expq3 (problem 3.4.2)

Average Error: 60.3 → 1.0

Time: 46.8s

Precision: binary64

Cost: 61768

\[-1 < \varepsilon \land \varepsilon < 1\]

Math

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{b} + \left(\left(-\varepsilon\right) + \frac{1}{a}\right)\\ \mathbf{elif}\;t_0 \leq 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

↓

(FPCore (a b eps)
 :precision binary64
 (let* ((t_0
         (/
          (* eps (- (exp (* (+ a b) eps)) 1.0))
          (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))))
   (if (<= t_0 -2e-105)
     (+ (/ 1.0 b) (+ (- eps) (/ 1.0 a)))
     (if (<= t_0 1e-160) t_0 (+ (/ 1.0 a) (/ 1.0 b))))))

C

double code(double a, double b, double eps) {
	return (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
}

↓

double code(double a, double b, double eps) {
	double t_0 = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
	double tmp;
	if (t_0 <= -2e-105) {
		tmp = (1.0 / b) + (-eps + (1.0 / a));
	} else if (t_0 <= 1e-160) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) + (1.0 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

↓

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
    if (t_0 <= (-2d-105)) then
        tmp = (1.0d0 / b) + (-eps + (1.0d0 / a))
    else if (t_0 <= 1d-160) then
        tmp = t_0
    else
        tmp = (1.0d0 / a) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	return (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
}

↓

public static double code(double a, double b, double eps) {
	double t_0 = (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
	double tmp;
	if (t_0 <= -2e-105) {
		tmp = (1.0 / b) + (-eps + (1.0 / a));
	} else if (t_0 <= 1e-160) {
		tmp = t_0;
	} else {
		tmp = (1.0 / a) + (1.0 / b);
	}
	return tmp;
}

Python

def code(a, b, eps):
	return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))

↓

def code(a, b, eps):
	t_0 = (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))
	tmp = 0
	if t_0 <= -2e-105:
		tmp = (1.0 / b) + (-eps + (1.0 / a))
	elif t_0 <= 1e-160:
		tmp = t_0
	else:
		tmp = (1.0 / a) + (1.0 / b)
	return tmp

Julia

function code(a, b, eps)
	return Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0)))
end

↓

function code(a, b, eps)
	t_0 = Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0)))
	tmp = 0.0
	if (t_0 <= -2e-105)
		tmp = Float64(Float64(1.0 / b) + Float64(Float64(-eps) + Float64(1.0 / a)));
	elseif (t_0 <= 1e-160)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / a) + Float64(1.0 / b));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
end

↓

function tmp_2 = code(a, b, eps)
	t_0 = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
	tmp = 0.0;
	if (t_0 <= -2e-105)
		tmp = (1.0 / b) + (-eps + (1.0 / a));
	elseif (t_0 <= 1e-160)
		tmp = t_0;
	else
		tmp = (1.0 / a) + (1.0 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, eps_] := Block[{t$95$0 = N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-105], N[(N[(1.0 / b), $MachinePrecision] + N[((-eps) + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-160], t$95$0, N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	60.3
Target	14.5
Herbie	1.0

\[\frac{a + b}{a \cdot b} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -1.99999999999999993e-105

   1. Initial program 58.9

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   2. Taylor expanded in a around 0 44.8

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   3. Simplified 41.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot {\left(\varepsilon \cdot a\right)}^{4}\right) + 0.5 \cdot {\left(\varepsilon \cdot a\right)}^{2}\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      Proof

      [Start] 44.8	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-41 [<=] 44.8	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \color{blue}{\left(\varepsilon \cdot a + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right)\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-41 [=>] 44.8	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot a + \left(\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right)\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-1 [=>] 44.8	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\color{blue}{\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right)\right)} + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 44.8	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{3}} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right)\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 42.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{4}}\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 41.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot {\left(\varepsilon \cdot a\right)}^{4}\right) + 0.5 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{2}}\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   4. Taylor expanded in a around 0 32.4

