Octave 3.8, jcobi/2

Average Accuracy: 62.4% → 97.1%

Time: 25.8s

Precision: binary64

Cost: 9796

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

Math

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

↓

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

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))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -1.0)
     (/ (/ (+ (- beta beta) (- (* i 4.0) (+ -2.0 (* beta -2.0)))) alpha) 2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta)))) t_1)
       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 t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -1.0) {
		tmp = (((beta - beta) + ((i * 4.0) - (-2.0 + (beta * -2.0)))) / alpha) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / t_1) + 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))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -1.0)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(Float64(i * 4.0) - Float64(-2.0 + Float64(beta * -2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / t_1) + 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]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -1.0], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(N[(i * 4.0), $MachinePrecision] - N[(-2.0 + N[(beta * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $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)) < -1

   1. Initial program 1.1%

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

   2. Taylor expanded in alpha around inf 90.6%

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

   if -1 < (/.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 80.3%

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

   2. Applied egg-rr 99.0%

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

      [Start] 80.3	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/l* [=>] 99.0	\[ \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r/ [=>] 99.0	\[ \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      fma-def [=>] 99.0	\[ \frac{\frac{\frac{\alpha + \beta}{\color{blue}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

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

Alternatives

Alternative 1
Accuracy	96.2%
Cost	2756

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

Alternative 2
Accuracy	89.0%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 3
Accuracy	85.3%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Accuracy	85.3%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	80.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	74.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.15 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\alpha}{4}}}{2}\\ \end{array} \]

Alternative 7
Accuracy	80.4%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	71.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	61.5%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023126 
(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))