Octave 3.8, jcobi/2

Average Error: 38.26% → 2.31%

Time: 26.3s

Precision: binary64

Cost: 22468

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.5:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \alpha + \beta, 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.5)
     (+ (/ beta alpha) (* 0.5 (+ (* 4.0 (/ i alpha)) (* 2.0 (/ 1.0 alpha)))))
     (/
      (fma
       (/
        (* (- beta alpha) (/ 1.0 (+ alpha (+ beta (fma 2.0 i 2.0)))))
        (+ alpha (fma 2.0 i beta)))
       (+ alpha beta)
       1.0)
      2.0))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.5) {
		tmp = (beta / alpha) + (0.5 * ((4.0 * (i / alpha)) + (2.0 * (1.0 / alpha))));
	} else {
		tmp = fma((((beta - alpha) * (1.0 / (alpha + (beta + fma(2.0, i, 2.0))))) / (alpha + fma(2.0, i, beta))), (alpha + beta), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) + 2.0)) + 1.0) / 2.0)
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -0.5)
		tmp = Float64(Float64(beta / alpha) + Float64(0.5 * Float64(Float64(4.0 * Float64(i / alpha)) + Float64(2.0 * Float64(1.0 / alpha)))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(beta - alpha) * Float64(1.0 / Float64(alpha + Float64(beta + fma(2.0, i, 2.0))))) / Float64(alpha + fma(2.0, i, beta))), Float64(alpha + beta), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(beta / alpha), $MachinePrecision] + N[(0.5 * N[(N[(4.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + beta), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 96.54

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

   2. Simplified 96.59

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

      [Start] 96.54	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l/ [=>] 96.59	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      associate-+l+ [=>] 96.59	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right)} \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2} \]
      associate-+l+ [=>] 96.59	\[ \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)\right) \cdot \color{blue}{\left(\alpha + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

   3. Taylor expanded in alpha around inf 9.87

      \[\leadsto \frac{\color{blue}{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}}{2} \]

   4. Taylor expanded in beta around inf 9.87

      \[\leadsto \color{blue}{\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 20.62

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

   2. Simplified 14.91

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

      [Start] 20.62	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 0.02

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

   4. Simplified 0.02

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

      [Start] 0.02	\[ \frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)} \cdot \frac{1}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)}, \alpha + \beta, 1\right)}{2} \]
      *-commutative [<=] 0.02	\[ \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}, \alpha + \beta, 1\right)}{2} \]
      associate-*r/ [=>] 0.02	\[ \frac{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\beta + \left(\alpha + \mathsf{fma}\left(2, i, 2\right)\right)} \cdot \left(\beta - \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}, \alpha + \beta, 1\right)}{2} \]
      associate-+r+ [=>] 0.02	\[ \frac{\mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\left(\beta + \alpha\right) + \mathsf{fma}\left(2, i, 2\right)}} \cdot \left(\beta - \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \alpha + \beta, 1\right)}{2} \]
      +-commutative [=>] 0.02	\[ \frac{\mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\left(\alpha + \beta\right)} + \mathsf{fma}\left(2, i, 2\right)} \cdot \left(\beta - \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \alpha + \beta, 1\right)}{2} \]
      associate-+l+ [=>] 0.02	\[ \frac{\mathsf{fma}\left(\frac{\frac{1}{\color{blue}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}} \cdot \left(\beta - \alpha\right)}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \alpha + \beta, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.31

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, \alpha + \beta, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.31%
Cost	9924

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}}{t_1}}{2}\\ \end{array} \]

Alternative 2
Error	2.31%
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	3.19%
Cost	2756

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \]

Alternative 4
Error	11.97%
Cost	1485

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.05 \cdot 10^{+25} \lor \neg \left(\alpha \leq 5.8 \cdot 10^{+119}\right) \land \alpha \leq 7.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta}{\alpha} + 0.5 \cdot \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\\ \end{array} \]

Alternative 5
Error	16.01%
Cost	1356

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{if}\;\alpha \leq 1.92 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.05 \cdot 10^{+118}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	23.2%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+24} \lor \neg \left(\alpha \leq 5.4 \cdot 10^{+120}\right) \land \alpha \leq 1.25 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \end{array} \]

Alternative 7
Error	21.38%
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.9 \cdot 10^{+117}:\\ \;\;\;\;\frac{\beta + 1}{\alpha}\\ \mathbf{elif}\;\alpha \leq 1.2 \cdot 10^{+158}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	29.21%
Cost	460

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.5 \cdot 10^{-141}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\beta \leq 7.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	36%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 6.4 \cdot 10^{+23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 10
Error	39.1%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))