expq3 (problem 3.4.2)

Percentage Accurate: 6.5% → 94.7%

Time: 12.7s

Alternatives: 9

Speedup: 107.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = 1.9999999999999999e-154
  3. eps = -1.9999999999999999e-77

Specification

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

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: 6.5% 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: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot t_0\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\varepsilon \cdot -0.5 + \frac{1}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (let* ((t_0 (/ (expm1 (* eps (+ b a))) (expm1 (* b eps)))))
   (if (<= b -2.6e+23)
     (* (/ eps (expm1 (* eps a))) t_0)
     (if (<= b 1.8e+108)
       (/ (+ (/ b a) 1.0) b)
       (* t_0 (+ (* eps -0.5) (/ 1.0 a)))))))

C

assert(a < b);
double code(double a, double b, double eps) {
	double t_0 = expm1((eps * (b + a))) / expm1((b * eps));
	double tmp;
	if (b <= -2.6e+23) {
		tmp = (eps / expm1((eps * a))) * t_0;
	} else if (b <= 1.8e+108) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = t_0 * ((eps * -0.5) + (1.0 / a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double t_0 = Math.expm1((eps * (b + a))) / Math.expm1((b * eps));
	double tmp;
	if (b <= -2.6e+23) {
		tmp = (eps / Math.expm1((eps * a))) * t_0;
	} else if (b <= 1.8e+108) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = t_0 * ((eps * -0.5) + (1.0 / a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	t_0 = math.expm1((eps * (b + a))) / math.expm1((b * eps))
	tmp = 0
	if b <= -2.6e+23:
		tmp = (eps / math.expm1((eps * a))) * t_0
	elif b <= 1.8e+108:
		tmp = ((b / a) + 1.0) / b
	else:
		tmp = t_0 * ((eps * -0.5) + (1.0 / a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	t_0 = Float64(expm1(Float64(eps * Float64(b + a))) / expm1(Float64(b * eps)))
	tmp = 0.0
	if (b <= -2.6e+23)
		tmp = Float64(Float64(eps / expm1(Float64(eps * a))) * t_0);
	elseif (b <= 1.8e+108)
		tmp = Float64(Float64(Float64(b / a) + 1.0) / b);
	else
		tmp = Float64(t_0 * Float64(Float64(eps * -0.5) + Float64(1.0 / a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := Block[{t$95$0 = N[(N[(Exp[N[(eps * N[(b + a), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[N[(b * eps), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.6e+23], N[(N[(eps / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[b, 1.8e+108], N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision], N[(t$95$0 * N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999992e23

   1. Initial program 26.2%

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

      1. times-frac 26.2%

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

      2. expm1-def 46.1%

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

      3. *-commutative 46.1%

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

      4. expm1-def 43.1%

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

      5. *-commutative 43.1%

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

      6. expm1-def 79.8%

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

      7. *-commutative 79.8%

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

   3. Simplified 79.8%

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

   if -2.59999999999999992e23 < b < 1.8e108

   1. Initial program 1.2%

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

      1. times-frac 1.2%

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

      2. expm1-def 1.8%

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

      3. *-commutative 1.8%

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

      4. expm1-def 3.9%

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

      5. *-commutative 3.9%

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

      6. expm1-def 50.1%

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

      7. *-commutative 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in eps around 0 49.2%

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

   5. Taylor expanded in eps around 0 99.3%

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

   if 1.8e108 < b

   1. Initial program 14.8%

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

      1. times-frac 14.8%

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

      2. expm1-def 43.2%

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

      3. *-commutative 43.2%

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

      4. expm1-def 42.8%

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

      5. *-commutative 42.8%

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

      6. expm1-def 68.1%

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

      7. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in eps around 0 92.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \varepsilon + \frac{1}{a}\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot -0.5 + \frac{1}{a}\right)\\ \end{array} \]

Alternative 2: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{expm1}\left(b \cdot \varepsilon\right)\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{t_0}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{t_0} \cdot \left(\varepsilon \cdot -0.5 + \frac{1}{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (let* ((t_0 (expm1 (* b eps))))
   (if (<= b -2.6e+23)
     (/ eps t_0)
     (if (<= b 7.5e+106)
       (/ (+ (/ b a) 1.0) b)
       (* (/ (expm1 (* eps (+ b a))) t_0) (+ (* eps -0.5) (/ 1.0 a)))))))

