Octave 3.8, jcobi/2

Average Error: 23.6 → 1.5

Time: 22.9s

Precision: binary64

Cost: 9924

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

Math

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

↓

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

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) -0.5)
     (/ (/ (+ 2.0 (+ (* 2.0 i) (* 2.0 (+ beta i)))) alpha) 2.0)
     (/
      (+
       1.0
       (/
        (*
         (+ alpha beta)
         (/ 1.0 (/ (fma 2.0 i (+ alpha beta)) (- beta alpha))))
        t_1))
      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) <= -0.5) {
		tmp = ((2.0 + ((2.0 * i) + (2.0 * (beta + i)))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * (1.0 / (fma(2.0, i, (alpha + beta)) / (beta - alpha)))) / t_1)) / 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) <= -0.5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(2.0 * i) + Float64(2.0 * Float64(beta + i)))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(alpha + beta) * Float64(1.0 / Float64(fma(2.0, i, Float64(alpha + beta)) / Float64(beta - alpha)))) / t_1)) / 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], -0.5], N[(N[(N[(2.0 + N[(N[(2.0 * i), $MachinePrecision] + N[(2.0 * N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(alpha + beta), $MachinePrecision] * N[(1.0 / N[(N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $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 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. Taylor expanded in alpha around inf 57.7

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

   3. Simplified 57.7

      \[\leadsto \frac{\frac{\color{blue}{\left(0 \cdot \beta - \alpha\right) - \left(\left(-\beta\right) + -2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      Proof

      [Start] 57.7	\[ \frac{\frac{\left(-1 \cdot \beta + \left(\beta + -1 \cdot \alpha\right)\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      mul-1-neg [=>] 57.7	\[ \frac{\frac{\left(-1 \cdot \beta + \left(\beta + \color{blue}{\left(-\alpha\right)}\right)\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      sub-neg [<=] 57.7	\[ \frac{\frac{\left(-1 \cdot \beta + \color{blue}{\left(\beta - \alpha\right)}\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-+r- [=>] 57.7	\[ \frac{\frac{\color{blue}{\left(\left(-1 \cdot \beta + \beta\right) - \alpha\right)} - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      distribute-lft1-in [=>] 57.7	\[ \frac{\frac{\left(\color{blue}{\left(-1 + 1\right) \cdot \beta} - \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      metadata-eval [=>] 57.7	\[ \frac{\frac{\left(\color{blue}{0} \cdot \beta - \alpha\right) - -1 \cdot \left(\beta + 2 \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      distribute-lft-in [=>] 57.7	\[ \frac{\frac{\left(0 \cdot \beta - \alpha\right) - \color{blue}{\left(-1 \cdot \beta + -1 \cdot \left(2 \cdot i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      mul-1-neg [=>] 57.7	\[ \frac{\frac{\left(0 \cdot \beta - \alpha\right) - \left(\color{blue}{\left(-\beta\right)} + -1 \cdot \left(2 \cdot i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-*r* [=>] 57.7	\[ \frac{\frac{\left(0 \cdot \beta - \alpha\right) - \left(\left(-\beta\right) + \color{blue}{\left(-1 \cdot 2\right) \cdot i}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      metadata-eval [=>] 57.7	\[ \frac{\frac{\left(0 \cdot \beta - \alpha\right) - \left(\left(-\beta\right) + \color{blue}{-2} \cdot i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   4. Taylor expanded in alpha around inf 6.8

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

   5. Simplified 6.8

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

      [Start] 6.8	\[ \frac{\frac{\left(2 + \left(2 \cdot \beta + 2 \cdot i\right)\right) - -2 \cdot i}{\alpha}}{2} \]
      associate--l+ [=>] 6.8	\[ \frac{\frac{\color{blue}{2 + \left(\left(2 \cdot \beta + 2 \cdot i\right) - -2 \cdot i\right)}}{\alpha}}{2} \]
      distribute-lft-out [=>] 6.8	\[ \frac{\frac{2 + \left(\color{blue}{2 \cdot \left(\beta + i\right)} - -2 \cdot i\right)}{\alpha}}{2} \]

   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 12.8

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

   2. Applied egg-rr 0.0

      \[\leadsto \frac{\frac{\color{blue}{\left(\alpha + \beta\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.5
Cost	9796

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

Alternative 2
Error	2.0
Cost	3268

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

Alternative 3
Error	7.1
Cost	1356

\[\begin{array}{l} t_0 := 2 + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 2.9 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{2 \cdot i + t_0}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.95 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(2 \cdot i + 2 \cdot \left(\beta + i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	8.9
Cost	1229

\[\begin{array}{l} t_0 := 2 + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+121} \lor \neg \left(\alpha \leq 1.85 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{\frac{2 \cdot i + t_0}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	17.4
Cost	576

\[\frac{1 + \frac{\beta}{\beta + 2}}{2} \]

Alternative 9
Error	17.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	24.0
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023032 
(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))