expq3 (problem 3.4.2)

Average Error: 60.0 → 0.3

Time: 60.0s

Precision: binary64

Cost: 61768

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ t_1 := \frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;\frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} + -1}{e^{\varepsilon \cdot b} + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (- (exp (* (+ a b) eps)) 1.0))
          (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))
        (t_1 (+ (/ 1.0 a) (/ 1.0 b))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 0.5)
       (*
        (/ eps (+ (exp (* eps a)) -1.0))
        (/ (+ (exp (* eps (+ a b))) -1.0) (+ (exp (* eps b)) -1.0)))
       t_1))))

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 * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
	double t_1 = (1.0 / a) + (1.0 / b);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = (eps / (exp((eps * a)) + -1.0)) * ((exp((eps * (a + b))) + -1.0) / (exp((eps * b)) + -1.0));
	} else {
		tmp = t_1;
	}
	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 * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
	double t_1 = (1.0 / a) + (1.0 / b);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= 0.5) {
		tmp = (eps / (Math.exp((eps * a)) + -1.0)) * ((Math.exp((eps * (a + b))) + -1.0) / (Math.exp((eps * b)) + -1.0));
	} else {
		tmp = t_1;
	}
	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 * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))
	t_1 = (1.0 / a) + (1.0 / b)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= 0.5:
		tmp = (eps / (math.exp((eps * a)) + -1.0)) * ((math.exp((eps * (a + b))) + -1.0) / (math.exp((eps * b)) + -1.0))
	else:
		tmp = t_1
	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(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)))
	t_1 = Float64(Float64(1.0 / a) + Float64(1.0 / b))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = Float64(Float64(eps / Float64(exp(Float64(eps * a)) + -1.0)) * Float64(Float64(exp(Float64(eps * Float64(a + b))) + -1.0) / Float64(exp(Float64(eps * b)) + -1.0)));
	else
		tmp = t_1;
	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)
	t_0 = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
	t_1 = (1.0 / a) + (1.0 / b);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= 0.5)
		tmp = (eps / (exp((eps * a)) + -1.0)) * ((exp((eps * (a + b))) + -1.0) / (exp((eps * b)) + -1.0));
	else
		tmp = t_1;
	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_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, 0.5], N[(N[(eps / N[(N[Exp[N[(eps * a), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Exp[N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Exp[N[(eps * b), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	60.0
Target	15.0
Herbie	0.3

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

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 0.5 < (/.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 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-46 [=>] 64.0	\[ \color{blue}{\frac{\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{e^{a \cdot \varepsilon} - 1}}{e^{b \cdot \varepsilon} - 1}} \]
      rational.json-simplify-49 [=>] 64.0	\[ \frac{\color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-49 [=>] 64.0	\[ \color{blue}{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1}} \]
      rational.json-simplify-16 [=>] 64.0	\[ \frac{\varepsilon}{\color{blue}{e^{a \cdot \varepsilon} + -1}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\varepsilon}{e^{\color{blue}{\varepsilon \cdot a}} + -1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-16 [=>] 64.0	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{\color{blue}{e^{\left(a + b\right) \cdot \varepsilon} + -1}}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\color{blue}{\varepsilon \cdot \left(a + b\right)}} + -1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-16 [=>] 64.0	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} + -1}{\color{blue}{e^{b \cdot \varepsilon} + -1}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} + -1}{e^{\color{blue}{\varepsilon \cdot b}} + -1} \]

   3. Taylor expanded in eps around 0 13.3

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

   4. Simplified 0.0

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

      [Start] 13.3	\[ \frac{a + b}{a \cdot b} \]
      rational.json-simplify-28 [=>] 0.0	\[ \color{blue}{\frac{1}{a} + \frac{1}{b}} \]

   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))) < 0.5

   1. Initial program 3.6

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

   2. Simplified 3.6

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

      [Start] 3.6	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-46 [=>] 3.6	\[ \color{blue}{\frac{\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{e^{a \cdot \varepsilon} - 1}}{e^{b \cdot \varepsilon} - 1}} \]
      rational.json-simplify-49 [=>] 3.6	\[ \frac{\color{blue}{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-49 [=>] 3.6	\[ \color{blue}{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1}} \]
      rational.json-simplify-16 [=>] 3.6	\[ \frac{\varepsilon}{\color{blue}{e^{a \cdot \varepsilon} + -1}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-2 [=>] 3.6	\[ \frac{\varepsilon}{e^{\color{blue}{\varepsilon \cdot a}} + -1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-16 [=>] 3.6	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{\color{blue}{e^{\left(a + b\right) \cdot \varepsilon} + -1}}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-2 [=>] 3.6	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\color{blue}{\varepsilon \cdot \left(a + b\right)}} + -1}{e^{b \cdot \varepsilon} - 1} \]
      rational.json-simplify-16 [=>] 3.6	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} + -1}{\color{blue}{e^{b \cdot \varepsilon} + -1}} \]
      rational.json-simplify-2 [=>] 3.6	\[ \frac{\varepsilon}{e^{\varepsilon \cdot a} + -1} \cdot \frac{e^{\varepsilon \cdot \left(a + b\right)} + -1}{e^{\color{blue}{\varepsilon \cdot b}} + -1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	3.6
Cost	448

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

Alternative 2
Error	13.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Error	32.9
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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))))