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

   5. Simplified 32.4

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

      [Start] 32.4	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot b}}{e^{\varepsilon \cdot b} - 1} + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \]
      rational.json-simplify-48 [=>] 32.4	\[ \color{blue}{\frac{1}{a} + \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot b}}{e^{\varepsilon \cdot b} - 1} - 0.5 \cdot \varepsilon\right)} \]
      rational.json-simplify-2 [=>] 32.4	\[ \frac{1}{a} + \left(\frac{\color{blue}{e^{\varepsilon \cdot b} \cdot \varepsilon}}{e^{\varepsilon \cdot b} - 1} - 0.5 \cdot \varepsilon\right) \]
      rational.json-simplify-49 [=>] 32.4	\[ \frac{1}{a} + \left(\color{blue}{\varepsilon \cdot \frac{e^{\varepsilon \cdot b}}{e^{\varepsilon \cdot b} - 1}} - 0.5 \cdot \varepsilon\right) \]
      rational.json-simplify-52 [=>] 32.4	\[ \frac{1}{a} + \color{blue}{\varepsilon \cdot \left(\frac{e^{\varepsilon \cdot b}}{e^{\varepsilon \cdot b} - 1} - 0.5\right)} \]
      rational.json-simplify-15 [<=] 32.4	\[ \frac{1}{a} + \varepsilon \cdot \left(\frac{e^{\varepsilon \cdot b}}{\color{blue}{e^{\varepsilon \cdot b} + -1}} - 0.5\right) \]

   6. Taylor expanded in eps around 0 32.6

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

   7. Taylor expanded in eps around 0 0.7

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

   8. Simplified 0.7

      \[\leadsto \color{blue}{\frac{1}{b} + \left(\left(-\varepsilon\right) + \frac{1}{a}\right)} \]
      Proof

      [Start] 0.7	\[ -1 \cdot \varepsilon + \left(\frac{1}{b} + \frac{1}{a}\right) \]
      rational.json-simplify-41 [=>] 0.7	\[ \color{blue}{\frac{1}{b} + \left(\frac{1}{a} + -1 \cdot \varepsilon\right)} \]
      rational.json-simplify-1 [=>] 0.7	\[ \frac{1}{b} + \color{blue}{\left(-1 \cdot \varepsilon + \frac{1}{a}\right)} \]
      rational.json-simplify-2 [=>] 0.7	\[ \frac{1}{b} + \left(\color{blue}{\varepsilon \cdot -1} + \frac{1}{a}\right) \]
      rational.json-simplify-9 [=>] 0.7	\[ \frac{1}{b} + \left(\color{blue}{\left(-\varepsilon\right)} + \frac{1}{a}\right) \]

   if -1.99999999999999993e-105 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 9.9999999999999999e-161

   1. Initial program 2.1

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   if 9.9999999999999999e-161 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

   1. Initial program 62.9

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   2. Taylor expanded in a around 0 60.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   3. Simplified 60.4

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot {\left(\varepsilon \cdot a\right)}^{4}\right) + 0.5 \cdot {\left(\varepsilon \cdot a\right)}^{2}\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      Proof

      [Start] 60.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-41 [<=] 60.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \color{blue}{\left(\varepsilon \cdot a + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right)\right)\right)}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-41 [=>] 60.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\varepsilon \cdot a + \left(\left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right)\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-1 [=>] 60.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\color{blue}{\left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right)\right)} + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 60.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{3}} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot {a}^{4}\right)\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 60.5	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{4}}\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      exponential.json-simplify-27 [=>] 60.4	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\varepsilon \cdot a + \left(\left(0.16666666666666666 \cdot {\left(\varepsilon \cdot a\right)}^{3} + 0.041666666666666664 \cdot {\left(\varepsilon \cdot a\right)}^{4}\right) + 0.5 \cdot \color{blue}{{\left(\varepsilon \cdot a\right)}^{2}}\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   4. Taylor expanded in eps around 0 14.6

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

   5. Simplified 1.0

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}} \]
      Proof

      [Start] 14.6	\[ \frac{a + b}{a \cdot b} \]
      rational.json-simplify-28 [=>] 1.0	\[ \color{blue}{\frac{1}{a} + \frac{1}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leq -2 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{b} + \left(\left(-\varepsilon\right) + \frac{1}{a}\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leq 10^{-160}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.9
Cost	640

\[\frac{1}{b} + \left(\left(-\varepsilon\right) + \frac{1}{a}\right) \]

Alternative 2
Error	25.7
Cost	588

\[\begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Error	3.3
Cost	448

\[\frac{1}{a} + \frac{1}{b} \]

Alternative 4
Error	32.8
Cost	192

\[\frac{1}{a} \]

Reproduce

herbie shell --seed 2023074 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))