Octave 3.8, jcobi/4

Average Accuracy: 15.4% → 96.7%

Time: 31.0s

Precision: binary64

Cost: 21440

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

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

Math

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

↓

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

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

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

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

Derivation

1. Initial program 15.4%

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

2. Applied egg-rr 40.4%

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

   [Start] 15.4	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   times-frac [=>] 37.4	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   difference-of-sqr-1 [=>] 37.4	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
   times-frac [=>] 40.3	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

3. Taylor expanded in alpha around 0 37.0%

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

4. Simplified 96.7%

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

   [Start] 37.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \left(\beta + i\right)}{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)} \]
   times-frac [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\left(\frac{i}{\left(\beta + 2 \cdot i\right) - 1} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right)} \]
   sub-neg [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{i}{\color{blue}{\left(\beta + 2 \cdot i\right) + \left(-1\right)}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \]
   *-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{i}{\left(\beta + \color{blue}{i \cdot 2}\right) + \left(-1\right)} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \]
   metadata-eval [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + \color{blue}{-1}} \cdot \frac{\beta + i}{\beta + 2 \cdot i}\right) \]
   *-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{\beta + i}{\beta + \color{blue}{i \cdot 2}}\right) \]

5. Applied egg-rr 96.7%

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

   [Start] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \frac{\beta + i}{\beta + i \cdot 2}\right) \]
   *-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\left(\frac{\beta + i}{\beta + i \cdot 2} \cdot \frac{i}{\left(\beta + i \cdot 2\right) + -1}\right)} \]
   clear-num [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \left(\frac{\beta + i}{\beta + i \cdot 2} \cdot \color{blue}{\frac{1}{\frac{\left(\beta + i \cdot 2\right) + -1}{i}}}\right) \]
   un-div-inv [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \color{blue}{\frac{\frac{\beta + i}{\beta + i \cdot 2}}{\frac{\left(\beta + i \cdot 2\right) + -1}{i}}} \]
   +-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{\color{blue}{i + \beta}}{\beta + i \cdot 2}}{\frac{\left(\beta + i \cdot 2\right) + -1}{i}} \]
   +-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\color{blue}{i \cdot 2 + \beta}}}{\frac{\left(\beta + i \cdot 2\right) + -1}{i}} \]
   fma-def [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}}{\frac{\left(\beta + i \cdot 2\right) + -1}{i}} \]
   associate-+l+ [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\color{blue}{\beta + \left(i \cdot 2 + -1\right)}}{i}} \]
   fma-def [=>] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\beta + \color{blue}{\mathsf{fma}\left(i, 2, -1\right)}}{i}} \]

6. Taylor expanded in alpha around 0 96.7%

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

7. Simplified 96.7%

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

   [Start] 96.7	\[ \frac{\frac{i}{\frac{\beta + 2 \cdot i}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{i}} \]
   *-commutative [=>] 96.7	\[ \frac{\frac{i}{\frac{\beta + \color{blue}{i \cdot 2}}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\beta + \mathsf{fma}\left(i, 2, -1\right)}{i}} \]

8. Final simplification 96.7%

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

Alternatives

Alternative 1
Accuracy	85.4%
Cost	8900

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.4 \cdot 10^{+179}:\\ \;\;\;\;\left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right) \cdot \frac{i \cdot 0.5 + \left(\beta + \alpha\right) \cdot 0.25}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2
Accuracy	96.7%
Cost	8896

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \frac{\frac{i}{\frac{t_0}{i + \beta}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \left(\frac{i}{t_0 + -1} \cdot \frac{i + \beta}{t_0}\right) \end{array} \]

Alternative 3
Accuracy	84.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+180}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	73.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+235}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \]

Alternative 5
Accuracy	75.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+235}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	82.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{+179}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	70.8%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023147 
(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)))