expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 99.9%

Time: 43.2s

Alternatives: 5

Speedup: 107.0×

Specification

\[\left(\left|a\right| \leq 710 \land \left|b\right| \leq 710\right) \land \left(10^{-27} \cdot \mathsf{min}\left(\left|a\right|, \left|b\right|\right) \leq \varepsilon \land \varepsilon \leq \mathsf{min}\left(\left|a\right|, \left|b\right|\right)\right)\]

Math

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

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

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));
}

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

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));
}

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))

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

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

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]

Initial Program: 0.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

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));
}

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

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));
}

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))

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

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

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]

Alternative 1: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ {\left(a \cdot \frac{b}{a + b}\right)}^{-1} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (pow (* a (/ b (+ a b))) -1.0))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return pow((a * (b / (a + b))), -1.0);
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (a * (b / (a + b))) ** (-1.0d0)
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return Math.pow((a * (b / (a + b))), -1.0);
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return math.pow((a * (b / (a + b))), -1.0)

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(a * Float64(b / Float64(a + b))) ^ -1.0
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = (a * (b / (a + b))) ^ -1.0;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[Power[N[(a * N[(b / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 0.0%

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

   2. associate-/l* 0.0%

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

   3. expm1-define 1.2%

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

   4. *-commutative 1.2%

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

   5. expm1-define 1.4%

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

   6. *-commutative 1.4%

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

   7. expm1-define 7.5%

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

   8. *-commutative 7.5%

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

3. Simplified 7.5%

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

5. Taylor expanded in eps around 0 58.7%

   \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
6. Step-by-step derivation

   1. clear-num 58.8%

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

   2. inv-pow 58.8%

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

   3. associate-/l* 99.9%

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

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(a \cdot \frac{b}{a + b}\right)}^{-1}} \]
8. Add Preprocessing

Alternative 2: 80.5% accurate, 40.0× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (if (<= b 2.3e-174) (/ 1.0 b) (/ 1.0 a)))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.3e-174) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this 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) :: tmp
    if (b <= 2.3d-174) then
        tmp = 1.0d0 / b
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.3e-174) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	tmp = 0
	if b <= 2.3e-174:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 2.3e-174)
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 2.3e-174)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 2.3e-174], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999999e-174

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/l* 0.0%

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

      3. expm1-define 1.1%

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

      4. *-commutative 1.1%

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

      5. expm1-define 0.5%

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

      6. *-commutative 0.5%

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

      7. expm1-define 5.4%

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

      8. *-commutative 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in b around 0 54.9%

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

   if 2.2999999999999999e-174 < b

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 0.0%

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

      2. associate-/l* 0.0%

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

      3. expm1-define 1.4%

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

      4. *-commutative 1.4%

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

      5. expm1-define 3.5%

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

      6. *-commutative 3.5%

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

      7. expm1-define 12.5%

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

      8. *-commutative 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in a around 0 78.3%

      \[\leadsto \color{blue}{\frac{1}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \frac{1 + \frac{b}{a}}{b} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (/ (+ 1.0 (/ b a)) b))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return (1.0 + (b / a)) / b;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (1.0d0 + (b / a)) / b
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return (1.0 + (b / a)) / b;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return (1.0 + (b / a)) / b

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(Float64(1.0 + Float64(b / a)) / b)
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = (1.0 + (b / a)) / b;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(N[(1.0 + N[(b / a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 0.0%

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

   2. associate-/l* 0.0%

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

   3. expm1-define 1.2%

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

   4. *-commutative 1.2%

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

   5. expm1-define 1.4%

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

   6. *-commutative 1.4%

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

   7. expm1-define 7.5%

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

   8. *-commutative 7.5%

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

3. Simplified 7.5%

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

5. Taylor expanded in eps around 0 58.7%

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

6. Taylor expanded in b around 0 99.8%

   \[\leadsto \color{blue}{\frac{1 + \frac{b}{a}}{b}} \]
7. Add Preprocessing

Alternative 4: 99.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \frac{1 + \frac{a}{b}}{a} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (/ (+ 1.0 (/ a b)) a))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return (1.0 + (a / b)) / a;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (1.0d0 + (a / b)) / a
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return (1.0 + (a / b)) / a;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return (1.0 + (a / b)) / a

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(Float64(1.0 + Float64(a / b)) / a)
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = (1.0 + (a / b)) / a;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(N[(1.0 + N[(a / b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 0.0%

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

   2. associate-/l* 0.0%

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

   3. expm1-define 1.2%

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

   4. *-commutative 1.2%

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

   5. expm1-define 1.4%

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

   6. *-commutative 1.4%

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

   7. expm1-define 7.5%

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

   8. *-commutative 7.5%

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

3. Simplified 7.5%

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

5. Taylor expanded in eps around 0 58.7%

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

6. Taylor expanded in a around 0 99.8%

   \[\leadsto \color{blue}{\frac{1 + \frac{a}{b}}{a}} \]
7. Add Preprocessing

Alternative 5: 50.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \frac{1}{a} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (/ 1.0 a))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return 1.0 / a;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = 1.0d0 / a
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return 1.0 / a;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return 1.0 / a

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(1.0 / a)
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = 1.0 / a;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(1.0 / a), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 0.0%

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

   2. associate-/l* 0.0%

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

   3. expm1-define 1.2%

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

   4. *-commutative 1.2%

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

   5. expm1-define 1.4%

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

   6. *-commutative 1.4%

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

   7. expm1-define 7.5%

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

   8. *-commutative 7.5%

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

3. Simplified 7.5%

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

5. Taylor expanded in a around 0 54.8%

   \[\leadsto \color{blue}{\frac{1}{a}} \]
6. Add Preprocessing

Developer Target 1: 99.9% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a} + \frac{1}{b} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (+ (/ 1.0 a) (/ 1.0 b)))

C

double code(double a, double b, double eps) {
	return (1.0 / a) + (1.0 / b);
}

Fortran

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

Java

public static double code(double a, double b, double eps) {
	return (1.0 / a) + (1.0 / b);
}

Python

def code(a, b, eps):
	return (1.0 / a) + (1.0 / b)

Julia

function code(a, b, eps)
	return Float64(Float64(1.0 / a) + Float64(1.0 / b))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (1.0 / a) + (1.0 / b);
end

Wolfram

code[a_, b_, eps_] := N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024113 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (and (<= (fabs a) 710.0) (<= (fabs b) 710.0)) (and (<= (* 1e-27 (fmin (fabs a) (fabs b))) eps) (<= eps (fmin (fabs a) (fabs b)))))

  :alt
  (! :herbie-platform default (+ (/ 1 a) (/ 1 b)))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))