expq3 (problem 3.4.2)

Average Error: 60.1 → 26.4

Time: 50.7s

Precision: binary64

Cost: 27212

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(a \cdot \varepsilon\right)\\ t_1 := \mathsf{expm1}\left(b \cdot \varepsilon\right)\\ t_2 := \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)\\ t_3 := \varepsilon \cdot t_2\\ t_4 := \frac{t_3}{t_1 \cdot t_0}\\ \mathbf{if}\;a \leq -9.5 \cdot 10^{-162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{\frac{t_2}{t_0}}{\frac{t_1}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-57}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{t_1}}{\mathsf{fma}\left(-0.5, \left(\varepsilon \cdot \varepsilon\right) \cdot a, \frac{a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_0 \cdot t_1}\\ \end{array}\\ \mathbf{elif}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{\frac{t_2}{\frac{t_0}{\varepsilon}}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (expm1 (* a eps)))
        (t_1 (expm1 (* b eps)))
        (t_2 (expm1 (* (+ a b) eps)))
        (t_3 (* eps t_2))
        (t_4 (/ t_3 (* t_1 t_0))))
   (if (<= a -9.5e-162)
     (if (!= eps 0.0) (/ (/ t_2 t_0) (/ t_1 eps)) t_4)
     (if (<= a 1.45e-57)
       (if (!= t_2 0.0)
         (/ (/ eps t_1) (fma -0.5 (* (* eps eps) a) (/ a b)))
         (/ t_3 (* t_0 t_1)))
       (if (!= eps 0.0) (/ (/ t_2 (/ t_0 eps)) t_1) t_4)))))

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 = expm1((a * eps));
	double t_1 = expm1((b * eps));
	double t_2 = expm1(((a + b) * eps));
	double t_3 = eps * t_2;
	double t_4 = t_3 / (t_1 * t_0);
	double tmp_1;
	if (a <= -9.5e-162) {
		double tmp_2;
		if (eps != 0.0) {
			tmp_2 = (t_2 / t_0) / (t_1 / eps);
		} else {
			tmp_2 = t_4;
		}
		tmp_1 = tmp_2;
	} else if (a <= 1.45e-57) {
		double tmp_3;
		if (t_2 != 0.0) {
			tmp_3 = (eps / t_1) / fma(-0.5, ((eps * eps) * a), (a / b));
		} else {
			tmp_3 = t_3 / (t_0 * t_1);
		}
		tmp_1 = tmp_3;
	} else if (eps != 0.0) {
		tmp_1 = (t_2 / (t_0 / eps)) / t_1;
	} else {
		tmp_1 = t_4;
	}
	return tmp_1;
}

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 = expm1(Float64(a * eps))
	t_1 = expm1(Float64(b * eps))
	t_2 = expm1(Float64(Float64(a + b) * eps))
	t_3 = Float64(eps * t_2)
	t_4 = Float64(t_3 / Float64(t_1 * t_0))
	tmp_1 = 0.0
	if (a <= -9.5e-162)
		tmp_2 = 0.0
		if (eps != 0.0)
			tmp_2 = Float64(Float64(t_2 / t_0) / Float64(t_1 / eps));
		else
			tmp_2 = t_4;
		end
		tmp_1 = tmp_2;
	elseif (a <= 1.45e-57)
		tmp_3 = 0.0
		if (t_2 != 0.0)
			tmp_3 = Float64(Float64(eps / t_1) / fma(-0.5, Float64(Float64(eps * eps) * a), Float64(a / b)));
		else
			tmp_3 = Float64(t_3 / Float64(t_0 * t_1));
		end
		tmp_1 = tmp_3;
	elseif (eps != 0.0)
		tmp_1 = Float64(Float64(t_2 / Float64(t_0 / eps)) / t_1);
	else
		tmp_1 = t_4;
	end
	return tmp_1
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[(Exp[N[(a * eps), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$1 = N[(Exp[N[(b * eps), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]] - 1), $MachinePrecision]}, Block[{t$95$3 = N[(eps * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9.5e-162], If[Unequal[eps, 0.0], N[(N[(t$95$2 / t$95$0), $MachinePrecision] / N[(t$95$1 / eps), $MachinePrecision]), $MachinePrecision], t$95$4], If[LessEqual[a, 1.45e-57], If[Unequal[t$95$2, 0.0], N[(N[(eps / t$95$1), $MachinePrecision] / N[(-0.5 * N[(N[(eps * eps), $MachinePrecision] * a), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], If[Unequal[eps, 0.0], N[(N[(t$95$2 / N[(t$95$0 / eps), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]]]

