Average Error: 60.5 → 28.8
Time: 18.1s
Precision: binary64
Cost: 20288
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
↓
\[\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}} \]
(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
 (/
  1.0
  (* (/ (expm1 (* eps b)) eps) (/ (expm1 (* eps a)) (expm1 (* eps (+ b 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) {
	return 1.0 / ((expm1((eps * b)) / eps) * (expm1((eps * a)) / expm1((eps * (b + a)))));
}

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) {
	return 1.0 / ((Math.expm1((eps * b)) / eps) * (Math.expm1((eps * a)) / Math.expm1((eps * (b + a)))));
}

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):
	return 1.0 / ((math.expm1((eps * b)) / eps) * (math.expm1((eps * a)) / math.expm1((eps * (b + a)))))

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)
	return Float64(1.0 / Float64(Float64(expm1(Float64(eps * b)) / eps) * Float64(expm1(Float64(eps * a)) / expm1(Float64(eps * Float64(b + a))))))
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_] := N[(1.0 / N[(N[(N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[N[(eps * N[(b + a), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $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)}
↓
\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.5
Target15.7
Herbie28.8
\[\frac{a + b}{a \cdot b} \]

Derivation

  1. Initial program 60.5

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

  2. Simplified29.1

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
    Proof
    (*.f64 (/.f64 (expm1.f64 (*.f64 eps (+.f64 a b))) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) eps))) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (expm1.f64 (*.f64 eps a))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 95 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 a eps)))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a eps)) 1))) (/.f64 eps (expm1.f64 (*.f64 eps b)))): 106 points increase in error, 6 points decrease in error
    (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 eps (expm1.f64 (Rewrite<= *-commutative_binary64 (*.f64 b eps))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) (-.f64 (exp.f64 (*.f64 a eps)) 1)) (/.f64 eps (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 24 points increase in error, 0 points decrease in error
    (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1) eps) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))): 0 points increase in error, 1 points decrease in error
    (/.f64 (Rewrite<= *-commutative_binary64 (*.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. Applied egg-rr28.8

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}}} \]

  4. Final simplification28.8

    \[\leadsto \frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}} \]

Alternatives

Alternative 1
Error28.8
Cost20288
\[\frac{1}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \frac{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}} \]
Alternative 2
Error39.6
Cost20160
\[\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)} \]
Alternative 3
Error39.2
Cost20160
\[\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)} \]
Alternative 4
Error29.0
Cost20160
\[\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
Alternative 5
Error29.1
Cost20160
\[\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \]
Alternative 6
Error28.9
Cost20160
\[\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
Alternative 7
Error28.9
Cost20160
\[\frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \]

Error

Reproduce

herbie shell --seed 2022334 
(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))))