Octave 3.8, jcobi/4

Average Error: 53.5 → 10.6

Time: 11.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := i \cdot t_2\\ t_4 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_5 := \frac{i}{t_4}\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_1 + -1} \leq \infty:\\ \;\;\;\;\left(t_5 \cdot t_2\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{{t_4}^{2} + -1}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 \cdot \log \left(e^{t_5}\right)\right) \cdot \left(\frac{1}{t_4} \cdot 0.25\right)\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* i t_2))
        (t_4 (+ alpha (fma i 2.0 beta)))
        (t_5 (/ i t_4)))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) (+ t_1 -1.0)) INFINITY)
     (*
      (* t_5 t_2)
      (/
       (/ (fma i (+ beta (+ i alpha)) (* alpha beta)) (+ (pow t_4 2.0) -1.0))
       t_4))
     (* (* t_2 (log (exp t_5))) (* (/ 1.0 t_4) 0.25)))))

C

double code(double alpha, double beta, double i) {
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i + (alpha + beta);
	double t_3 = i * t_2;
	double t_4 = alpha + fma(i, 2.0, beta);
	double t_5 = i / t_4;
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = (t_5 * t_2) * ((fma(i, (beta + (i + alpha)), (alpha * beta)) / (pow(t_4, 2.0) + -1.0)) / t_4);
	} else {
		tmp = (t_2 * log(exp(t_5))) * ((1.0 / t_4) * 0.25);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(i * t_2)
	t_4 = Float64(alpha + fma(i, 2.0, beta))
	t_5 = Float64(i / t_4)
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(t_5 * t_2) * Float64(Float64(fma(i, Float64(beta + Float64(i + alpha)), Float64(alpha * beta)) / Float64((t_4 ^ 2.0) + -1.0)) / t_4));
	else
		tmp = Float64(Float64(t_2 * log(exp(t_5))) * Float64(Float64(1.0 / t_4) * 0.25));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(i / t$95$4), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$5 * t$95$2), $MachinePrecision] * N[(N[(N[(i * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[Log[N[Exp[t$95$5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / t$95$4), $MachinePrecision] * 0.25), $MachinePrecision]), $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 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 33.8

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

   2. Simplified 25.4

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 64.0

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

   2. Simplified 60.7

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

   3. Applied egg-rr 60.7

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

   4. Taylor expanded in i around inf 22.3

      \[\leadsto \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \color{blue}{0.25}\right) \]

   5. Applied egg-rr 16.1

      \[\leadsto \left(\color{blue}{\log \left(e^{\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot 0.25\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(i + \left(\alpha + \beta\right)\right) \cdot \log \left(e^{\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot 0.25\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))