Octave 3.8, jcobi/2

Average Error: 24.2 → 1.2

Time: 9.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \mathsf{fma}\left(2, i, 2\right)\\ t_1 := \frac{\beta + \mathsf{fma}\left(2, i, \beta\right)}{\alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := \mathsf{fma}\left(\beta, -1, t_0\right)\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{2 + t_2} \leq -0.99998:\\ \;\;\;\;\frac{\left(t_1 \cdot \frac{i \cdot -2}{\alpha} - \left(\frac{t_3}{\alpha} \cdot \frac{t_0}{\alpha} + \frac{\mathsf{fma}\left(2, i, \beta\right)}{\alpha} \cdot t_1\right)\right) + \frac{\mathsf{fma}\left(2, i, \beta\right) + \left(\beta + t_3\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right) + 2\right)}, \frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}, 1\right)}{2}\\ \end{array} \]

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 (+ beta (fma 2.0 i 2.0)))
        (t_1 (/ (+ beta (fma 2.0 i beta)) alpha))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (fma beta -1.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.99998)
     (/
      (+
       (-
        (* t_1 (/ (* i -2.0) alpha))
        (+ (* (/ t_3 alpha) (/ t_0 alpha)) (* (/ (fma 2.0 i beta) alpha) t_1)))
       (/ (+ (fma 2.0 i beta) (+ beta t_3)) alpha))
      2.0)
     (/
      (fma
       (/ (+ alpha beta) (fma 2.0 i (+ (+ alpha beta) 2.0)))
       (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
       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 = beta + fma(2.0, i, 2.0);
	double t_1 = (beta + fma(2.0, i, beta)) / alpha;
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = fma(beta, -1.0, t_0);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.99998) {
		tmp = (((t_1 * ((i * -2.0) / alpha)) - (((t_3 / alpha) * (t_0 / alpha)) + ((fma(2.0, i, beta) / alpha) * t_1))) + ((fma(2.0, i, beta) + (beta + t_3)) / alpha)) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / fma(2.0, i, ((alpha + beta) + 2.0))), ((beta - alpha) / (alpha + fma(2.0, i, beta))), 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(beta + fma(2.0, i, 2.0))
	t_1 = Float64(Float64(beta + fma(2.0, i, beta)) / alpha)
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = fma(beta, -1.0, t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(2.0 + t_2)) <= -0.99998)
		tmp = Float64(Float64(Float64(Float64(t_1 * Float64(Float64(i * -2.0) / alpha)) - Float64(Float64(Float64(t_3 / alpha) * Float64(t_0 / alpha)) + Float64(Float64(fma(2.0, i, beta) / alpha) * t_1))) + Float64(Float64(fma(2.0, i, beta) + Float64(beta + t_3)) / alpha)) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / fma(2.0, i, Float64(Float64(alpha + beta) + 2.0))), Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))), 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[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta * -1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.99998], N[(N[(N[(N[(t$95$1 * N[(N[(i * -2.0), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$3 / alpha), $MachinePrecision] * N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * i + beta), $MachinePrecision] / alpha), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * i + beta), $MachinePrecision] + N[(beta + t$95$3), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

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. Simplified 54.7

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

   3. Taylor expanded in alpha around -inf 12.1

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

   4. Simplified 5.2

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

   5. Taylor expanded in i around inf 5.2

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

   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 12.9

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

   2. Simplified 0.0

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

4. Final simplification 1.2

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

Reproduce

herbie shell --seed 2022178 
(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))