Octave 3.8, jcobi/2

Average Error: 23.8 → 2.5

Time: 22.5s

Precision: binary64

Cost: 2628

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

Math

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

↓

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

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)
     (/ (/ (+ (* 4.0 i) (+ 2.0 (* 2.0 beta))) alpha) 2.0)
     (/ (+ (/ 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 = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 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) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 + 2.0d0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.5d0)) then
        tmp = (((4.0d0 * i) + (2.0d0 + (2.0d0 * beta))) / alpha) / 2.0d0
    else
        tmp = ((beta / t_1) + 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 t_1 = t_0 + 2.0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	} else {
		tmp = ((beta / t_1) + 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)
	t_1 = t_0 + 2.0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5:
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0
	else:
		tmp = ((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(4.0 * i) + Float64(2.0 + Float64(2.0 * beta))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(beta / t_1) + 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);
	t_1 = t_0 + 2.0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = (((4.0 * i) + (2.0 + (2.0 * beta))) / alpha) / 2.0;
	else
		tmp = ((beta / t_1) + 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]}, 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[(4.0 * i), $MachinePrecision] + N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(beta / 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 61.5

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

   2. Taylor expanded in alpha around inf 14.0

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

   3. Simplified 14.0

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

      [Start] 14.0	\[ \frac{\left(\frac{\beta}{\alpha} + \left(-1 \cdot \frac{\beta}{\alpha} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      rational_best-simplify-2 [=>] 14.0	\[ \frac{\left(\frac{\beta}{\alpha} + \left(\color{blue}{\frac{\beta}{\alpha} \cdot -1} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      rational_best-simplify-12 [=>] 14.0	\[ \frac{\left(\frac{\beta}{\alpha} + \left(\color{blue}{\left(-\frac{\beta}{\alpha}\right)} + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + \frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      rational_best-simplify-43 [=>] 14.0	\[ \frac{\left(\frac{\beta}{\alpha} + \left(\left(-\frac{\beta}{\alpha}\right) + \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right) - \color{blue}{\left(\frac{\left(\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right) \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{{\alpha}^{2}} + \left(-1 \cdot \frac{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}} + -1 \cdot \frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}\right)\right)}}{2} \]

   4. Taylor expanded in alpha around inf 7.1

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

   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 12.3

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

   2. Taylor expanded in beta around inf 1.0

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

4. Final simplification 2.5

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

Alternatives

Alternative 1
Error	13.2
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 3.7 \cdot 10^{+149}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-\frac{2 - i \cdot -4}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 2
Error	13.1
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + \alpha\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 9.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+148}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-\frac{2 - i \cdot -4}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 3
Error	9.6
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 7.8 \cdot 10^{+148}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(-\frac{2 - i \cdot -4}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 4
Error	7.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + \left(2 + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4 \cdot i + \left(2 + 2 \cdot \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6
Error	14.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \beta}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	17.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	24.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023094 
(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))