Octave 3.8, jcobi/2

Average Error: 24.0 → 1.2

Time: 23.9s

Precision: binary64

Cost: 22340

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.995)
     (/
      (+
       (* -2.0 (* (/ beta alpha) (/ beta 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 t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.995) {
		tmp = ((-2.0 * ((beta / alpha) * (beta / 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)
	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)) <= -0.995)
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(beta / alpha) * Float64(beta / 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_] := 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], -0.995], N[(N[(N[(-2.0 * N[(N[(beta / alpha), $MachinePrecision] * N[(beta / 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)) < -0.994999999999999996

   1. Initial program 61.7

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

   2. Simplified 55.3

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Taylor expanded in alpha around inf 12.9

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

   4. Taylor expanded in beta around inf 8.8

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

   5. Simplified 5.5

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

      [Start] 8.8	\[ \frac{-2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 8.8	\[ \frac{-2 \cdot \frac{\color{blue}{\beta \cdot \beta}}{{\alpha}^{2}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      unpow2 [=>] 8.8	\[ \frac{-2 \cdot \frac{\beta \cdot \beta}{\color{blue}{\alpha \cdot \alpha}} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]
      times-frac [=>] 5.5	\[ \frac{-2 \cdot \color{blue}{\left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)} - -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2} \]

   if -0.994999999999999996 < (/.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 12.9

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

   2. Simplified 0.0

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

      [Start] 12.9	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 13.0	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 0.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 0.0	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 0.0	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	12296

\[\begin{array}{l} t_0 := i \cdot 4 + \left(2 + \beta \cdot 2\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1}\\ \mathbf{if}\;t_2 \leq -0.995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{t_0}{\alpha}}{2}\\ \mathbf{elif}\;t_2 \leq 0.95:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\alpha \cdot t_0 + \left({\alpha}^{2} + \left(\beta - i \cdot -2\right) \cdot \left(\beta - \left(-2 + i \cdot -2\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 2
Error	2.5
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0}\\ \mathbf{if}\;t_1 \leq -0.995:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 0.95:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \left(\alpha - \beta\right)}{\left(\left(i \cdot -2 - \beta\right) - \alpha\right) \cdot \left(\left(-2 + i \cdot -2\right) - \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{-2}{\beta}}{2}\\ \end{array} \]

Alternative 3
Error	10.4
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right) + \frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	10.4
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+89}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{i}{\alpha} + 2 \cdot \frac{\beta}{\alpha}\right) + \frac{2 + 2 \cdot i}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	10.4
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 10^{+88}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - i \cdot -2\right) + \left(\beta - \left(-2 + i \cdot -2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	10.3
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot \frac{i}{\alpha} + \frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	16.7
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;i \leq 4.9 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;i \leq 9.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;i \leq 8.5 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	14.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.15 \cdot 10^{+155}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	13.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	18.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+110}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	24.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023066 
(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))