Average Error: 60.2 → 0.3
Time: 20.9s
Precision: binary64
Cost: 61384
\[-1 < \varepsilon \land \varepsilon < 1\]
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
↓
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;t_1 \leq 10^{-8}:\\ \;\;\;\;\frac{\varepsilon \cdot \frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]
(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 (+ a b)))
        (t_1
         (/
          (* eps (+ (exp t_0) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (<= t_1 (- INFINITY))
     (/ (/ (+ a b) b) a)
     (if (<= t_1 1e-8)
       (/ (* eps (/ (expm1 t_0) (expm1 (* eps a)))) (expm1 (* eps b)))
       (+ (/ 1.0 b) (/ 1.0 a))))))
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 * (a + b);
	double t_1 = (eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((a + b) / b) / a;
	} else if (t_1 <= 1e-8) {
		tmp = (eps * (expm1(t_0) / expm1((eps * a)))) / expm1((eps * b));
	} else {
		tmp = (1.0 / b) + (1.0 / a);
	}
	return tmp;
}

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 * (a + b);
	double t_1 = (eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((a + b) / b) / a;
	} else if (t_1 <= 1e-8) {
		tmp = (eps * (Math.expm1(t_0) / Math.expm1((eps * a)))) / Math.expm1((eps * b));
	} else {
		tmp = (1.0 / b) + (1.0 / a);
	}
	return tmp;
}

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 * (a + b)
	t_1 = (eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((a + b) / b) / a
	elif t_1 <= 1e-8:
		tmp = (eps * (math.expm1(t_0) / math.expm1((eps * a)))) / math.expm1((eps * b))
	else:
		tmp = (1.0 / b) + (1.0 / a)
	return tmp

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(eps * Float64(a + b))
	t_1 = Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(a + b) / b) / a);
	elseif (t_1 <= 1e-8)
		tmp = Float64(Float64(eps * Float64(expm1(t_0) / expm1(Float64(eps * a)))) / expm1(Float64(eps * b)));
	else
		tmp = Float64(Float64(1.0 / b) + Float64(1.0 / a));
	end
	return tmp
end

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[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * N[(N[Exp[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(eps * a), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Exp[N[(eps * b), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(a + b), $MachinePrecision] / b), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 1e-8], N[(N[(eps * N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]]]]]

\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 := \varepsilon \cdot \left(a + b\right)\\
t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\frac{\frac{a + b}{b}}{a}\\

\mathbf{elif}\;t_1 \leq 10^{-8}:\\
\;\;\;\;\frac{\varepsilon \cdot \frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.2
Target15.5
Herbie0.3
\[\frac{a + b}{a \cdot b} \]

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))) < -inf.0

    1. Initial program 64.0

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

    2. Simplified17.1

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 53 points increase in error, 1 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 67 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.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)))): 2 points increase in error, 1 points decrease in error

    3. Taylor expanded in eps around 0 7.5

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

    4. Applied egg-rr28.5

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

    5. Applied egg-rr2.1

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

    if -inf.0 < (/.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))) < 1e-8

    1. Initial program 4.2

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 53 points increase in error, 1 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 67 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.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)))): 2 points increase in error, 1 points decrease in error

    3. Taylor expanded in eps around inf 4.2

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

    4. Simplified0.1

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      Proof
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 (+.f64 a b) eps)) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps (+.f64 a b)))) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 eps (Rewrite=> +-commutative_binary64 (+.f64 b a)))) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 eps (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 a)))))) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 eps (+.f64 b (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 a)))))) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (expm1.f64 (*.f64 eps (Rewrite<= sub-neg_binary64 (-.f64 b (*.f64 -1 a))))) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1)) (expm1.f64 (*.f64 a eps))) eps) (expm1.f64 (*.f64 b eps))): 113 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps a)))) eps) (expm1.f64 (*.f64 b eps))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 eps a)) 1))) eps) (expm1.f64 (*.f64 b eps))): 125 points increase in error, 4 points decrease in error
      (/.f64 (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) (-.f64 (exp.f64 (*.f64 eps a)) 1)) eps) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) (-.f64 (exp.f64 (*.f64 eps a)) 1)) eps) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 eps b)) 1))): 25 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (/.f64 eps (-.f64 (exp.f64 (*.f64 eps b)) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1) eps) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1)))): 1 points increase in error, 2 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 eps (-.f64 (exp.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))) 1))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (Rewrite=> expm1-def_binary64 (expm1.f64 (*.f64 eps (-.f64 b (*.f64 -1 a)))))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 b (*.f64 -1 a)) eps)))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (*.f64 (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 b (*.f64 (neg.f64 -1) a))) eps))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (*.f64 (+.f64 b (*.f64 (Rewrite=> metadata-eval 1) a)) eps))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (*.f64 (+.f64 b (Rewrite=> *-lft-identity_binary64 a)) eps))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b)) eps))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 eps (+.f64 a b))))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 eps (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 eps (+.f64 a b))) 1))) (*.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (-.f64 (exp.f64 (*.f64 eps b)) 1))): 1 points increase in error, 0 points decrease in error

    if 1e-8 < (/.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 63.9

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

    2. Simplified46.6

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      Proof
      (*.f64 eps (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (expm1.f64 (*.f64 eps a)) (expm1.f64 (*.f64 eps b))))): 53 points increase in error, 1 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps))) (expm1.f64 (*.f64 eps b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1)) (expm1.f64 (*.f64 eps b))))): 67 points increase in error, 3 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 eps (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1))))): 24 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.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)))): 2 points increase in error, 1 points decrease in error

    3. Taylor expanded in eps around 0 14.7

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

    4. Taylor expanded in a around 0 0.1

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 10^{-8}:\\ \;\;\;\;\frac{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error15.1
Cost844
\[\begin{array}{l} \mathbf{if}\;b \leq 5.551263307660544 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.188978128409926 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 4.178900223638555 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]
Alternative 2
Error15.0
Cost844
\[\begin{array}{l} \mathbf{if}\;b \leq 5.551263307660544 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.188978128409926 \cdot 10^{-37}:\\ \;\;\;\;\left(1 + \frac{1}{a}\right) + -1\\ \mathbf{elif}\;b \leq 4.178900223638555 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]
Alternative 3
Error14.9
Cost844
\[\begin{array}{l} \mathbf{if}\;b \leq 5.551263307660544 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{elif}\;b \leq 1.188978128409926 \cdot 10^{-37}:\\ \;\;\;\;\left(1 + \frac{1}{a}\right) + -1\\ \mathbf{elif}\;b \leq 4.178900223638555 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]
Alternative 4
Error3.1
Cost704
\[\left(\frac{1}{b} + \frac{1}{a}\right) + \varepsilon \cdot -0.5 \]
Alternative 5
Error15.2
Cost588
\[\begin{array}{l} \mathbf{if}\;b \leq 5.551263307660544 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.188978128409926 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 4.178900223638555 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]
Alternative 6
Error5.2
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \end{array} \]
Alternative 7
Error3.4
Cost448
\[\frac{1}{b} + \frac{1}{a} \]
Alternative 8
Error61.9
Cost192
\[\varepsilon \cdot -0.5 \]
Alternative 9
Error32.5
Cost192
\[\frac{1}{b} \]

Error

Reproduce

herbie shell --seed 2022294 
(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))))