C

assert(a < b);
double code(double a, double b, double eps) {
	double t_0 = expm1((b * eps));
	double tmp;
	if (b <= -2.6e+23) {
		tmp = eps / t_0;
	} else if (b <= 7.5e+106) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (expm1((eps * (b + a))) / t_0) * ((eps * -0.5) + (1.0 / a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double t_0 = Math.expm1((b * eps));
	double tmp;
	if (b <= -2.6e+23) {
		tmp = eps / t_0;
	} else if (b <= 7.5e+106) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (Math.expm1((eps * (b + a))) / t_0) * ((eps * -0.5) + (1.0 / a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	t_0 = math.expm1((b * eps))
	tmp = 0
	if b <= -2.6e+23:
		tmp = eps / t_0
	elif b <= 7.5e+106:
		tmp = ((b / a) + 1.0) / b
	else:
		tmp = (math.expm1((eps * (b + a))) / t_0) * ((eps * -0.5) + (1.0 / a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	t_0 = expm1(Float64(b * eps))
	tmp = 0.0
	if (b <= -2.6e+23)
		tmp = Float64(eps / t_0);
	elseif (b <= 7.5e+106)
		tmp = Float64(Float64(Float64(b / a) + 1.0) / b);
	else
		tmp = Float64(Float64(expm1(Float64(eps * Float64(b + a))) / t_0) * Float64(Float64(eps * -0.5) + Float64(1.0 / a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := Block[{t$95$0 = N[(Exp[N[(b * eps), $MachinePrecision]] - 1), $MachinePrecision]}, If[LessEqual[b, -2.6e+23], N[(eps / t$95$0), $MachinePrecision], If[LessEqual[b, 7.5e+106], N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(Exp[N[(eps * N[(b + a), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999992e23

   1. Initial program 26.2%

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

      1. expm1-def 26.7%

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

      2. *-commutative 26.7%

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

      3. expm1-def 42.7%

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

      4. *-commutative 42.7%

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

      5. expm1-def 72.2%

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

      6. *-commutative 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in a around inf 27.1%

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

   5. Taylor expanded in eps around inf 21.0%

      \[\leadsto \color{blue}{\frac{\varepsilon}{e^{\varepsilon \cdot b} - 1}} \]
   6. Step-by-step derivation

      1. expm1-def 31.6%

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

      2. *-commutative 31.6%

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

   7. Simplified 31.6%

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

   if -2.59999999999999992e23 < b < 7.50000000000000058e106

   1. Initial program 1.2%

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

      1. times-frac 1.2%

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

      2. expm1-def 1.8%

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

      3. *-commutative 1.8%

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

      4. expm1-def 3.9%

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

      5. *-commutative 3.9%

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

      6. expm1-def 50.1%

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

      7. *-commutative 50.1%

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

   3. Simplified 50.1%

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

   4. Taylor expanded in eps around 0 49.2%

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

   5. Taylor expanded in eps around 0 99.3%

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

   if 7.50000000000000058e106 < b

   1. Initial program 14.8%

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

      1. times-frac 14.8%

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

      2. expm1-def 43.2%

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

      3. *-commutative 43.2%

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

      4. expm1-def 42.8%

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

      5. *-commutative 42.8%

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

      6. expm1-def 68.1%

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

      7. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in eps around 0 92.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \varepsilon + \frac{1}{a}\right)} \cdot \frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(\varepsilon \cdot -0.5 + \frac{1}{a}\right)\\ \end{array} \]

Alternative 3: 93.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b -2.6e+23)
   (/ eps (expm1 (* b eps)))
   (if (<= b 6e+78) (/ (+ (/ b a) 1.0) b) (+ (* eps -0.5) (/ 1.0 a)))))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= -2.6e+23) {
		tmp = eps / expm1((b * eps));
	} else if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (eps * -0.5) + (1.0 / a);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= -2.6e+23) {
		tmp = eps / Math.expm1((b * eps));
	} else if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (eps * -0.5) + (1.0 / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= -2.6e+23:
		tmp = eps / math.expm1((b * eps))
	elif b <= 6e+78:
		tmp = ((b / a) + 1.0) / b
	else:
		tmp = (eps * -0.5) + (1.0 / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= -2.6e+23)
		tmp = Float64(eps / expm1(Float64(b * eps)));
	elseif (b <= 6e+78)
		tmp = Float64(Float64(Float64(b / a) + 1.0) / b);
	else
		tmp = Float64(Float64(eps * -0.5) + Float64(1.0 / a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, -2.6e+23], N[(eps / N[(Exp[N[(b * eps), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e+78], N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision], N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999992e23

