expq3 (problem 3.4.2)

Average Error: 60.5 → 5.4

Time: 1.2min

Precision: binary64

Cost: 976

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a + b}{a \cdot b}\\ t_1 := \frac{1 + \frac{b}{a}}{b}\\ \mathbf{if}\;b \leq -4.9 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+192}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \varepsilon + \frac{1}{a}\\ \end{array} \]

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 (/ (+ a b) (* a b))) (t_1 (/ (+ 1.0 (/ b a)) b)))
   (if (<= b -4.9e+100)
     t_0
     (if (<= b 5.2e+48)
       t_1
       (if (<= b 1.2e+191)
         t_0
         (if (<= b 9e+192) t_1 (+ (* -0.5 eps) (/ 1.0 a))))))))

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 = (a + b) / (a * b);
	double t_1 = (1.0 + (b / a)) / b;
	double tmp;
	if (b <= -4.9e+100) {
		tmp = t_0;
	} else if (b <= 5.2e+48) {
		tmp = t_1;
	} else if (b <= 1.2e+191) {
		tmp = t_0;
	} else if (b <= 9e+192) {
		tmp = t_1;
	} else {
		tmp = (-0.5 * eps) + (1.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

↓

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a + b) / (a * b)
    t_1 = (1.0d0 + (b / a)) / b
    if (b <= (-4.9d+100)) then
        tmp = t_0
    else if (b <= 5.2d+48) then
        tmp = t_1
    else if (b <= 1.2d+191) then
        tmp = t_0
    else if (b <= 9d+192) then
        tmp = t_1
    else
        tmp = ((-0.5d0) * eps) + (1.0d0 / a)
    end if
    code = tmp
end function

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 = (a + b) / (a * b);
	double t_1 = (1.0 + (b / a)) / b;
	double tmp;
	if (b <= -4.9e+100) {
		tmp = t_0;
	} else if (b <= 5.2e+48) {
		tmp = t_1;
	} else if (b <= 1.2e+191) {
		tmp = t_0;
	} else if (b <= 9e+192) {
		tmp = t_1;
	} else {
		tmp = (-0.5 * eps) + (1.0 / a);
	}
	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 = (a + b) / (a * b)
	t_1 = (1.0 + (b / a)) / b
	tmp = 0
	if b <= -4.9e+100:
		tmp = t_0
	elif b <= 5.2e+48:
		tmp = t_1
	elif b <= 1.2e+191:
		tmp = t_0
	elif b <= 9e+192:
		tmp = t_1
	else:
		tmp = (-0.5 * eps) + (1.0 / a)
	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(a + b) / Float64(a * b))
	t_1 = Float64(Float64(1.0 + Float64(b / a)) / b)
	tmp = 0.0
	if (b <= -4.9e+100)
		tmp = t_0;
	elseif (b <= 5.2e+48)
		tmp = t_1;
	elseif (b <= 1.2e+191)
		tmp = t_0;
	elseif (b <= 9e+192)
		tmp = t_1;
	else
		tmp = Float64(Float64(-0.5 * eps) + Float64(1.0 / a));
	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 = (a + b) / (a * b);
	t_1 = (1.0 + (b / a)) / b;
	tmp = 0.0;
	if (b <= -4.9e+100)
		tmp = t_0;
	elseif (b <= 5.2e+48)
		tmp = t_1;
	elseif (b <= 1.2e+191)
		tmp = t_0;
	elseif (b <= 9e+192)
		tmp = t_1;
	else
		tmp = (-0.5 * eps) + (1.0 / a);
	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[(a + b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, -4.9e+100], t$95$0, If[LessEqual[b, 5.2e+48], t$95$1, If[LessEqual[b, 1.2e+191], t$95$0, If[LessEqual[b, 9e+192], t$95$1, N[(N[(-0.5 * eps), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	60.5
Target	14.7
Herbie	5.4

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

Derivation

1. Split input into 3 regimes

2. if b < -4.89999999999999967e100 or 5.1999999999999999e48 < b < 1.19999999999999993e191

   1. Initial program 55.4

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

   2. Simplified 19.1

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

   3. Taylor expanded in eps around 0 13.5

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

   if -4.89999999999999967e100 < b < 5.1999999999999999e48 or 1.19999999999999993e191 < b < 9e192

   1. Initial program 63.4

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

   2. Simplified 40.2

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

   3. Taylor expanded in eps around 0 14.7

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

   4. Applied egg-rr 1.5

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

   5. Simplified 1.5

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

   if 9e192 < b

   1. Initial program 50.2

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

   2. Simplified 19.3

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

   3. Taylor expanded in a around 0 22.9

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

   4. Taylor expanded in eps around 0 13.7

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

Alternatives

Alternative 1
Error	5.8
Cost	14152

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \varepsilon + \frac{1}{a}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)\\ \end{array} \]

Alternative 2
Error	5.9
Cost	13896

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+98}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)\\ \end{array} \]

Alternative 3
Error	18.6
Cost	976

\[\begin{array}{l} t_0 := -0.5 \cdot \varepsilon + \frac{1}{a}\\ \mathbf{if}\;b \leq -9.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	18.6
Cost	720

\[\begin{array}{l} \mathbf{if}\;b \leq -750000:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 5
Error	6.1
Cost	712

\[\begin{array}{l} t_0 := -0.5 \cdot \varepsilon + \frac{1}{a}\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9 \cdot 10^{+46}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	62.0
Cost	192

\[\varepsilon \cdot -0.5 \]

Alternative 7
Error	33.5
Cost	192

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

Reproduce

herbie shell --seed 2023033 
(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))))