expq3 (problem 3.4.2)

Average Error: 93.83% → 0.14%

Time: 17.9s

Precision: binary64

Cost: 61384

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

Math

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

↓

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -100000000000:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

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 (<= t_1 -100000000000.0)
     (+ (/ 1.0 a) (+ (/ 1.0 b) (* eps -0.5)))
     (if (<= t_1 5e-5)
       (* eps (/ (expm1 t_0) (* (expm1 (* eps a)) (expm1 (* eps b)))))
       (+ (/ 1.0 a) (/ 1.0 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 <= -100000000000.0) {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	} else if (t_1 <= 5e-5) {
		tmp = eps * (expm1(t_0) / (expm1((eps * a)) * expm1((eps * b))));
	} else {
		tmp = (1.0 / a) + (1.0 / 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 <= -100000000000.0) {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	} else if (t_1 <= 5e-5) {
		tmp = eps * (Math.expm1(t_0) / (Math.expm1((eps * a)) * Math.expm1((eps * b))));
	} else {
		tmp = (1.0 / a) + (1.0 / 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 <= -100000000000.0:
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5))
	elif t_1 <= 5e-5:
		tmp = eps * (math.expm1(t_0) / (math.expm1((eps * a)) * math.expm1((eps * b))))
	else:
		tmp = (1.0 / a) + (1.0 / 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 <= -100000000000.0)
		tmp = Float64(Float64(1.0 / a) + Float64(Float64(1.0 / b) + Float64(eps * -0.5)));
	elseif (t_1 <= 5e-5)
		tmp = Float64(eps * Float64(expm1(t_0) / Float64(expm1(Float64(eps * a)) * expm1(Float64(eps * b)))));
	else
		tmp = Float64(Float64(1.0 / a) + Float64(1.0 / 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[LessEqual[t$95$1, -100000000000.0], N[(N[(1.0 / a), $MachinePrecision] + N[(N[(1.0 / b), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-5], N[(eps * N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] * N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	93.83%
Target	23.58%
Herbie	0.14%

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

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))) < -1e11

   1. Initial program 100

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

   2. Simplified 26.72

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

      [Start] 100	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      times-frac [=>] 100	\[ \color{blue}{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1}} \]
      expm1-def [=>] 64.11	\[ \frac{\varepsilon}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 64.11	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right)} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 66.77	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{e^{b \cdot \varepsilon} - 1} \]
      *-commutative [=>] 66.77	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{e^{b \cdot \varepsilon} - 1} \]
      expm1-def [=>] 26.72	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 26.72	\[ \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in b around 0 48.83

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

   4. Simplified 12.84

      \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{\mathsf{expm1}\left(a \cdot \varepsilon\right)}{e^{a \cdot \varepsilon}}} + \left(\frac{1}{b} - \varepsilon \cdot 0.5\right)} \]
      Proof

      [Start] 48.83	\[ \left(\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
      associate--l+ [=>] 48.83	\[ \color{blue}{\frac{\varepsilon \cdot e^{\varepsilon \cdot a}}{e^{\varepsilon \cdot a} - 1} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right)} \]
      associate-/l* [=>] 48.83	\[ \color{blue}{\frac{\varepsilon}{\frac{e^{\varepsilon \cdot a} - 1}{e^{\varepsilon \cdot a}}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      expm1-def [=>] 12.84	\[ \frac{\varepsilon}{\frac{\color{blue}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{e^{\varepsilon \cdot a}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      *-commutative [=>] 12.84	\[ \frac{\varepsilon}{\frac{\mathsf{expm1}\left(\color{blue}{a \cdot \varepsilon}\right)}{e^{\varepsilon \cdot a}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      *-commutative [=>] 12.84	\[ \frac{\varepsilon}{\frac{\mathsf{expm1}\left(a \cdot \varepsilon\right)}{e^{\color{blue}{a \cdot \varepsilon}}}} + \left(\frac{1}{b} - 0.5 \cdot \varepsilon\right) \]
      *-commutative [=>] 12.84	\[ \frac{\varepsilon}{\frac{\mathsf{expm1}\left(a \cdot \varepsilon\right)}{e^{a \cdot \varepsilon}}} + \left(\frac{1}{b} - \color{blue}{\varepsilon \cdot 0.5}\right) \]

   5. Taylor expanded in eps around 0 0

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

   if -1e11 < (/.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))) < 5.00000000000000024e-5

   1. Initial program 4.44

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

   2. Simplified 0.14

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

      [Start] 4.44	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*r/ [<=] 4.45	\[ \color{blue}{\varepsilon \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 4.45	\[ \varepsilon \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 4.45	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 2.51	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 2.51	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 0.14	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 0.14	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   if 5.00000000000000024e-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 99.86

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

   2. Simplified 73.46

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

      [Start] 99.86	\[ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      associate-*r/ [<=] 99.86	\[ \color{blue}{\varepsilon \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]
      expm1-def [=>] 99.86	\[ \varepsilon \cdot \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 99.86	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(a + b\right)}\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 94.46	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      *-commutative [=>] 94.46	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
      expm1-def [=>] 73.46	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \]
      *-commutative [=>] 73.46	\[ \varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b}\right)} \]

   3. Taylor expanded in eps around 0 22.83

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

   4. Taylor expanded in a around 0 0.16

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -100000000000:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.23%
Cost	704

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

Alternative 2
Error	5.66%
Cost	448

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

Alternative 3
Error	39.59%
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Error	96.84%
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 5
Error	52.87%
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(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))))