   1. Initial program 26.2%

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

      1. expm1-def 26.7%

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

      2. *-commutative 26.7%

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

      3. expm1-def 42.7%

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

      4. *-commutative 42.7%

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

      5. expm1-def 72.2%

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

      6. *-commutative 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in a around inf 27.1%

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

   5. Taylor expanded in eps around inf 21.0%

      \[\leadsto \color{blue}{\frac{\varepsilon}{e^{\varepsilon \cdot b} - 1}} \]
   6. Step-by-step derivation

      1. expm1-def 31.6%

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

      2. *-commutative 31.6%

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

   7. Simplified 31.6%

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

   if -2.59999999999999992e23 < b < 5.99999999999999964e78

   1. Initial program 1.3%

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

      1. times-frac 1.3%

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

      2. expm1-def 1.9%

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

      3. *-commutative 1.9%

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

      4. expm1-def 4.0%

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

      5. *-commutative 4.0%

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

      6. expm1-def 50.7%

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

      7. *-commutative 50.7%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in eps around 0 48.6%

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

   5. Taylor expanded in eps around 0 99.3%

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

   if 5.99999999999999964e78 < b

   1. Initial program 13.3%

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

      1. times-frac 13.3%

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

      2. expm1-def 38.6%

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

      3. *-commutative 38.6%

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

      4. expm1-def 38.5%

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

      5. *-commutative 38.5%

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

      6. expm1-def 63.8%

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

      7. *-commutative 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in a around 0 58.1%

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

   5. Taylor expanded in eps around 0 87.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{a}\\ \end{array} \]

Alternative 4: 85.1% accurate, 24.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{b}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+271}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 9.2e-176)
   (+ (* eps -0.5) (/ 1.0 b))
   (if (<= b 2.3e-158)
     (/ 1.0 a)
     (if (<= b 2.05e+271) (/ (+ b a) (* b a)) (/ 1.0 a)))))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = (eps * -0.5) + (1.0 / b);
	} else if (b <= 2.3e-158) {
		tmp = 1.0 / a;
	} else if (b <= 2.05e+271) {
		tmp = (b + a) / (b * a);
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

NOTE: a and b 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 <= 9.2d-176) then
        tmp = (eps * (-0.5d0)) + (1.0d0 / b)
    else if (b <= 2.3d-158) then
        tmp = 1.0d0 / a
    else if (b <= 2.05d+271) then
        tmp = (b + a) / (b * a)
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = (eps * -0.5) + (1.0 / b);
	} else if (b <= 2.3e-158) {
		tmp = 1.0 / a;
	} else if (b <= 2.05e+271) {
		tmp = (b + a) / (b * a);
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= 9.2e-176:
		tmp = (eps * -0.5) + (1.0 / b)
	elif b <= 2.3e-158:
		tmp = 1.0 / a
	elif b <= 2.05e+271:
		tmp = (b + a) / (b * a)
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 9.2e-176)
		tmp = Float64(Float64(eps * -0.5) + Float64(1.0 / b));
	elseif (b <= 2.3e-158)
		tmp = Float64(1.0 / a);
	elseif (b <= 2.05e+271)
		tmp = Float64(Float64(b + a) / Float64(b * a));
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 9.2e-176)
		tmp = (eps * -0.5) + (1.0 / b);
	elseif (b <= 2.3e-158)
		tmp = 1.0 / a;
	elseif (b <= 2.05e+271)
		tmp = (b + a) / (b * a);
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 9.2e-176], N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-158], N[(1.0 / a), $MachinePrecision], If[LessEqual[b, 2.05e+271], N[(N[(b + a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.2000000000000005e-176

   1. Initial program 9.2%

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

      1. expm1-def 9.9%

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

      2. *-commutative 9.9%

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

      3. expm1-def 15.4%

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

      4. *-commutative 15.4%

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

      5. expm1-def 37.1%

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

      6. *-commutative 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in a around inf 20.9%

