Octave 3.8, jcobi/2

Average Error: 23.6 → 1.4

Time: 19.6s

Precision: binary64

Cost: 28996

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

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)) -0.99999)
     (/ (/ (+ (- beta beta) (+ (* i 4.0) (+ 2.0 (* beta 2.0)))) alpha) 2.0)
     (/
      (pow
       (pow
        (+
         (*
          (/ (+ alpha beta) (+ (+ alpha beta) (fma 2.0 i 2.0)))
          (/ (- beta alpha) (fma 2.0 i (+ alpha beta))))
         1.0)
        3.0)
       0.3333333333333333)
      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)) <= -0.99999) {
		tmp = (((beta - beta) + ((i * 4.0) + (2.0 + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = pow(pow(((((alpha + beta) / ((alpha + beta) + fma(2.0, i, 2.0))) * ((beta - alpha) / fma(2.0, i, (alpha + beta)))) + 1.0), 3.0), 0.3333333333333333) / 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)) <= -0.99999)
		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(alpha + beta) / Float64(Float64(alpha + beta) + fma(2.0, i, 2.0))) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta)))) + 1.0) ^ 3.0) ^ 0.3333333333333333) / 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], -0.99999], 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[Power[N[Power[N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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)) < -0.999990000000000046

   1. Initial program 62.0

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

   2. Taylor expanded in alpha around inf 6.0

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

   if -0.999990000000000046 < (/.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 12.2

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

   2. Applied egg-rr 0.0

      \[\leadsto \frac{\color{blue}{{\left({\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1\right)}^{3}\right)}^{0.3333333333333333}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.99999:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1}}{2}\\ \end{array} \]

Alternative 2
Error	1.9
Cost	3140

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

Alternative 3
Error	9.0
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.855073009704979 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(2 - i \cdot -4\right)\right) - \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	6.6
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.4985332828503704 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(1 + 2 \cdot \frac{i}{\beta}\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	10.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.4985332828503704 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.855073009704979 \cdot 10^{+142}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(2 - i \cdot -4\right)\right) - \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	15.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 1.5028311985372568 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	13.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.4985332828503704 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	17.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.4448536108606096 \cdot 10^{+35}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	24.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022296 
(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))