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(FPCore (a b)
 :precision binary64
 (log (+ (* (sinh a) b) (sqrt (+ 1.0 (pow (* (sinh a) b) 2.0))))))

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (sinh a) b)))
   (if (<= t_0 -4000.0)
     (log (/ (/ -0.5 b) (sinh a)))
     (if (<= t_0 2e-6)
       (* b (* 0.5 (- (exp a) (exp (- a)))))
       (log (+ t_0 (hypot 1.0 t_0)))))))

double code(double a, double b) {
	return log(((sinh(a) * b) + sqrt((1.0 + pow((sinh(a) * b), 2.0)))));
}

double code(double a, double b) {
	double t_0 = sinh(a) * b;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = log(((-0.5 / b) / sinh(a)));
	} else if (t_0 <= 2e-6) {
		tmp = b * (0.5 * (exp(a) - exp(-a)));
	} else {
		tmp = log((t_0 + hypot(1.0, t_0)));
	}
	return tmp;
}

public static double code(double a, double b) {
	return Math.log(((Math.sinh(a) * b) + Math.sqrt((1.0 + Math.pow((Math.sinh(a) * b), 2.0)))));
}

public static double code(double a, double b) {
	double t_0 = Math.sinh(a) * b;
	double tmp;
	if (t_0 <= -4000.0) {
		tmp = Math.log(((-0.5 / b) / Math.sinh(a)));
	} else if (t_0 <= 2e-6) {
		tmp = b * (0.5 * (Math.exp(a) - Math.exp(-a)));
	} else {
		tmp = Math.log((t_0 + Math.hypot(1.0, t_0)));
	}
	return tmp;
}

def code(a, b):
	return math.log(((math.sinh(a) * b) + math.sqrt((1.0 + math.pow((math.sinh(a) * b), 2.0)))))

def code(a, b):
	t_0 = math.sinh(a) * b
	tmp = 0
	if t_0 <= -4000.0:
		tmp = math.log(((-0.5 / b) / math.sinh(a)))
	elif t_0 <= 2e-6:
		tmp = b * (0.5 * (math.exp(a) - math.exp(-a)))
	else:
		tmp = math.log((t_0 + math.hypot(1.0, t_0)))
	return tmp

function code(a, b)
	return log(Float64(Float64(sinh(a) * b) + sqrt(Float64(1.0 + (Float64(sinh(a) * b) ^ 2.0)))))
end

function code(a, b)
	t_0 = Float64(sinh(a) * b)
	tmp = 0.0
	if (t_0 <= -4000.0)
		tmp = log(Float64(Float64(-0.5 / b) / sinh(a)));
	elseif (t_0 <= 2e-6)
		tmp = Float64(b * Float64(0.5 * Float64(exp(a) - exp(Float64(-a)))));
	else
		tmp = log(Float64(t_0 + hypot(1.0, t_0)));
	end
	return tmp
end

function tmp = code(a, b)
	tmp = log(((sinh(a) * b) + sqrt((1.0 + ((sinh(a) * b) ^ 2.0)))));
end

function tmp_2 = code(a, b)
	t_0 = sinh(a) * b;
	tmp = 0.0;
	if (t_0 <= -4000.0)
		tmp = log(((-0.5 / b) / sinh(a)));
	elseif (t_0 <= 2e-6)
		tmp = b * (0.5 * (exp(a) - exp(-a)));
	else
		tmp = log((t_0 + hypot(1.0, t_0)));
	end
	tmp_2 = tmp;
end

code[a_, b_] := N[Log[N[(N[(N[Sinh[a], $MachinePrecision] * b), $MachinePrecision] + N[Sqrt[N[(1.0 + N[Power[N[(N[Sinh[a], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

code[a_, b_] := Block[{t$95$0 = N[(N[Sinh[a], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$0, -4000.0], N[Log[N[(N[(-0.5 / b), $MachinePrecision] / N[Sinh[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[(b * N[(0.5 * N[(N[Exp[a], $MachinePrecision] - N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(t$95$0 + N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

\log \left(\sinh a \cdot b + \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right)

\begin{array}{l}
t_0 := \sinh a \cdot b\\
\mathbf{if}\;t_0 \leq -4000:\\
\;\;\;\;\log \left(\frac{\frac{-0.5}{b}}{\sinh a}\right)\\

\mathbf{elif}\;t_0 \leq 2 \cdot 10^{-6}:\\
\;\;\;\;b \cdot \left(0.5 \cdot \left(e^{a} - e^{-a}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(t_0 + \mathsf{hypot}\left(1, t_0\right)\right)\\


\end{array}

Error?