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

   5. Taylor expanded in eps around 0 58.1%

      \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \frac{1}{b}} \]

   if 9.2000000000000005e-176 < b < 2.2999999999999999e-158 or 2.05000000000000012e271 < b

   1. Initial program 6.2%

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

      1. associate-*l/ 6.2%

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

      2. *-commutative 6.2%

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

      3. expm1-def 7.0%

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

      4. *-commutative 7.0%

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

      5. expm1-def 46.2%

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

      6. *-commutative 46.2%

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

      7. expm1-def 55.3%

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

      8. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in a around 0 84.9%

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

   if 2.2999999999999999e-158 < b < 2.05000000000000012e271

   1. Initial program 7.3%

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

      1. associate-*l/ 7.3%

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

      2. *-commutative 7.3%

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

      3. expm1-def 8.9%

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

      4. *-commutative 8.9%

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

      5. expm1-def 15.3%

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

      6. *-commutative 15.3%

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

      7. expm1-def 44.1%

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

      8. *-commutative 44.1%

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

   3. Simplified 44.1%

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

   4. Taylor expanded in eps around 0 85.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{b}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{+271}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 5: 78.5% accurate, 35.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 9.2e-176) (+ (* eps -0.5) (/ 1.0 b)) (/ 1.0 a)))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = (eps * -0.5) + (1.0 / b);
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

NOTE: a and b 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 <= 9.2d-176) then
        tmp = (eps * (-0.5d0)) + (1.0d0 / b)
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = (eps * -0.5) + (1.0 / b);
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= 9.2e-176:
		tmp = (eps * -0.5) + (1.0 / b)
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 9.2e-176)
		tmp = Float64(Float64(eps * -0.5) + Float64(1.0 / b));
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 9.2e-176)
		tmp = (eps * -0.5) + (1.0 / b);
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 9.2e-176], N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.2000000000000005e-176

   1. Initial program 9.2%

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

      1. expm1-def 9.9%

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

      2. *-commutative 9.9%

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

      3. expm1-def 15.4%

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

      4. *-commutative 15.4%

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

      5. expm1-def 37.1%

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

      6. *-commutative 37.1%

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

   3. Simplified 37.1%

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

   4. Taylor expanded in a around inf 20.9%

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

   5. Taylor expanded in eps around 0 58.1%

      \[\leadsto \color{blue}{-0.5 \cdot \varepsilon + \frac{1}{b}} \]

   if 9.2000000000000005e-176 < b

   1. Initial program 7.1%

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

      1. associate-*l/ 7.1%

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

      2. *-commutative 7.1%

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

      3. expm1-def 8.6%

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

      4. *-commutative 8.6%

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

      5. expm1-def 20.2%

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

      6. *-commutative 20.2%

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

      7. expm1-def 45.8%

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

      8. *-commutative 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in a around 0 63.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 6: 92.1% accurate, 35.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 6e+78) (/ (+ (/ b a) 1.0) b) (/ 1.0 a)))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

NOTE: a and b 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 <= 6d+78) then
        tmp = ((b / a) + 1.0d0) / b
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= 6e+78:
		tmp = ((b / a) + 1.0) / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 6e+78)
		tmp = Float64(Float64(Float64(b / a) + 1.0) / b);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 6e+78)
		tmp = ((b / a) + 1.0) / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 6e+78], N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999964e78

   1. Initial program 7.7%

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

      1. times-frac 7.7%

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

      2. expm1-def 13.2%

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

      3. *-commutative 13.2%

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

      4. expm1-def 14.0%

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

      5. *-commutative 14.0%

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

      6. expm1-def 58.2%

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

      7. *-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in eps around 0 54.7%

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

   5. Taylor expanded in eps around 0 88.0%

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

   if 5.99999999999999964e78 < b

   1. Initial program 13.3%

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

      1. associate-*l/ 13.3%

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

      2. *-commutative 13.3%

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

      3. expm1-def 14.8%

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

      4. *-commutative 14.8%

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

      5. expm1-def 38.5%

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

      6. *-commutative 38.5%

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

      7. expm1-def 57.8%

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

      8. *-commutative 57.8%

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

   3. Simplified 57.8%

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

   4. Taylor expanded in a around 0 86.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 7: 92.3% accurate, 35.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 6e+78) (/ (+ (/ b a) 1.0) b) (+ (* eps -0.5) (/ 1.0 a))))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (eps * -0.5) + (1.0 / a);
	}
	return tmp;
}

