?

?

(FPCore (a c b)
 :precision binary64
 (/ (- (* a c) (pow b 2.0)) (- (+ a c) (* 2.0 b))))

(FPCore (a c b)
 :precision binary64
 (let* ((t_0 (- (* a a) (pow (fma b -2.0 c) 2.0))))
   (if (or (<= b -3.2e-184) (not (<= b 2.85e-202)))
     (* (- (/ a (/ t_0 c)) (/ b (/ t_0 b))) (+ a (- (* b 2.0) c)))
     (* a (/ c (+ a c))))))

double code(double a, double c, double b) {
	return ((a * c) - pow(b, 2.0)) / ((a + c) - (2.0 * b));
}

double code(double a, double c, double b) {
	double t_0 = (a * a) - pow(fma(b, -2.0, c), 2.0);
	double tmp;
	if ((b <= -3.2e-184) || !(b <= 2.85e-202)) {
		tmp = ((a / (t_0 / c)) - (b / (t_0 / b))) * (a + ((b * 2.0) - c));
	} else {
		tmp = a * (c / (a + c));
	}
	return tmp;
}

function code(a, c, b)
	return Float64(Float64(Float64(a * c) - (b ^ 2.0)) / Float64(Float64(a + c) - Float64(2.0 * b)))
end

function code(a, c, b)
	t_0 = Float64(Float64(a * a) - (fma(b, -2.0, c) ^ 2.0))
	tmp = 0.0
	if ((b <= -3.2e-184) || !(b <= 2.85e-202))
		tmp = Float64(Float64(Float64(a / Float64(t_0 / c)) - Float64(b / Float64(t_0 / b))) * Float64(a + Float64(Float64(b * 2.0) - c)));
	else
		tmp = Float64(a * Float64(c / Float64(a + c)));
	end
	return tmp
end

code[a_, c_, b_] := N[(N[(N[(a * c), $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(a + c), $MachinePrecision] - N[(2.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

code[a_, c_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] - N[Power[N[(b * -2.0 + c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[b, -3.2e-184], N[Not[LessEqual[b, 2.85e-202]], $MachinePrecision]], N[(N[(N[(a / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision] - N[(b / N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + N[(N[(b * 2.0), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(c / N[(a + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\frac{a \cdot c - {b}^{2}}{\left(a + c\right) - 2 \cdot b}

\begin{array}{l}
t_0 := a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}\\
\mathbf{if}\;b \leq -3.2 \cdot 10^{-184} \lor \neg \left(b \leq 2.85 \cdot 10^{-202}\right):\\
\;\;\;\;\left(\frac{a}{\frac{t_0}{c}} - \frac{b}{\frac{t_0}{b}}\right) \cdot \left(a + \left(b \cdot 2 - c\right)\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot \frac{c}{a + c}\\


\end{array}

Error?

Derivation?

1. Split input into 2 regimes

2. if b < -3.2e-184 or 2.8500000000000002e-202 < b

   1. Initial program 5.3

      \[\frac{a \cdot c - {b}^{2}}{\left(a + c\right) - 2 \cdot b} \]

   2. Simplified5.3

      \[\leadsto \color{blue}{\frac{a \cdot c - b \cdot b}{\left(a + c\right) + -2 \cdot b}} \]
      Proof

      [Start]5.3	\[ \frac{a \cdot c - {b}^{2}}{\left(a + c\right) - 2 \cdot b} \]
      unpow2 [=>]5.3	\[ \frac{a \cdot c - \color{blue}{b \cdot b}}{\left(a + c\right) - 2 \cdot b} \]
      cancel-sign-sub-inv [=>]5.3	\[ \frac{a \cdot c - b \cdot b}{\color{blue}{\left(a + c\right) + \left(-2\right) \cdot b}} \]
      metadata-eval [=>]5.3	\[ \frac{a \cdot c - b \cdot b}{\left(a + c\right) + \color{blue}{-2} \cdot b} \]

