expq3 (problem 3.4.2)

Average Error: 60.2 → 0.8

Time: 13.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{a} + \frac{1}{b}\\ t_1 := \varepsilon \cdot \left(a + b\right)\\ t_2 := \frac{\varepsilon \cdot \left(e^{t_1} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 1.902505775443173 \cdot 10^{-239}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(t_1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (+ (/ 1.0 a) (/ 1.0 b)))
        (t_1 (* eps (+ a b)))
        (t_2
         (/
          (* eps (+ (exp t_1) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 1.902505775443173e-239)
       (/ eps (* (expm1 (* eps b)) (/ (expm1 (* eps a)) (expm1 t_1))))
       t_0))))

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 = (1.0 / a) + (1.0 / b);
	double t_1 = eps * (a + b);
	double t_2 = (eps * (exp(t_1) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= 1.902505775443173e-239) {
		tmp = eps / (expm1((eps * b)) * (expm1((eps * a)) / expm1(t_1)));
	} else {
		tmp = t_0;
	}
	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 = (1.0 / a) + (1.0 / b);
	double t_1 = eps * (a + b);
	double t_2 = (eps * (Math.exp(t_1) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_2 <= 1.902505775443173e-239) {
		tmp = eps / (Math.expm1((eps * b)) * (Math.expm1((eps * a)) / Math.expm1(t_1)));
	} else {
		tmp = t_0;
	}
	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 = (1.0 / a) + (1.0 / b)
	t_1 = eps * (a + b)
	t_2 = (eps * (math.exp(t_1) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_0
	elif t_2 <= 1.902505775443173e-239:
		tmp = eps / (math.expm1((eps * b)) * (math.expm1((eps * a)) / math.expm1(t_1)))
	else:
		tmp = t_0
	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(1.0 / a) + Float64(1.0 / b))
	t_1 = Float64(eps * Float64(a + b))
	t_2 = Float64(Float64(eps * Float64(exp(t_1) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= 1.902505775443173e-239)
		tmp = Float64(eps / Float64(expm1(Float64(eps * b)) * Float64(expm1(Float64(eps * a)) / expm1(t_1))));
	else
		tmp = t_0;
	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[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps * N[(N[Exp[t$95$1], $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$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, 1.902505775443173e-239], N[(eps / N[(N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision] * N[(N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[t$95$1] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	15.2
Herbie	0.8

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

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 1.9025057754431729e-239 < (/.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 63.0

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

   2. Simplified 32.2

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

   3. Taylor expanded in eps around 0 3.9

      \[\leadsto \color{blue}{0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot a\right) + \left(\frac{1}{b} + \left(\frac{1}{a} + 0.08333333333333333 \cdot \left({\varepsilon}^{2} \cdot b\right)\right)\right)} \]

   4. Simplified 3.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, a \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{b}\right) + \mathsf{fma}\left(0.08333333333333333, b \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a}\right)} \]

   5. Taylor expanded in a around 0 2.4

      \[\leadsto \color{blue}{\frac{1}{b}} + \mathsf{fma}\left(0.08333333333333333, b \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a}\right) \]

   6. Applied egg-rr 3.6

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{b} + \mathsf{fma}\left(0.08333333333333333, b \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a}\right)} \cdot {\left(\sqrt[3]{\frac{1}{b} + \mathsf{fma}\left(0.08333333333333333, b \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{1}{a}\right)}\right)}^{2}} \]

   7. Taylor expanded in b around 0 0.9

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

   8. Simplified 0.9

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

   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))) < 1.9025057754431729e-239

   1. Initial program 3.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 1.902505775443173 \cdot 10^{-239}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot a\right)}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))))