Fortran

NOTE: a and b 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 <= 6d+78) then
        tmp = ((b / a) + 1.0d0) / b
    else
        tmp = (eps * (-0.5d0)) + (1.0d0 / a)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 6e+78) {
		tmp = ((b / a) + 1.0) / b;
	} else {
		tmp = (eps * -0.5) + (1.0 / a);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= 6e+78:
		tmp = ((b / a) + 1.0) / b
	else:
		tmp = (eps * -0.5) + (1.0 / a)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 6e+78)
		tmp = Float64(Float64(Float64(b / a) + 1.0) / b);
	else
		tmp = Float64(Float64(eps * -0.5) + Float64(1.0 / a));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 6e+78)
		tmp = ((b / a) + 1.0) / b;
	else
		tmp = (eps * -0.5) + (1.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 6e+78], N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision], N[(N[(eps * -0.5), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.99999999999999964e78

   1. Initial program 7.7%

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

      1. times-frac 7.7%

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

      2. expm1-def 13.2%

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

      3. *-commutative 13.2%

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

      4. expm1-def 14.0%

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

      5. *-commutative 14.0%

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

      6. expm1-def 58.2%

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

      7. *-commutative 58.2%

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

   3. Simplified 58.2%

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

   4. Taylor expanded in eps around 0 54.7%

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

   5. Taylor expanded in eps around 0 88.0%

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

   if 5.99999999999999964e78 < b

   1. Initial program 13.3%

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

      1. times-frac 13.3%

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

      2. expm1-def 38.6%

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

      3. *-commutative 38.6%

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

      4. expm1-def 38.5%

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

      5. *-commutative 38.5%

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

      6. expm1-def 63.8%

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

      7. *-commutative 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in a around 0 58.1%

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

   5. Taylor expanded in eps around 0 87.3%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \varepsilon + \frac{1}{a}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{b}{a} + 1}{b}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot -0.5 + \frac{1}{a}\\ \end{array} \]

Alternative 8: 78.3% accurate, 63.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (if (<= b 9.2e-176) (/ 1.0 b) (/ 1.0 a)))

C

assert(a < b);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

NOTE: a and b 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 <= 9.2d-176) then
        tmp = 1.0d0 / b
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 9.2e-176) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b, eps):
	tmp = 0
	if b <= 9.2e-176:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b = sort([a, b])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 9.2e-176)
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 9.2e-176)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 9.2e-176], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.2000000000000005e-176

   1. Initial program 9.2%

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

      1. associate-*l/ 9.2%

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

      2. *-commutative 9.2%

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

      3. expm1-def 10.7%

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

      4. *-commutative 10.7%

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

      5. expm1-def 16.3%

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

      6. *-commutative 16.3%

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

      7. expm1-def 38.6%

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

      8. *-commutative 38.6%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in b around 0 57.2%

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

   if 9.2000000000000005e-176 < b

   1. Initial program 7.1%

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

      1. associate-*l/ 7.1%

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

      2. *-commutative 7.1%

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

      3. expm1-def 8.6%

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

      4. *-commutative 8.6%

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

      5. expm1-def 20.2%

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

      6. *-commutative 20.2%

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

      7. expm1-def 45.8%

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

      8. *-commutative 45.8%

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

   3. Simplified 45.8%

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

   4. Taylor expanded in a around 0 63.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 9: 48.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.5%

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

   1. associate-*l/ 8.5%

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

   2. *-commutative 8.5%

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

   3. expm1-def 9.9%

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

   4. *-commutative 9.9%

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

   5. expm1-def 17.7%

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

   6. *-commutative 17.7%

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

   7. expm1-def 41.1%

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

   8. *-commutative 41.1%

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

3. Simplified 41.1%

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

4. Taylor expanded in a around 0 46.2%

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

5. Final simplification 46.2%

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

Developer target: 77.5% accurate, 45.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double eps) {
	return (a + b) / (a * 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 = (a + b) / (a * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023187 
(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))))