 2 \cdot i\right) + \left(-1\right)}} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   +-commutative [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\color{blue}{\left(2 \cdot i + \beta\right)} + \left(-1\right)} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   *-commutative [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\left(\color{blue}{i \cdot 2} + \beta\right) + \left(-1\right)} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   fma-udef [<=] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \left(-1\right)} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   metadata-eval [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{-1}} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   +-commutative [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{i + \beta}{\beta + 2 \cdot i}\right) \]
   +-commutative [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{-1 + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{2 \cdot i + \beta}}\right) \]
   *-commutative [=>] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{-1 + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{i \cdot 2} + \beta}\right) \]
   fma-udef [<=] 96.8	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{-1 + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}\right) \]

7. Applied egg-rr 96.8

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

8. Final simplification 96.8

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

Alternatives

Alternative 1
Error	96.8%
Cost	27712.00

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

Alternative 2
Error	90.0%
Cost	27396.00

\[\begin{array}{l} \mathbf{if}\;i \leq 1.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \beta\right)\right)}^{2}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}}{1 + \left(\mathsf{fma}\left(i, 2, \beta\right) + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \end{array} \]

Alternative 3
Error	84.5%
Cost	14544.00

\[\begin{array}{l} t_0 := \left(\beta + \alpha\right) \cdot \frac{0.125}{i}\\ \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+128}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 5.3 \cdot 10^{+143}:\\ \;\;\;\;\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+162}:\\ \;\;\;\;\frac{{\left(0.0625 + t_0\right)}^{2} + t_0 \cdot \left(\left(\beta + \alpha\right) \cdot \frac{-0.125}{i}\right)}{0.0625 + \left(t_0 + t_0\right)}\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+206}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + -1} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot {\beta}^{-1}}{\beta \cdot \frac{1}{i + \alpha}}\\ \end{array} \]

Alternative 4
Error	85.1%
Cost	14532.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i}{t_0}\\ \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+128}:\\ \;\;\;\;\left(t_1 \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	85.0%
Cost	7748.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+128}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot {\beta}^{-1}}{\beta \cdot \frac{1}{i + \alpha}}\\ \end{array} \]

Alternative 6
Error	84.9%
Cost	7300.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+128}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot {\beta}^{-1}}{\beta \cdot \frac{1}{i + \alpha}}\\ \end{array} \]

Alternative 7
Error	84.9%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+128}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	77.1%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.3 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	83.1%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	73.5%
Cost	196.00

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+214}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	10.1%
Cost	64.00

\[0 \]

Reproduce

herbie shell --seed 2023093 
(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)))