expq3 (problem 3.4.2)

Percentage Accurate: 1.4% → 99.6%

Time: 31.2s

Precision: binary64

Cost: 704

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. a = 1.6336217463793145e+169
  2. b = 3.5443913197730517e+251
  3. eps = 2.573518807610884e-178

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

Math

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

↓

\[\left(\frac{1}{b} + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \]

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 (- (+ (/ 1.0 b) (/ 1.0 a)) (* 0.5 eps)))

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) {
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
}

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
    code = ((1.0d0 / b) + (1.0d0 / a)) - (0.5d0 * eps)
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) {
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
}

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):
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps)

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)
	return Float64(Float64(Float64(1.0 / b) + Float64(1.0 / a)) - Float64(0.5 * eps))
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 = code(a, b, eps)
	tmp = ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
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_] := N[(N[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]

Target

Original	1.4%
Target	79.7%
Herbie	99.6%

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

Derivation

1. Initial program 1.2%

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

2. Simplified 34.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
   Step-by-step derivation

   [Start] 1.2%	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   associate-*l/ [<=] 1.2%	\[ \color{blue}{\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)} \]
   *-commutative [=>] 1.2%	\[ \color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
   expm1-def [=>] 3.1%	\[ \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   *-commutative [=>] 3.1%	\[ \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   expm1-def [=>] 11.3%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   *-commutative [=>] 11.3%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   expm1-def [=>] 34.4%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
   *-commutative [=>] 34.4%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

3. Taylor expanded in b around 0 14.4%

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

4. Applied egg-rr 3.5%

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

   [Start] 14.4%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   sub-neg [=>] 14.4%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{e^{\varepsilon \cdot a} + \left(-1\right)}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   exp-prod [=>] 3.5%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{{\left(e^{\varepsilon}\right)}^{a}} + \left(-1\right)} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   metadata-eval [=>] 3.5%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{{\left(e^{\varepsilon}\right)}^{a} + \color{blue}{-1}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]

5. Simplified 70.4%

   \[\leadsto \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   Step-by-step derivation

   [Start] 3.5%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{{\left(e^{\varepsilon}\right)}^{a} + -1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   metadata-eval [<=] 3.5%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{{\left(e^{\varepsilon}\right)}^{a} + \color{blue}{\left(-1\right)}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   sub-neg [<=] 3.5%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{{\left(e^{\varepsilon}\right)}^{a} - 1}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   exp-prod [<=] 14.4%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{e^{\varepsilon \cdot a}} - 1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   expm1-def [=>] 70.4%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   *-commutative [=>] 70.4%	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\mathsf{expm1}\left(\color{blue}{a \cdot \varepsilon}\right)} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]

6. Taylor expanded in eps around 0 99.8%

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

7. Simplified 99.8%

   \[\leadsto \color{blue}{\left(\frac{1}{b} + \frac{1}{a}\right)} - 0.5 \cdot \varepsilon \]
   Step-by-step derivation

   [Start] 99.8%	\[ \left(\frac{1}{a} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   +-commutative [=>] 99.8%	\[ \color{blue}{\left(\frac{1}{b} + \frac{1}{a}\right)} - 0.5 \cdot \varepsilon \]

8. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	704

\[\left(\frac{1}{b} + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \]

Alternative 2
Accuracy	99.5%
Cost	448

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

Alternative 3
Accuracy	61.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Accuracy	50.0%
Cost	192

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

Reproduce

herbie shell --seed 2023277 
(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))))