Octave 3.8, jcobi/4

Average Accuracy: 15.1% → 96.6%

Time: 29.2s

Precision: binary64

Cost: 27840

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

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

Math

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

↓

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

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)) (+ i beta)) (+ 1.0 (fma i 2.0 (+ beta alpha))))
  (/
   i
   (* (+ alpha (+ (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)) * (i + beta)) / (1.0 + fma(i, 2.0, (beta + alpha)))) * (i / ((alpha + (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(i + beta)) / Float64(1.0 + fma(i, 2.0, Float64(beta + alpha)))) * Float64(i / Float64(Float64(alpha + 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[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $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 15.0%

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

3. Simplified 34.3%

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

   [Start] 15.0	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   associate-/l* [=>] 34.3	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   unpow2 [=>] 34.3	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   *-commutative [=>] 34.3	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

4. Applied egg-rr 96.7%

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

   [Start] 34.3	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   add-sqr-sqrt [=>] 34.3	\[ \frac{\color{blue}{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}} \cdot \sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   difference-of-sqr-1 [=>] 34.3	\[ \frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}} \cdot \sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
   times-frac [=>] 34.3	\[ \color{blue}{\frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\sqrt{\frac{i \cdot i}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

5. Simplified 96.6%

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

   [Start] 96.7	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   associate-/r/ [=>] 96.7	\[ \frac{\color{blue}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(\beta + i\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   +-commutative [=>] 96.7	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(\beta + i\right)}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
   associate-/l/ [=>] 96.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(\beta + i\right)}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \color{blue}{\frac{i}{\left(\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}} \]
   sub-neg [=>] 96.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(\beta + i\right)}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\left(\alpha + \color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \left(-1\right)\right)}\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]
   metadata-eval [=>] 96.6	\[ \frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(\beta + i\right)}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\left(\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{-1}\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]

6. Final simplification 96.6%

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

Alternatives

Alternative 1
Accuracy	96.6%
Cost	21440

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

Alternative 2
Accuracy	96.7%
Cost	21312

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

Alternative 3
Accuracy	85.1%
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+170}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	85.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+170}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	82.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+170}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	73.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+219}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	9.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023131 
(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)))