Average Error: 29.0 → 11.3
Time: 54.9s
Precision: binary64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
↓
\[\begin{array}{l} t_1 := \left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \mathbf{if}\;y \leq -1.0562119916037821 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.6332404320936567 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left({y}^{3}, x, 230661.510616 + \left(y \cdot 27464.7644705 + z \cdot {y}^{2}\right)\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))
↓
(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- (+ (/ z y) x) (/ (* x a) y))))
   (if (<= y -1.0562119916037821e+77)
     t_1
     (if (<= y 5.6332404320936567e+51)
       (/
        (fma
         y
         (fma
          (pow y 3.0)
          x
          (+ 230661.510616 (+ (* y 27464.7644705) (* z (pow y 2.0)))))
         t)
        (fma y (fma y (fma y (+ y a) b) c) i))
       t_1))))
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + x) - ((x * a) / y);
	double tmp;
	if (y <= -1.0562119916037821e+77) {
		tmp = t_1;
	} else if (y <= 5.6332404320936567e+51) {
		tmp = fma(y, fma(pow(y, 3.0), x, (230661.510616 + ((y * 27464.7644705) + (z * pow(y, 2.0))))), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end
↓
function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + x) - Float64(Float64(x * a) / y))
	tmp = 0.0
	if (y <= -1.0562119916037821e+77)
		tmp = t_1;
	elseif (y <= 5.6332404320936567e+51)
		tmp = Float64(fma(y, fma((y ^ 3.0), x, Float64(230661.510616 + Float64(Float64(y * 27464.7644705) + Float64(z * (y ^ 2.0))))), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + x), $MachinePrecision] - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0562119916037821e+77], t$95$1, If[LessEqual[y, 5.6332404320936567e+51], N[(N[(y * N[(N[Power[y, 3.0], $MachinePrecision] * x + N[(230661.510616 + N[(N[(y * 27464.7644705), $MachinePrecision] + N[(z * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\begin{array}{l}
t_1 := \left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\
\mathbf{if}\;y \leq -1.0562119916037821 \cdot 10^{+77}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;y \leq 5.6332404320936567 \cdot 10^{+51}:\\
\;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left({y}^{3}, x, 230661.510616 + \left(y \cdot 27464.7644705 + z \cdot {y}^{2}\right)\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if y < -1.0562119916037821e77 or 5.6332404320936567e51 < y

    1. Initial program 63.0

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

    2. Simplified63.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}} \]

    3. Taylor expanded in y around inf 19.7

      \[\leadsto \color{blue}{\left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y}} \]

    if -1.0562119916037821e77 < y < 5.6332404320936567e51

    1. Initial program 5.5

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

    2. Simplified5.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}} \]

    3. Taylor expanded in y around 0 5.5

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{{y}^{3} \cdot x + \left(230661.510616 + \left(27464.7644705 \cdot y + {y}^{2} \cdot z\right)\right)}, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]

    4. Applied fma-def_binary645.5

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left({y}^{3}, x, 230661.510616 + \left(27464.7644705 \cdot y + {y}^{2} \cdot z\right)\right)}, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.0562119916037821 \cdot 10^{+77}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \mathbf{elif}\;y \leq 5.6332404320936567 \cdot 10^{+51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left({y}^{3}, x, 230661.510616 + \left(y \cdot 27464.7644705 + z \cdot {y}^{2}\right)\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + x\right) - \frac{x \cdot a}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))