expq3 (problem 3.4.2)

Average Error: 60.4 → 0.6

Time: 50.1s

Precision: binary64

Cost: 61768

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ t_1 := \frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 - 0.5 \cdot \varepsilon\\ \end{array} \]

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 1e-111) t_0 (- t_1 (* 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) {
	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 <= 1e-111) {
		tmp = t_0;
	} else {
		tmp = t_1 - (0.5 * eps);
	}
	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 <= 1e-111) {
		tmp = t_0;
	} else {
		tmp = t_1 - (0.5 * eps);
	}
	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 <= 1e-111:
		tmp = t_0
	else:
		tmp = t_1 - (0.5 * eps)
	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 <= 1e-111)
		tmp = t_0;
	else
		tmp = Float64(t_1 - Float64(0.5 * eps));
	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 <= 1e-111)
		tmp = t_0;
	else
		tmp = t_1 - (0.5 * eps);
	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, 1e-111], t$95$0, N[(t$95$1 - N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	60.4
Target	14.6
Herbie	0.6

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

Derivation

1. Split input into 3 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

   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. Taylor expanded in b around 0 43.1

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\varepsilon \cdot b + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)}} \]

   3. Simplified 41.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(0.5 \cdot {\left(\varepsilon \cdot b\right)}^{2} + 0.16666666666666666 \cdot {\left(\varepsilon \cdot b\right)}^{3}\right)\right)}} \]
      Proof

      [Start] 43.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \left(\varepsilon \cdot b + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)} \]
      rational.json-simplify-41 [=>] 43.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\varepsilon \cdot b + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + 0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right)\right)\right)}} \]
      rational.json-simplify-1 [=>] 43.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + \color{blue}{\left(0.5 \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)}\right)} \]
      exponential.json-simplify-27 [=>] 43.1	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + \left(0.5 \cdot \color{blue}{{\left(\varepsilon \cdot b\right)}^{2}} + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right)\right)\right)} \]
      exponential.json-simplify-27 [=>] 41.3	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(\varepsilon \cdot b + \left(0.5 \cdot {\left(\varepsilon \cdot b\right)}^{2} + 0.16666666666666666 \cdot \color{blue}{{\left(\varepsilon \cdot b\right)}^{3}}\right)\right)} \]

   4. Taylor expanded in eps around 0 6.5

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

   5. Simplified 0.0

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

      [Start] 6.5	\[ \frac{a + b}{a \cdot b} \]
      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))) < 1.00000000000000009e-111

   1. Initial program 3.8

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

   if 1.00000000000000009e-111 < (/.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 63.3

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

   2. Taylor expanded in a around 0 60.2

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

   3. Taylor expanded in eps around 0 63.2

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

   4. Simplified 63.2

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

      [Start] 63.2	\[ \frac{\varepsilon \cdot \left(\varepsilon \cdot \left(a + b\right)\right)}{\left(\varepsilon \cdot a\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      rational.json-simplify-1 [=>] 63.2	\[ \frac{\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right)}{\left(\varepsilon \cdot a\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   5. Taylor expanded in b around 0 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \leq 10^{-111}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	704

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

Alternative 2
Error	27.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3
Error	3.2
Cost	448

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

Alternative 4
Error	62.0
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 5
Error	32.7
Cost	192

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

Reproduce

herbie shell --seed 2023075 
(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))))