Average Error: 60.5 → 0.2
Time: 20.1s
Precision: binary64
Cost: 61384
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{b} + \frac{1}{a}\\ t_1 := \varepsilon \cdot \left(a + b\right)\\ t_2 := \frac{\varepsilon \cdot \left(e^{t_1} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(t_1\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(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 (+ (/ 1.0 b) (/ 1.0 a)))
        (t_1 (* eps (+ a b)))
        (t_2
         (/
          (* eps (+ (exp t_1) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 5e-27)
       (* eps (/ (expm1 t_1) (* (expm1 (* eps a)) (expm1 (* eps b)))))
       t_0))))
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 = (1.0 / b) + (1.0 / a);
	double t_1 = eps * (a + b);
	double t_2 = (eps * (exp(t_1) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= 5e-27) {
		tmp = eps * (expm1(t_1) / (expm1((eps * a)) * expm1((eps * b))));
	} else {
		tmp = t_0;
	}
	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 = (1.0 / b) + (1.0 / a);
	double t_1 = eps * (a + b);
	double t_2 = (eps * (Math.exp(t_1) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_2 <= 5e-27) {
		tmp = eps * (Math.expm1(t_1) / (Math.expm1((eps * a)) * Math.expm1((eps * b))));
	} else {
		tmp = t_0;
	}
	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 = (1.0 / b) + (1.0 / a)
	t_1 = eps * (a + b)
	t_2 = (eps * (math.exp(t_1) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_0
	elif t_2 <= 5e-27:
		tmp = eps * (math.expm1(t_1) / (math.expm1((eps * a)) * math.expm1((eps * b))))
	else:
		tmp = t_0
	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(Float64(1.0 / b) + Float64(1.0 / a))
	t_1 = Float64(eps * Float64(a + b))
	t_2 = Float64(Float64(eps * Float64(exp(t_1) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= 5e-27)
		tmp = Float64(eps * Float64(expm1(t_1) / Float64(expm1(Float64(eps * a)) * expm1(Float64(eps * b)))));
	else
		tmp = t_0;
	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[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps * N[(N[Exp[t$95$1], $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$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, 5e-27], N[(eps * N[(N[(Exp[t$95$1] - 1), $MachinePrecision] / N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] * N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$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)}
↓
\begin{array}{l}
t_0 := \frac{1}{b} + \frac{1}{a}\\
t_1 := \varepsilon \cdot \left(a + b\right)\\
t_2 := \frac{\varepsilon \cdot \left(e^{t_1} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\
\mathbf{if}\;t_2 \leq -\infty:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_2 \leq 5 \cdot 10^{-27}:\\
\;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(t_1\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.5
Target15.1
Herbie0.2
\[\frac{a + b}{a \cdot b} \]

Derivation

  1. Split input into 2 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 or 5.0000000000000002e-27 < (/.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 63.7

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

    2. Simplified42.5

      \[\leadsto \color{blue}{\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      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))))): 38 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))))): 52 points increase in error, 6 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))))): 33 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, 0 points decrease in error

    3. Taylor expanded in eps around 0 13.8

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

    4. Simplified7.7

      \[\leadsto \color{blue}{\frac{\frac{a + b}{a}}{b}} \]
      Proof
      (/.f64 (/.f64 (+.f64 a b) a) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 a b) (*.f64 a b))): 101 points increase in error, 47 points decrease in error

    5. Taylor expanded in a around 0 0.3

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

    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))) < 5.0000000000000002e-27

    1. Initial program 3.2

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

    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))))): 38 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))))): 52 points increase in error, 6 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))))): 33 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, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error3.5
Cost836
\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+262}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{a}{b}}{a}\\ \end{array} \]
Alternative 2
Error3.7
Cost712
\[\begin{array}{l} \mathbf{if}\;b \leq 10^{+152}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+262}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{a}{b}}{a}\\ \end{array} \]
Alternative 3
Error8.1
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{a}{b}}{a}\\ \end{array} \]
Alternative 4
Error8.0
Cost580
\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{a}{b}}{a}\\ \end{array} \]
Alternative 5
Error3.2
Cost448
\[\frac{1}{b} + \frac{1}{a} \]
Alternative 6
Error27.0
Cost324
\[\begin{array}{l} \mathbf{if}\;b \leq 4.012859772327977 \cdot 10^{-166}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]
Alternative 7
Error62.0
Cost192
\[\varepsilon \cdot -0.5 \]
Alternative 8
Error33.4
Cost192
\[\frac{1}{a} \]

Error

Reproduce

herbie shell --seed 2022291 
(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))))