expq3 (problem 3.4.2)

Percentage Accurate: 6.5% → 95.1%

Time: 14.4s

Precision: binary64

Cost: 34244

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

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = 1.9999999999999999e-154
  3. eps = -1.9999999999999999e-77

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 0:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

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
 (let* ((t_0 (* eps (+ a b))))
   (if (<=
        (/
         (* eps (+ (exp t_0) -1.0))
         (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))
        0.0)
     (* (/ 1.0 a) (/ (expm1 t_0) (expm1 (* eps b))))
     (+ (/ 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 t_0 = eps * (a + b);
	double tmp;
	if (((eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0))) <= 0.0) {
		tmp = (1.0 / a) * (expm1(t_0) / expm1((eps * b)));
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	}
	return tmp;
}

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 t_0 = eps * (a + b);
	double tmp;
	if (((eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0))) <= 0.0) {
		tmp = (1.0 / a) * (Math.expm1(t_0) / Math.expm1((eps * b)));
	} 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):
	t_0 = eps * (a + b)
	tmp = 0
	if ((eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))) <= 0.0:
		tmp = (1.0 / a) * (math.expm1(t_0) / math.expm1((eps * b)))
	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)
	t_0 = Float64(eps * Float64(a + b))
	tmp = 0.0
	if (Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0))) <= 0.0)
		tmp = Float64(Float64(1.0 / a) * Float64(expm1(t_0) / expm1(Float64(eps * b))));
	else
		tmp = Float64(Float64(1.0 / a) + Float64(Float64(1.0 / b) + Float64(eps * -0.5)));
	end
	return 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_] := Block[{t$95$0 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 0.0], N[(N[(1.0 / a), $MachinePrecision] * N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] + N[(N[(1.0 / b), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.5%
Target	77.7%
Herbie	95.1%

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

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))) < -0.0

   1. Initial program 23.0%

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

   2. Simplified 86.8%

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

      [Start] 23.0	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      times-frac [=>] 23.0	\[ \color{blue}{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1}} \]
      expm1-def [=>] 45.8	\[ \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 45.8	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right)} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 44.1	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 44.1	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 86.8	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 86.8	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in eps around 0 48.1%

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

   if -0.0 < (/.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)))

   1. Initial program 1.7%

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

   2. Simplified 32.6%

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

      [Start] 1.7	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 2.8	\[ \frac{\varepsilon \cdot \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 2.8	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 9.7	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 9.7	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 32.6	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 32.6	\[ \frac{\varepsilon \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in b around 0 10.9%

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

   4. Simplified 11.8%

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

      [Start] 10.9	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
      associate--l+ [=>] 10.9	\[ \color{blue}{\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right)} \]
      expm1-def [=>] 69.2	\[ \frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{\color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      associate-/l* [=>] 69.2	\[ \color{blue}{\frac{\varepsilon}{\frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{e^{\varepsilon \cdot a}}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      expm1-def [<=] 10.9	\[ \frac{\varepsilon}{\frac{\color{blue}{e^{\varepsilon \cdot a} - 1}}{e^{\varepsilon \cdot a}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      div-sub [=>] 4.1	\[ \frac{\varepsilon}{\color{blue}{\frac{e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a}} - \frac{1}{e^{\varepsilon \cdot a}}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      *-inverses [=>] 11.8	\[ \frac{\varepsilon}{\color{blue}{1} - \frac{1}{e^{\varepsilon \cdot a}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      rec-exp [=>] 11.8	\[ \frac{\varepsilon}{1 - \color{blue}{e^{-\varepsilon \cdot a}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      *-commutative [=>] 11.8	\[ \frac{\varepsilon}{1 - e^{-\color{blue}{a \cdot \varepsilon}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      sub-neg [=>] 11.8	\[ \frac{\varepsilon}{1 - e^{-a \cdot \varepsilon}} + \color{blue}{\left(\frac{1}{b} + \left(-0.5 \cdot \varepsilon\right)\right)} \]
      *-commutative [=>] 11.8	\[ \frac{\varepsilon}{1 - e^{-a \cdot \varepsilon}} + \left(\frac{1}{b} + \left(-\color{blue}{\varepsilon \cdot 0.5}\right)\right) \]
      distribute-rgt-neg-in [=>] 11.8	\[ \frac{\varepsilon}{1 - e^{-a \cdot \varepsilon}} + \left(\frac{1}{b} + \color{blue}{\varepsilon \cdot \left(-0.5\right)}\right) \]
      metadata-eval [=>] 11.8	\[ \frac{\varepsilon}{1 - e^{-a \cdot \varepsilon}} + \left(\frac{1}{b} + \varepsilon \cdot \color{blue}{-0.5}\right) \]

   5. Taylor expanded in eps around 0 99.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 0:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.1%
Cost	704

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

Alternative 2
Accuracy	78.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{b} + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Accuracy	94.7%
Cost	448

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

Alternative 4
Accuracy	78.7%
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 5
Accuracy	3.1%
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 6
Accuracy	48.0%
Cost	192

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

Reproduce

herbie shell --seed 2023162 
(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))))