Try it out?

Derivation?

1. Split input into 3 regimes

2. if (*.f64 (sinh.f64 a) b) < -4e3

   1. Initial program 62.8

      \[\log \left(\sinh a \cdot b + \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right) \]

   2. Taylor expanded in b around -inf 20.7

      \[\leadsto \log \color{blue}{\left(-1 \cdot \left(b \cdot \left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right) + -0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right)\right) - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right)} \]

   3. Simplified20.7

      \[\leadsto \log \color{blue}{\left(b \cdot 0 - \frac{\frac{1}{b}}{e^{a} - e^{-a}}\right)} \]
      Proof

      [Start]20.7	\[ \log \left(-1 \cdot \left(b \cdot \left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right) + -0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right)\right) - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      mul-1-neg [=>]20.7	\[ \log \left(\color{blue}{\left(-b \cdot \left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right) + -0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right)\right)} - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      distribute-rgt-neg-in [=>]20.7	\[ \log \left(\color{blue}{b \cdot \left(-\left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right) + -0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right)\right)} - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      distribute-rgt-out [=>]20.7	\[ \log \left(b \cdot \left(-\color{blue}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot \left(0.5 + -0.5\right)}\right) - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      metadata-eval [=>]20.7	\[ \log \left(b \cdot \left(-\left(e^{a} - \frac{1}{e^{a}}\right) \cdot \color{blue}{0}\right) - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      mul0-rgt [=>]20.7	\[ \log \left(b \cdot \left(-\color{blue}{0}\right) - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      metadata-eval [=>]20.7	\[ \log \left(b \cdot \color{blue}{0} - \frac{1}{\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b}\right) \]
      *-commutative [=>]20.7	\[ \log \left(b \cdot 0 - \frac{1}{\color{blue}{b \cdot \left(e^{a} - \frac{1}{e^{a}}\right)}}\right) \]
      associate-/r* [=>]20.7	\[ \log \left(b \cdot 0 - \color{blue}{\frac{\frac{1}{b}}{e^{a} - \frac{1}{e^{a}}}}\right) \]
      rec-exp [=>]20.7	\[ \log \left(b \cdot 0 - \frac{\frac{1}{b}}{e^{a} - \color{blue}{e^{-a}}}\right) \]

   4. Taylor expanded in b around 0 64.0

      \[\leadsto \color{blue}{\log \left(-\frac{1}{e^{a} - e^{-a}}\right) + -1 \cdot \log b} \]

   5. Simplified20.7

      \[\leadsto \color{blue}{\log \left(\frac{\frac{-0.5}{b}}{\sinh a}\right)} \]
      Proof

      [Start]64.0	\[ \log \left(-\frac{1}{e^{a} - e^{-a}}\right) + -1 \cdot \log b \]
      neg-mul-1 [=>]64.0	\[ \log \color{blue}{\left(-1 \cdot \frac{1}{e^{a} - e^{-a}}\right)} + -1 \cdot \log b \]
      log-prod [=>]64.0	\[ \color{blue}{\left(\log -1 + \log \left(\frac{1}{e^{a} - e^{-a}}\right)\right)} + -1 \cdot \log b \]
      mul-1-neg [=>]64.0	\[ \left(\log -1 + \log \left(\frac{1}{e^{a} - e^{-a}}\right)\right) + \color{blue}{\left(-\log b\right)} \]
      log-rec [<=]64.0	\[ \left(\log -1 + \log \left(\frac{1}{e^{a} - e^{-a}}\right)\right) + \color{blue}{\log \left(\frac{1}{b}\right)} \]
      associate-+l+ [=>]64.0	\[ \color{blue}{\log -1 + \left(\log \left(\frac{1}{e^{a} - e^{-a}}\right) + \log \left(\frac{1}{b}\right)\right)} \]
      metadata-eval [<=]64.0	\[ \log -1 + \left(\log \left(\frac{\color{blue}{0.5 \cdot 2}}{e^{a} - e^{-a}}\right) + \log \left(\frac{1}{b}\right)\right) \]
      associate-*l/ [<=]64.0	\[ \log -1 + \left(\log \color{blue}{\left(\frac{0.5}{e^{a} - e^{-a}} \cdot 2\right)} + \log \left(\frac{1}{b}\right)\right) \]
      associate-/r/ [<=]64.0	\[ \log -1 + \left(\log \color{blue}{\left(\frac{0.5}{\frac{e^{a} - e^{-a}}{2}}\right)} + \log \left(\frac{1}{b}\right)\right) \]
      metadata-eval [<=]64.0	\[ \log -1 + \left(\log \left(\frac{\color{blue}{\frac{1}{2}}}{\frac{e^{a} - e^{-a}}{2}}\right) + \log \left(\frac{1}{b}\right)\right) \]
      sinh-def [<=]64.0	\[ \log -1 + \left(\log \left(\frac{\frac{1}{2}}{\color{blue}{\sinh a}}\right) + \log \left(\frac{1}{b}\right)\right) \]
      associate-/r* [<=]64.0	\[ \log -1 + \left(\log \color{blue}{\left(\frac{1}{2 \cdot \sinh a}\right)} + \log \left(\frac{1}{b}\right)\right) \]
      log-rec [=>]64.0	\[ \log -1 + \left(\log \left(\frac{1}{2 \cdot \sinh a}\right) + \color{blue}{\left(-\log b\right)}\right) \]
      sub-neg [<=]64.0	\[ \log -1 + \color{blue}{\left(\log \left(\frac{1}{2 \cdot \sinh a}\right) - \log b\right)} \]
      log-div [<=]64.0	\[ \log -1 + \color{blue}{\log \left(\frac{\frac{1}{2 \cdot \sinh a}}{b}\right)} \]
      associate-/r* [<=]64.0	\[ \log -1 + \log \color{blue}{\left(\frac{1}{\left(2 \cdot \sinh a\right) \cdot b}\right)} \]

