Octave 3.8, jcobi/2

Average Error: 23.9 → 1.4

Time: 24.4s

Precision: binary64

Cost: 22468

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t_0}}{2 + t_0} \leq -0.99998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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 (+ (* 2.0 i) (+ alpha beta))))
   (if (<= (/ (/ (* (- beta alpha) (+ alpha beta)) t_0) (+ 2.0 t_0)) -0.99998)
     (/ (/ (+ (- beta beta) (+ (* i 4.0) (+ 2.0 (* beta 2.0)))) alpha) 2.0)
     (/
      (fma
       (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta))))
       (/ 1.0 (+ alpha (+ beta (fma 2.0 i 2.0))))
       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 = (2.0 * i) + (alpha + beta);
	double tmp;
	if (((((beta - alpha) * (alpha + beta)) / t_0) / (2.0 + t_0)) <= -0.99998) {
		tmp = (((beta - beta) + ((i * 4.0) + (2.0 + (beta * 2.0)))) / alpha) / 2.0;
	} else {
		tmp = fma(((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))), (1.0 / (alpha + (beta + fma(2.0, i, 2.0)))), 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(2.0 * i) + Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) * Float64(alpha + beta)) / t_0) / Float64(2.0 + t_0)) <= -0.99998)
		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(fma(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))), Float64(1.0 / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 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[(2.0 * i), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.99998], 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[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $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)) < -0.99997999999999998

   1. Initial program 62.3

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

   2. Taylor expanded in alpha around inf 5.9

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

   if -0.99997999999999998 < (/.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. Applied egg-rr 0.0

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{2 \cdot i + \left(\alpha + \beta\right)}}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)} \leq -0.99998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, \frac{1}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	3524

\[\begin{array}{l} t_0 := 2 \cdot i + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\left(\beta - \alpha\right) \cdot \left(\alpha + \beta\right)}{t_0}}{2 + t_0} \leq -0.99998:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\left(\alpha + \beta\right) \cdot \frac{\alpha - \beta}{t_0}}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}{2}\\ \end{array} \]

Alternative 2
Error	7.2
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta}{\beta - i \cdot -2}}{\beta + \left(\alpha + \left(2 + 2 \cdot i\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	7.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	7.3
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 - \frac{\alpha - \beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	15.7
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -9.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq -1.1 \cdot 10^{-301}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 1.8 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	15.2
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq -2 \cdot 10^{-109}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq -1.06 \cdot 10^{-301}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.85 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	10.2
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(2 \cdot i + \left(\alpha + \beta\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	16.0
Cost	973

\[\begin{array}{l} \mathbf{if}\;i \leq 6 \cdot 10^{+104} \lor \neg \left(i \leq 9.4 \cdot 10^{+129}\right) \land i \leq 10^{+193}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10
Error	16.1
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq -9.6 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq -1 \cdot 10^{-301}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\alpha \leq 2.65 \cdot 10^{+175}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta - \left(-2 - \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	10.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.6 \cdot 10^{+175}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\beta - i \cdot -2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 12
Error	18.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+82}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	24.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023045 
(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))