Octave 3.8, jcobi/2

Average Error: 23.9 → 1.7

Time: 31.5s

Precision: binary64

Cost: 9796

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

Math

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

↓

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

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 (+ t_0 2.0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/
      (/
       (- (+ (* -1.0 beta) beta) (* -1.0 (+ (* 4.0 i) (+ 2.0 (* 2.0 beta)))))
       alpha)
      2.0)
     (/
      (+
       (/ (* (/ (- beta alpha) (fma 2.0 i (+ alpha beta))) (+ 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 = t_0 + 2.0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = ((((-1.0 * beta) + beta) - (-1.0 * ((4.0 * i) + (2.0 + (2.0 * beta))))) / alpha) / 2.0;
	} else {
		tmp = (((((beta - alpha) / fma(2.0, i, (alpha + beta))) * (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(t_0 + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(-1.0 * beta) + beta) - Float64(-1.0 * Float64(Float64(4.0 * i) + Float64(2.0 + Float64(2.0 * beta))))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(beta - alpha) / fma(2.0, i, Float64(alpha + beta))) * 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[(t$95$0 + 2.0), $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[(N[(N[(-1.0 * beta), $MachinePrecision] + beta), $MachinePrecision] - N[(-1.0 * N[(N[(4.0 * i), $MachinePrecision] + N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(alpha + beta), $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)) < -0.5

   1. Initial program 61.4

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

   2. Taylor expanded in alpha around inf 7.1

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

   if -0.5 < (/.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 12.5

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

   2. Applied egg-rr 0.0

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

Alternatives

Alternative 1
Error	2.8
Cost	5192

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2}\\ \mathbf{if}\;t_1 \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t_1 \leq 0.99999995:\\ \;\;\;\;\frac{t_1 + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha}{\beta \cdot -0.5} + 2}{2}\\ \end{array} \]

Alternative 2
Error	11.0
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	11.0
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + 2 \cdot i\right) - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} \cdot 0.5\\ \end{array} \]

Alternative 4
Error	15.3
Cost	1092

\[\begin{array}{l} \mathbf{if}\;i \leq 1.42 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{1}{-2 - \left(\alpha + \beta\right)} \cdot \left(\alpha - \beta\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;i \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(2 + \alpha\right) + \beta} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2}\\ \end{array} \]

Alternative 6
Error	15.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 2.9 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2}\\ \end{array} \]

Alternative 7
Error	23.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+226}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\alpha}\\ \end{array} \]

Alternative 8
Error	17.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.22 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2}\\ \end{array} \]

Alternative 9
Error	24.4
Cost	192

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

Reproduce

herbie shell --seed 2023010 
(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))