   if -4e3 < (*.f64 (sinh.f64 a) b) < 1.99999999999999991e-6

   1. Initial program 59.0

      \[\log \left(\sinh a \cdot b + \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right) \]

   2. Taylor expanded in b around 0 0.9

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b\right)} \]

   3. Simplified0.9

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \left(e^{a} - e^{-a}\right)\right)} \]
      Proof

      [Start]0.9	\[ 0.5 \cdot \left(\left(e^{a} - \frac{1}{e^{a}}\right) \cdot b\right) \]
      associate-*r* [=>]0.9	\[ \color{blue}{\left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right) \cdot b} \]
      *-commutative [<=]0.9	\[ \color{blue}{b \cdot \left(0.5 \cdot \left(e^{a} - \frac{1}{e^{a}}\right)\right)} \]
      rec-exp [=>]0.9	\[ b \cdot \left(0.5 \cdot \left(e^{a} - \color{blue}{e^{-a}}\right)\right) \]

   if 1.99999999999999991e-6 < (*.f64 (sinh.f64 a) b)

   1. Initial program 23.5

      \[\log \left(\sinh a \cdot b + \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right) \]

   2. Simplified18.2

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\sinh a, b, \mathsf{hypot}\left(1, \sinh a \cdot b\right)\right)\right)} \]
      Proof

      [Start]23.5	\[ \log \left(\sinh a \cdot b + \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right) \]
      fma-def [=>]23.5	\[ \log \color{blue}{\left(\mathsf{fma}\left(\sinh a, b, \sqrt{1 + {\left(\sinh a \cdot b\right)}^{2}}\right)\right)} \]
      unpow2 [=>]23.5	\[ \log \left(\mathsf{fma}\left(\sinh a, b, \sqrt{1 + \color{blue}{\left(\sinh a \cdot b\right) \cdot \left(\sinh a \cdot b\right)}}\right)\right) \]
      hypot-1-def [=>]18.2	\[ \log \left(\mathsf{fma}\left(\sinh a, b, \color{blue}{\mathsf{hypot}\left(1, \sinh a \cdot b\right)}\right)\right) \]

   3. Applied egg-rr18.1

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(1, \sinh a \cdot b\right) + \sinh a \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh a \cdot b \leq -4000:\\ \;\;\;\;\log \left(\frac{\frac{-0.5}{b}}{\sinh a}\right)\\ \mathbf{elif}\;\sinh a \cdot b \leq 2 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \left(0.5 \cdot \left(e^{a} - e^{-a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sinh a \cdot b + \mathsf{hypot}\left(1, \sinh a \cdot b\right)\right)\\ \end{array} \]

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (a b)
  :name "log(sinh(a) * b + sqrt(1 + (sinh(a) * b) ^ 2))"
  :precision binary64
  :pre (and (and (<= 1.0 a) (<= a 1000.0)) (and (<= -1.0 b) (<= b 1.0)))
  (log (+ (* (sinh a) b) (sqrt (+ 1.0 (pow (* (sinh a) b) 2.0))))))