Octave 3.8, jcobi/2

Average Accuracy: 60.9% → 97.5%

Time: 32.6s

Precision: binary64

Cost: 22340

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)} \leq -0.9998:\\ \;\;\;\;\frac{-12 \cdot \left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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
 (if (<=
      (/
       (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
       (- 2.0 (- (* i -2.0) (+ alpha beta))))
      -0.9998)
   (/
    (+
     (* -12.0 (* (/ i alpha) (/ i alpha)))
     (/ (+ (* i 4.0) (+ 2.0 (* beta 2.0))) alpha))
    2.0)
   (/
    (fma
     (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
     (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
     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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (2.0 - ((i * -2.0) - (alpha + beta)))) <= -0.9998) {
		tmp = ((-12.0 * ((i / alpha) * (i / alpha))) + (((i * 4.0) + (2.0 + (beta * 2.0))) / alpha)) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (beta + fma(2.0, i, alpha))), ((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))), 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)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / Float64(Float64(alpha + beta) + Float64(2.0 * i))) / Float64(2.0 - Float64(Float64(i * -2.0) - Float64(alpha + beta)))) <= -0.9998)
		tmp = Float64(Float64(Float64(-12.0 * Float64(Float64(i / alpha) * Float64(i / alpha))) + Float64(Float64(Float64(i * 4.0) + Float64(2.0 + Float64(beta * 2.0))) / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))), Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 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_] := If[LessEqual[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[(2.0 - N[(N[(i * -2.0), $MachinePrecision] - N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.9998], N[(N[(N[(-12.0 * N[(N[(i / alpha), $MachinePrecision] * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.99980000000000002

   1. Initial program 3.2%

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

   2. Simplified 14.5%

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

      [Start] 3.2	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 78.7%

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

   4. Taylor expanded in i around inf 81.5%

      \[\leadsto \frac{\color{blue}{-12 \cdot \frac{{i}^{2}}{{\alpha}^{2}}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   5. Simplified 89.7%

      \[\leadsto \frac{\color{blue}{-12 \cdot \left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      Proof

      [Start] 81.5	\[ \frac{-12 \cdot \frac{{i}^{2}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 81.5	\[ \frac{-12 \cdot \frac{\color{blue}{i \cdot i}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 81.5	\[ \frac{-12 \cdot \frac{i \cdot i}{\color{blue}{\alpha \cdot \alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      times-frac [=>] 89.7	\[ \frac{-12 \cdot \color{blue}{\left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   if -0.99980000000000002 < (/.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 78.7%

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

   2. Simplified 100.0%

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

      [Start] 78.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 78.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 100.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 100.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 100.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)} \leq -0.9998:\\ \;\;\;\;\frac{-12 \cdot \left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	9796

\[\begin{array}{l} t_0 := 2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{t_0} \leq -0.9998:\\ \;\;\;\;\frac{-12 \cdot \left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_0}}{2}\\ \end{array} \]

Alternative 2
Accuracy	97.0%
Cost	3268

\[\begin{array}{l} t_0 := 2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{t_0} \leq -0.5:\\ \;\;\;\;\frac{-12 \cdot \left(\frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t_0}}{2}\\ \end{array} \]

Alternative 3
Accuracy	88.9%
Cost	1732

\[\begin{array}{l} t_0 := 2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\alpha \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 3.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 2.85 \cdot 10^{+141}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + i \cdot 4\right) + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Accuracy	82.5%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.1 \cdot 10^{+76} \lor \neg \left(\alpha \leq 7 \cdot 10^{+125}\right) \land \alpha \leq 2.75 \cdot 10^{+141}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Accuracy	88.2%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5 \cdot 10^{+76} \lor \neg \left(\alpha \leq 3.4 \cdot 10^{+126}\right) \land \alpha \leq 2.85 \cdot 10^{+141}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + i \cdot 4\right) + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Accuracy	88.2%
Cost	1356

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{2 - \left(i \cdot -2 - \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{if}\;\alpha \leq 8.8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{i \cdot 4}{\alpha} + \frac{\beta + 2}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 2.65 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + i \cdot 4\right) + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Accuracy	72.5%
Cost	1109

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 1.22 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3.4 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 6.2 \cdot 10^{+171} \lor \neg \left(\alpha \leq 4.2 \cdot 10^{+262}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Accuracy	79.8%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+69} \lor \neg \left(\alpha \leq 8.5 \cdot 10^{+125}\right) \land \alpha \leq 2.85 \cdot 10^{+141}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Accuracy	77.2%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+69} \lor \neg \left(\alpha \leq 1.05 \cdot 10^{+126}\right) \land \alpha \leq 2.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Accuracy	72.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+68}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	60.4%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023138 
(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))