Octave 3.8, jcobi/2

Average Error: 23.7 → 2.0

Time: 20.0s

Precision: binary64

Cost: 2884

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 + 2\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(-\beta\right)\right) - \left(-\left(\beta \cdot 2 + \left(2 + i \cdot 4\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{t_1} + 1}{2}\\ \end{array} \]

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) -0.999995)
     (/
      (/ (- (+ beta (- beta)) (- (+ (* beta 2.0) (+ 2.0 (* i 4.0))))) alpha)
      2.0)
     (/ (+ (/ (- beta alpha) 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) <= -0.999995) {
		tmp = (((beta + -beta) - -((beta * 2.0) + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) / 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) <= (-0.999995d0)) then
        tmp = (((beta + -beta) - -((beta * 2.0d0) + (2.0d0 + (i * 4.0d0)))) / alpha) / 2.0d0
    else
        tmp = (((beta - alpha) / 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) <= -0.999995) {
		tmp = (((beta + -beta) - -((beta * 2.0) + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	} else {
		tmp = (((beta - alpha) / 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) <= -0.999995:
		tmp = (((beta + -beta) - -((beta * 2.0) + (2.0 + (i * 4.0)))) / alpha) / 2.0
	else:
		tmp = (((beta - alpha) / 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) <= -0.999995)
		tmp = Float64(Float64(Float64(Float64(beta + Float64(-beta)) - Float64(-Float64(Float64(beta * 2.0) + Float64(2.0 + Float64(i * 4.0))))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(beta - alpha) / 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) <= -0.999995)
		tmp = (((beta + -beta) - -((beta * 2.0) + (2.0 + (i * 4.0)))) / alpha) / 2.0;
	else
		tmp = (((beta - alpha) / 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], -0.999995], N[(N[(N[(N[(beta + (-beta)), $MachinePrecision] - (-N[(N[(beta * 2.0), $MachinePrecision] + N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(beta - alpha), $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)) < -0.99999499999999997

   1. Initial program 62.1

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

   2. Taylor expanded in alpha around inf 6.2

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

   3. Simplified 6.2

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

      [Start] 6.2	\[ \frac{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2} \]
      rational_best.json-simplify-1 [<=] 6.2	\[ \frac{\frac{\color{blue}{\left(\beta + -1 \cdot \beta\right)} - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2} \]
      rational_best.json-simplify-2 [=>] 6.2	\[ \frac{\frac{\left(\beta + \color{blue}{\beta \cdot -1}\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2} \]
      rational_best.json-simplify-12 [=>] 6.2	\[ \frac{\frac{\left(\beta + \color{blue}{\left(-\beta\right)}\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}{2} \]
      rational_best.json-simplify-2 [=>] 6.2	\[ \frac{\frac{\left(\beta + \left(-\beta\right)\right) - \color{blue}{\left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right) \cdot -1}}{\alpha}}{2} \]
      rational_best.json-simplify-12 [=>] 6.2	\[ \frac{\frac{\left(\beta + \left(-\beta\right)\right) - \color{blue}{\left(-\left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)\right)}}{\alpha}}{2} \]
      rational_best.json-simplify-43 [=>] 6.2	\[ \frac{\frac{\left(\beta + \left(-\beta\right)\right) - \left(-\color{blue}{\left(2 \cdot \beta + \left(2 + 4 \cdot i\right)\right)}\right)}{\alpha}}{2} \]
      rational_best.json-simplify-2 [=>] 6.2	\[ \frac{\frac{\left(\beta + \left(-\beta\right)\right) - \left(-\left(\color{blue}{\beta \cdot 2} + \left(2 + 4 \cdot i\right)\right)\right)}{\alpha}}{2} \]
      rational_best.json-simplify-2 [=>] 6.2	\[ \frac{\frac{\left(\beta + \left(-\beta\right)\right) - \left(-\left(\beta \cdot 2 + \left(2 + \color{blue}{i \cdot 4}\right)\right)\right)}{\alpha}}{2} \]

   if -0.99999499999999997 < (/.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 12.4

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

   2. Taylor expanded in i around 0 0.8

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(-\beta\right)\right) - \left(-\left(\beta \cdot 2 + \left(2 + i \cdot 4\right)\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	1484

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \left(i + \beta\right)}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 9.8 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 9 \cdot 10^{+201}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.5
Cost	1356

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \left(i + \beta\right)}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.05 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.6 \cdot 10^{+201}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.7
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{\beta + 2}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 5.8 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	14.1
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \left(i + \beta\right)}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 5.3 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.6 \cdot 10^{+201}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.6
Cost	1100

\[\begin{array}{l} t_0 := \frac{\frac{2 + 2 \cdot \left(i + \beta\right)}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{elif}\;\alpha \leq 1.28 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 4.5 \cdot 10^{+201}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.0
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{\beta + 2}{\alpha}}{2}\\ t_1 := \frac{\frac{\beta}{\beta + 2} + 1}{2}\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3.8 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 7.5 \cdot 10^{+202}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.6
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	43.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))