Octave 3.8, jcobi/2

Average Error: 24.1 → 1.5

Time: 30.3s

Precision: binary64

Cost: 3524

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

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))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) -0.999998)
     (/ (/ (+ (* 4.0 i) (+ 2.0 (* 2.0 beta))) alpha) 2.0)
     (/
      (+
       (*
        (/ (+ alpha beta) (+ beta (+ alpha (* 2.0 i))))
        (/ (- beta alpha) (+ (+ alpha beta) (+ 2.0 (* 2.0 i)))))
       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) / (t_0 + 2.0)) <= -0.999998) {
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0d0 * i))) / (((alpha + beta) + (2.0d0 * i)) + 2.0d0)) + 1.0d0) / 2.0d0
end function

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) <= (-0.999998d0)) then
        tmp = (((4.0d0 * i) + (2.0d0 + (2.0d0 * beta))) / alpha) / 2.0d0
    else
        tmp = ((((alpha + beta) / (beta + (alpha + (2.0d0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0d0 + (2.0d0 * i))))) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static 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;
}

↓

public static 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) / (t_0 + 2.0)) <= -0.999998) {
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	} else {
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0

↓

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.999998:
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0
	else:
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 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(t_0 + 2.0)) <= -0.999998)
		tmp = Float64(Float64(Float64(Float64(4.0 * i) + Float64(2.0 + Float64(2.0 * beta))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) / Float64(beta + Float64(alpha + Float64(2.0 * i)))) * Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + Float64(2.0 + Float64(2.0 * i))))) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
end

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) <= -0.999998)
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	else
		tmp = ((((alpha + beta) / (beta + (alpha + (2.0 * i)))) * ((beta - alpha) / ((alpha + beta) + (2.0 + (2.0 * i))))) + 1.0) / 2.0;
	end
	tmp_2 = 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[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision], -0.999998], N[(N[(N[(N[(4.0 * i), $MachinePrecision] + N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 + N[(2.0 * i), $MachinePrecision]), $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)) < -0.999998000000000054

   1. Initial program 62.4

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

   2. Simplified 62.4

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

      [Start] 62.4	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-33 [=>] 62.4	\[ \frac{\frac{\frac{\color{blue}{\beta \cdot \beta - \alpha \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 62.4	\[ \frac{\frac{\frac{\beta \cdot \beta - \alpha \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]

   3. Taylor expanded in alpha around inf 6.4

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

   if -0.999998000000000054 < (/.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 12.7

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

   2. Simplified 0.0

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

      [Start] 12.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-49 [=>] 0.0	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{2 \cdot i + \color{blue}{\left(\beta + \alpha\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-41 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1}{2} \]
      rational.json-simplify-41 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}} + 1}{2} \]
      rational.json-simplify-1 [=>] 0.0	\[ \frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \color{blue}{\left(2 + 2 \cdot i\right)}} + 1}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.999998:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + \beta}{\beta + \left(\alpha + 2 \cdot i\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	2120

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2 \cdot i\right)\\ \mathbf{if}\;\beta \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{elif}\;\beta \leq 7.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{t_0 \cdot \left(2 + t_0\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	11.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1950000000000:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\beta + \left(\alpha + 2\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	15.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \end{array} \]

Alternative 4
Error	17.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 5
Error	15.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5800:\\ \;\;\;\;\frac{\frac{2}{2 + \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 - \frac{2}{\beta}}{2}\\ \end{array} \]

Alternative 6
Error	17.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	24.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023073 
(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))