Average Error: 60.2 → 0.5
Time: 20.7s
Precision: binary64
Cost: 74248
\[-1 < \varepsilon \land \varepsilon < 1\]
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \sqrt[3]{{\left(\frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]
(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 5e-13)
       (*
        eps
        (cbrt
         (pow (/ (expm1 t_0) (* (expm1 (* eps a)) (expm1 (* eps b)))) 3.0)))
       (+ (/ 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 <= 5e-13) {
		tmp = eps * cbrt(pow((expm1(t_0) / (expm1((eps * a)) * expm1((eps * b)))), 3.0));
	} 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 <= 5e-13) {
		tmp = eps * Math.cbrt(Math.pow((Math.expm1(t_0) / (Math.expm1((eps * a)) * Math.expm1((eps * b)))), 3.0));
	} 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 <= 5e-13)
		tmp = Float64(eps * cbrt((Float64(expm1(t_0) / Float64(expm1(Float64(eps * a)) * expm1(Float64(eps * b)))) ^ 3.0)));
	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, 5e-13], N[(eps * N[Power[N[Power[N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] * N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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 5 \cdot 10^{-13}:\\
\;\;\;\;\varepsilon \cdot \sqrt[3]{{\left(\frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{3}}\\

\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
Target14.9
Herbie0.5
\[\frac{a + b}{a \cdot b} \]

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. Taylor expanded in eps around 0 6.5

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
    3. Simplified6.5

      \[\leadsto \color{blue}{\frac{b + a}{b \cdot a}} \]
      Proof
      (/.f64 (+.f64 b a) (*.f64 b a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b)) (*.f64 b a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 a b) (Rewrite<= *-commutative_binary64 (*.f64 a b))): 0 points increase in error, 0 points decrease in error
    4. Applied egg-rr2.4

      \[\leadsto \color{blue}{\frac{b + a}{b} \cdot \frac{1}{a}} \]
    5. Applied egg-rr2.4

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

    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))) < 4.9999999999999999e-13

    1. Initial program 2.5

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
    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))))): 30 points increase in error, 0 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))))): 39 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))))): 26 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)))): 0 points increase in error, 2 points decrease in error
    3. Applied egg-rr0.2

      \[\leadsto \varepsilon \cdot \color{blue}{\sqrt[3]{{\left(\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{3}}} \]

    if 4.9999999999999999e-13 < (/.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. Taylor expanded in b around 0 55.0

      \[\leadsto \color{blue}{\left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon} \]
    3. Simplified54.6

      \[\leadsto \color{blue}{\frac{\varepsilon}{1 - e^{-\varepsilon \cdot a}} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)} \]
      Proof
      (+.f64 (/.f64 eps (-.f64 1 (exp.f64 (neg.f64 (*.f64 eps a))))) (+.f64 (/.f64 1 b) (*.f64 eps -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 eps (-.f64 (Rewrite<= *-inverses_binary64 (/.f64 (exp.f64 (*.f64 eps a)) (exp.f64 (*.f64 eps a)))) (exp.f64 (neg.f64 (*.f64 eps a))))) (+.f64 (/.f64 1 b) (*.f64 eps -1/2))): 37 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 eps (-.f64 (/.f64 (exp.f64 (*.f64 eps a)) (exp.f64 (*.f64 eps a))) (Rewrite<= rec-exp_binary64 (/.f64 1 (exp.f64 (*.f64 eps a)))))) (+.f64 (/.f64 1 b) (*.f64 eps -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 eps (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (exp.f64 (*.f64 eps a)) 1) (exp.f64 (*.f64 eps a))))) (+.f64 (/.f64 1 b) (*.f64 eps -1/2))): 0 points increase in error, 36 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1))) (+.f64 (/.f64 1 b) (*.f64 eps -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (+.f64 (/.f64 1 b) (*.f64 eps (Rewrite<= metadata-eval (neg.f64 1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (+.f64 (/.f64 1 b) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 eps 1/2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (+.f64 (/.f64 1 b) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 eps))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 b) (*.f64 1/2 eps)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (*.f64 eps (exp.f64 (*.f64 eps a))) (-.f64 (exp.f64 (*.f64 eps a)) 1)) (/.f64 1 b)) (*.f64 1/2 eps))): 0 points increase in error, 0 points decrease in error
    4. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}} \]
    5. Simplified0.2

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}} \]
      Proof
      (+.f64 (/.f64 1 b) (/.f64 1 a)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 1 a) (/.f64 1 b))): 0 points increase in error, 0 points decrease in error
  3. Recombined 3 regimes into one program.
  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\varepsilon \cdot \sqrt[3]{{\left(\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error0.5
Cost61384
\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]
Alternative 2
Error3.2
Cost704
\[\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right) \]
Alternative 3
Error14.1
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -1.631208889103719 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]
Alternative 4
Error14.0
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -1.631208889103719 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]
Alternative 5
Error5.6
Cost580
\[\begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+140}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \varepsilon \cdot -0.5\\ \end{array} \]
Alternative 6
Error3.7
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -8.86491277646649 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a + b}{b}}{a}\\ \end{array} \]
Alternative 7
Error3.5
Cost448
\[\frac{1}{b} + \frac{1}{a} \]
Alternative 8
Error14.3
Cost324
\[\begin{array}{l} \mathbf{if}\;a \leq -1.631208889103719 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]
Alternative 9
Error62.0
Cost192
\[\varepsilon \cdot -0.5 \]
Alternative 10
Error33.5
Cost192
\[\frac{1}{b} \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))