expq3 (problem 3.4.2)

Percentage Accurate: 3.2% → 96.7%

Time: 18.7s

Precision: binary64

Cost: 964

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

• could not determine a ground truth

  1. a = -9.055673953277239e+247
  2. b = -3.3728681062022286e+175
  3. eps = -7.182659804349337e-169

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

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{1}{b \cdot \left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5\\ \end{array} \]

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
 (if (<= a -8.2e+183)
   (* (* (+ a b) eps) (/ 1.0 (* b (* a eps))))
   (- (+ (/ 1.0 a) (/ 1.0 b)) (* eps 0.5))))

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) {
	double tmp;
	if (a <= -8.2e+183) {
		tmp = ((a + b) * eps) * (1.0 / (b * (a * eps)));
	} else {
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-8.2d+183)) then
        tmp = ((a + b) * eps) * (1.0d0 / (b * (a * eps)))
    else
        tmp = ((1.0d0 / a) + (1.0d0 / b)) - (eps * 0.5d0)
    end if
    code = tmp
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) {
	double tmp;
	if (a <= -8.2e+183) {
		tmp = ((a + b) * eps) * (1.0 / (b * (a * eps)));
	} else {
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
	}
	return tmp;
}

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):
	tmp = 0
	if a <= -8.2e+183:
		tmp = ((a + b) * eps) * (1.0 / (b * (a * eps)))
	else:
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5)
	return tmp

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)
	tmp = 0.0
	if (a <= -8.2e+183)
		tmp = Float64(Float64(Float64(a + b) * eps) * Float64(1.0 / Float64(b * Float64(a * eps))));
	else
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(1.0 / b)) - Float64(eps * 0.5));
	end
	return tmp
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_2 = code(a, b, eps)
	tmp = 0.0;
	if (a <= -8.2e+183)
		tmp = ((a + b) * eps) * (1.0 / (b * (a * eps)));
	else
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[a, -8.2e+183], N[(N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision] * N[(1.0 / N[(b * N[(a * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]]

Target

Original	3.2%
Target	80.6%
Herbie	96.7%

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

Derivation

1. Split input into 2 regimes

2. if a < -8.20000000000000029e183

   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. Simplified 48.8%

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

      [Start] 4.2%	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*l/ [<=] 4.2%	\[ \color{blue}{\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 4.2%	\[ \color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 4.6%	\[ \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 4.6%	\[ \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 4.5%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 4.5%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 48.8%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 48.8%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in eps around 0 22.1%

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

   4. Simplified 22.9%

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

      [Start] 22.1%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{\varepsilon \cdot \left(a \cdot b\right)} \]
      associate-*r* [=>] 22.9%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{\color{blue}{\left(\varepsilon \cdot a\right) \cdot b}} \]
      *-commutative [=>] 22.9%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{\color{blue}{\left(a \cdot \varepsilon\right)} \cdot b} \]

   5. Taylor expanded in eps around 0 99.6%

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

   6. Simplified 99.6%

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

      [Start] 99.6%	\[ \left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{\left(a \cdot \varepsilon\right) \cdot b} \]
      *-commutative [=>] 99.6%	\[ \color{blue}{\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{1}{\left(a \cdot \varepsilon\right) \cdot b} \]

   if -8.20000000000000029e183 < a

   1. Initial program 3.7%

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

   2. Simplified 35.2%

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

      [Start] 3.7%	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*l/ [<=] 3.7%	\[ \color{blue}{\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 3.7%	\[ \color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 4.9%	\[ \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 4.9%	\[ \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 16.8%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 16.8%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 35.2%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 35.2%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in a around 0 16.6%

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

   4. Simplified 41.3%

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

      [Start] 16.6%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{a \cdot \left(e^{\varepsilon \cdot b} - 1\right)} \]
      expm1-def [=>] 41.3%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{a \cdot \color{blue}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      associate-/r* [=>] 41.3%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \color{blue}{\frac{\frac{1}{a}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}} \]
      *-commutative [=>] 41.3%	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{1}{a}}{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon}\right)} \]

   5. Taylor expanded in eps around 0 40.2%

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

   6. Simplified 40.2%

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

      [Start] 59.0%	\[ \left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{1}{\left(a \cdot \varepsilon\right) \cdot b} \]
      *-commutative [=>] 59.0%	\[ \color{blue}{\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{1}{\left(a \cdot \varepsilon\right) \cdot b} \]

   7. Taylor expanded in b around 0 95.1%

      \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{1}{b \cdot \left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.7%
Cost	964

\[\begin{array}{l} \mathbf{if}\;a \leq -8.2 \cdot 10^{+183}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{1}{b \cdot \left(a \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	78.2%
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 2050000000000:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{a}{b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Accuracy	78.2%
Cost	589

\[\begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-115} \lor \neg \left(b \leq 1750000000000\right) \land b \leq 2.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Accuracy	94.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+278}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \end{array} \]

Alternative 5
Accuracy	95.3%
Cost	448

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

Alternative 6
Accuracy	48.3%
Cost	192

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

Reproduce

herbie shell --seed 2023272 
(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))))