   3. Applied egg-rr5.4

      \[\leadsto \color{blue}{\frac{a \cdot c - b \cdot b}{a \cdot a - \left(c + b \cdot -2\right) \cdot \left(c + b \cdot -2\right)} \cdot \left(a - \left(c + b \cdot -2\right)\right)} \]

   4. Applied egg-rr5.4

      \[\leadsto \color{blue}{\left(\frac{a \cdot c}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}} + \left(-\frac{b \cdot b}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}\right)\right)} \cdot \left(a - \left(c + b \cdot -2\right)\right) \]

   5. Simplified2.9

      \[\leadsto \color{blue}{\left(\frac{a}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{c}} - \frac{b}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{b}}\right)} \cdot \left(a - \left(c + b \cdot -2\right)\right) \]
      Proof

      [Start]5.4	\[ \left(\frac{a \cdot c}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}} + \left(-\frac{b \cdot b}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}\right)\right) \cdot \left(a - \left(c + b \cdot -2\right)\right) \]
      sub-neg [<=]5.4	\[ \color{blue}{\left(\frac{a \cdot c}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}} - \frac{b \cdot b}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}\right)} \cdot \left(a - \left(c + b \cdot -2\right)\right) \]
      associate-/l* [=>]4.5	\[ \left(\color{blue}{\frac{a}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{c}}} - \frac{b \cdot b}{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}\right) \cdot \left(a - \left(c + b \cdot -2\right)\right) \]
      associate-/l* [=>]2.9	\[ \left(\frac{a}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{c}} - \color{blue}{\frac{b}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{b}}}\right) \cdot \left(a - \left(c + b \cdot -2\right)\right) \]

   if -3.2e-184 < b < 2.8500000000000002e-202

   1. Initial program 28.2

      \[\frac{a \cdot c - {b}^{2}}{\left(a + c\right) - 2 \cdot b} \]

   2. Simplified28.2

      \[\leadsto \color{blue}{\frac{a \cdot c - b \cdot b}{\left(a + c\right) + -2 \cdot b}} \]
      Proof

      [Start]28.2	\[ \frac{a \cdot c - {b}^{2}}{\left(a + c\right) - 2 \cdot b} \]
      unpow2 [=>]28.2	\[ \frac{a \cdot c - \color{blue}{b \cdot b}}{\left(a + c\right) - 2 \cdot b} \]
      cancel-sign-sub-inv [=>]28.2	\[ \frac{a \cdot c - b \cdot b}{\color{blue}{\left(a + c\right) + \left(-2\right) \cdot b}} \]
      metadata-eval [=>]28.2	\[ \frac{a \cdot c - b \cdot b}{\left(a + c\right) + \color{blue}{-2} \cdot b} \]

   3. Taylor expanded in b around 0 28.2

      \[\leadsto \color{blue}{\frac{c \cdot a}{c + a}} \]

   4. Simplified6.2

      \[\leadsto \color{blue}{\frac{c}{c + a} \cdot a} \]
      Proof

      [Start]28.2	\[ \frac{c \cdot a}{c + a} \]
      associate-/l* [=>]6.2	\[ \color{blue}{\frac{c}{\frac{c + a}{a}}} \]
      associate-/r/ [=>]6.2	\[ \color{blue}{\frac{c}{c + a} \cdot a} \]

3. Recombined 2 regimes into one program.

4. Final simplification4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-184} \lor \neg \left(b \leq 2.85 \cdot 10^{-202}\right):\\ \;\;\;\;\left(\frac{a}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{c}} - \frac{b}{\frac{a \cdot a - {\left(\mathsf{fma}\left(b, -2, c\right)\right)}^{2}}{b}}\right) \cdot \left(a + \left(b \cdot 2 - c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{c}{a + c}\\ \end{array} \]

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (a c b)
  :name "(a*c - b^2)/(a + c - 2*b)"
  :precision binary64
  :pre (and (and (and (<= -1.0 a) (<= a 1.0)) (and (<= -1.0 c) (<= c 1.0))) (and (<= -1.0 b) (<= b 1.0)))
  (/ (- (* a c) (pow b 2.0)) (- (+ a c) (* 2.0 b))))