Octave 3.8, jcobi/4

Average Error: 53.8 → 0.3

Time: 21.3s

Precision: binary64

Cost: 28352

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

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

Math

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

↓

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

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
 (let* ((t_0 (fma i 2.0 (+ alpha beta))))
   (/
    (* (+ i alpha) (/ i (* (+ t_0 1.0) (/ t_0 (+ (+ i alpha) beta)))))
    (* (+ t_0 -1.0) (/ t_0 (+ 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) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	return ((i + alpha) * (i / ((t_0 + 1.0) * (t_0 / ((i + alpha) + beta))))) / ((t_0 + -1.0) * (t_0 / (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)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	return Float64(Float64(Float64(i + alpha) * Float64(i / Float64(Float64(t_0 + 1.0) * Float64(t_0 / Float64(Float64(i + alpha) + beta))))) / Float64(Float64(t_0 + -1.0) * Float64(t_0 / 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_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(i + alpha), $MachinePrecision] * N[(i / N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[(t$95$0 / N[(N[(i + alpha), $MachinePrecision] + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + -1.0), $MachinePrecision] * N[(t$95$0 / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.8

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

2. Taylor expanded in beta around 0 53.8

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

3. Simplified 53.8

   \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \color{blue}{\left(\left(i + \alpha\right) \cdot \left(\beta + i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   Proof
   (*.f64 (+.f64 i alpha) (+.f64 beta i)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 beta (+.f64 i alpha)) (*.f64 i (+.f64 i alpha)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.3

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

5. Applied egg-rr 0.3

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

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.2
Cost	22592

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

Alternative 2
Error	0.4
Cost	15424

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

Alternative 3
Error	8.7
Cost	15172

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\\ \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;\frac{i}{\frac{\beta + i \cdot 2}{\frac{i + \beta}{\beta + \mathsf{fma}\left(i, 2, -1\right)}}} \cdot \frac{i \cdot 0.5 + \left(\alpha + \beta\right) \cdot 0.25}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{t_0} \cdot \frac{i}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternative 4
Error	2.0
Cost	15168

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

Alternative 5
Error	8.9
Cost	14276

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i}{\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta}{i + \alpha}}}{\beta}\\ \end{array} \]

Alternative 7
Error	9.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	10.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 9
Error	10.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.816745240177177 \cdot 10^{+176}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 10
Error	16.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.7551182699050205 \cdot 10^{+230}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Error	18.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022295 
(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)))