Target

Original	60.1
Target	15.1
Herbie	26.4

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

Derivation

1. Split input into 3 regimes

2. if a < -9.5000000000000004e-162

   1. Initial program 59.0

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

   2. Simplified 35.1

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

   3. Applied egg-rr 38.2

      \[\leadsto \color{blue}{\frac{\varepsilon}{\sqrt[3]{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \cdot \frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\sqrt[3]{{\left(\mathsf{expm1}\left(b \cdot \varepsilon\right)\right)}^{2}}}} \]

   4. Applied egg-rr 21.0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\frac{\mathsf{expm1}\left(b \cdot \varepsilon\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ } \end{array}} \]

   if -9.5000000000000004e-162 < a < 1.45000000000000013e-57

   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 56.2

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

   3. Applied egg-rr 46.9

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

   4. Applied egg-rr 46.2

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{\frac{\mathsf{expm1}\left(a \cdot \varepsilon\right)}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ } \end{array}} \]

   5. Taylor expanded in a around 0 62.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{\color{blue}{\frac{a}{e^{\varepsilon \cdot b} - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array} \]

   6. Simplified 62.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{\color{blue}{\frac{a}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array} \]
      Proof

   7. Taylor expanded in b around 0 38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{\color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot a\right) + \frac{a}{b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array} \]

   8. Simplified 38.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \ne 0:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{\color{blue}{\mathsf{fma}\left(-0.5, \left(\varepsilon \cdot \varepsilon\right) \cdot a, \frac{a}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array} \]
      Proof

   if 1.45000000000000013e-57 < a

   1. Initial program 56.6

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

   2. Simplified 29.9

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

   3. Applied egg-rr 36.4

      \[\leadsto \color{blue}{\frac{\varepsilon}{\sqrt[3]{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \cdot \frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\sqrt[3]{{\left(\mathsf{expm1}\left(b \cdot \varepsilon\right)\right)}^{2}}}} \]

   4. Applied egg-rr 18.7

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\frac{\mathsf{expm1}\left(b \cdot \varepsilon\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ } \end{array}} \]

   5. Applied egg-rr 18.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\color{blue}{\frac{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\frac{\mathsf{expm1}\left(a \cdot \varepsilon\right)}{\varepsilon}}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ \end{array} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	38.9
Cost	20560

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(\varepsilon \cdot b\right)\\ t_1 := \mathsf{expm1}\left(a \cdot \varepsilon\right)\\ t_2 := \frac{\varepsilon}{t_1}\\ t_3 := \frac{\varepsilon}{t_0}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -8 \cdot 10^{+49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}}{t_1} \cdot t_0\\ \end{array} \]

Alternative 2
Error	29.3
Cost	20292

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(b \cdot \varepsilon\right)\\ t_1 := \mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)\\ t_2 := \mathsf{expm1}\left(a \cdot \varepsilon\right)\\ \mathbf{if}\;\varepsilon \ne 0:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{\frac{t_0}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot t_1}{t_0 \cdot t_2}\\ \end{array} \]

Alternative 3
Error	30.6
Cost	20160

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

Alternative 4
Error	29.3
Cost	20160

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

Alternative 5
Error	29.4
Cost	20160

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

Alternative 6
Error	29.2
Cost	20160

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

Alternative 7
Error	38.8
Cost	7248

\[\begin{array}{l} t_0 := \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}\\ t_1 := \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	45.5
Cost	6720

\[\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

Alternative 9
Error	61.5
Cost	192

\[\frac{\varepsilon}{a} \]

Alternative 10
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023010 
(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))))