expq3 (problem 3.4.2)

Average Error: 60.4 → 0.2

Time: 17.3s

Precision: binary64

Cost: 61385

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

Math

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

↓

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(t_0\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \]

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)))
        (t_1
         (/
          (* eps (+ (exp t_0) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-12)))
     (+ (/ 1.0 b) (/ 1.0 a))
     (* (expm1 t_0) (/ (/ eps (expm1 (* eps a))) (expm1 (* eps b)))))))

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 t_1 = (eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-12)) {
		tmp = (1.0 / b) + (1.0 / a);
	} else {
		tmp = expm1(t_0) * ((eps / expm1((eps * a))) / expm1((eps * b)));
	}
	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 t_1 = (eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e-12)) {
		tmp = (1.0 / b) + (1.0 / a);
	} else {
		tmp = Math.expm1(t_0) * ((eps / Math.expm1((eps * a))) / Math.expm1((eps * b)));
	}
	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)
	t_1 = (eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-12):
		tmp = (1.0 / b) + (1.0 / a)
	else:
		tmp = math.expm1(t_0) * ((eps / math.expm1((eps * a))) / math.expm1((eps * b)))
	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))
	t_1 = Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-12))
		tmp = Float64(Float64(1.0 / b) + Float64(1.0 / a));
	else
		tmp = Float64(expm1(t_0) * Float64(Float64(eps / expm1(Float64(eps * a))) / expm1(Float64(eps * b))));
	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]}, Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-12]], $MachinePrecision]], N[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[t$95$0] - 1), $MachinePrecision] * N[(N[(eps / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	60.4
Target	15.2
Herbie	0.2

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

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 4.9999999999999997e-12 < (/.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 63.9

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

   2. Simplified 43.3

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

      [Start] 63.9	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*r/ [<=] 63.9	\[ \color{blue}{\varepsilon \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 63.9	\[ \varepsilon \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 63.9	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 58.3	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 58.3	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 43.3	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 43.3	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in eps around 0 13.7

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

   4. Taylor expanded in a around 0 0.2

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

   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))) < 4.9999999999999997e-12

   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. Simplified 0.1

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

      [Start] 3.2	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*l/ [<=] 3.2	\[ \color{blue}{\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 3.2	\[ \color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      associate-/r* [=>] 3.2	\[ \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \color{blue}{\frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{e^{b \cdot \varepsilon} - 1}} \]
      expm1-def [=>] 3.2	\[ \color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 3.2	\[ \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right) \cdot \frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 2.1	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 2.1	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right)}}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 0.1	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 0.1	\[ \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq 6.5 \cdot 10^{+89}:\\ \;\;\;\;\left(\frac{1}{b} + \varepsilon \cdot 0.5\right) + \left(\frac{1}{a} + \varepsilon \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \end{array} \]

Alternative 2
Error	2.0
Cost	960

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

Alternative 3
Error	27.4
Cost	854

\[\begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+40} \lor \neg \left(a \leq -7 \cdot 10^{-26}\right) \land \left(a \leq -1.45 \cdot 10^{-55} \lor \neg \left(a \leq -7.5 \cdot 10^{-139}\right) \land a \leq -1.1 \cdot 10^{-174}\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Error	3.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{1}{b} + \varepsilon \cdot 0.5\right) + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternative 5
Error	3.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;\left(\frac{1}{b} + \varepsilon \cdot 0.5\right) + \varepsilon \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \left(\frac{1}{a} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 6
Error	28.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{+236}:\\ \;\;\;\;\frac{b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 7
Error	3.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+247}:\\ \;\;\;\;\left(1 + \frac{1}{b}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternative 8
Error	3.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+247}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array} \]

Alternative 9
Error	3.3
Cost	448

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

Alternative 10
Error	62.0
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 11
Error	33.4
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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))))