expq3 (problem 3.4.2)

Percentage Accurate: 6.5% → 99.9%

Time: 11.0s

Alternatives: 7

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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{-5}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(t_0\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (let* ((t_0 (* eps (+ a b)))
        (t_1
         (/
          (* eps (+ (exp t_0) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e-5)))
     (+ (/ 1.0 a) (/ 1.0 b))
     (* (expm1 t_0) (/ (/ eps (expm1 (* eps b))) (expm1 (* eps a)))))))

C

double code(double a, double b, double eps) {
	double t_0 = eps * (a + b);
	double t_1 = (eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e-5)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = expm1(t_0) * ((eps / expm1((eps * b))) / expm1((eps * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double eps) {
	double t_0 = eps * (a + b);
	double t_1 = (eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e-5)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = Math.expm1(t_0) * ((eps / Math.expm1((eps * b))) / Math.expm1((eps * a)));
	}
	return tmp;
}

Python

def code(a, b, eps):
	t_0 = eps * (a + b)
	t_1 = (eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e-5):
		tmp = (1.0 / a) + (1.0 / b)
	else:
		tmp = math.expm1(t_0) * ((eps / math.expm1((eps * b))) / math.expm1((eps * a)))
	return tmp

Julia

function code(a, b, eps)
	t_0 = Float64(eps * Float64(a + b))
	t_1 = Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e-5))
		tmp = Float64(Float64(1.0 / a) + Float64(1.0 / b));
	else
		tmp = Float64(expm1(t_0) * Float64(Float64(eps / expm1(Float64(eps * b))) / expm1(Float64(eps * a))));
	end
	return tmp
end

Wolfram

code[a_, b_, eps_] := Block[{t$95$0 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * N[(N[Exp[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(eps * a), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Exp[N[(eps * b), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e-5]], $MachinePrecision]], N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[t$95$0] - 1), $MachinePrecision] * N[(N[(eps / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.0 or 1.00000000000000008e-5 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1)))

   1. Initial program 0.6%

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

      1. *-commutative 0.6%

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

      2. associate-*l/ 0.6%

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

      3. *-commutative 0.6%

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

      4. expm1-def 2.3%

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

      5. *-commutative 2.3%

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

      6. associate-/r* 2.3%

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

      7. expm1-def 13.0%

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

      8. *-commutative 13.0%

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

      9. expm1-def 53.5%

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

      10. *-commutative 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in eps around 0 81.6%

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

   5. Taylor expanded in a around 0 100.0%

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

   if -inf.0 < (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < 1.00000000000000008e-5

   1. Initial program 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-*l/ 92.5%

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

      3. *-commutative 92.5%

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

      4. expm1-def 92.5%

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

      5. *-commutative 92.5%

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

      6. associate-/r* 92.5%

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

      7. expm1-def 99.8%

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

      8. *-commutative 99.8%

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

      9. expm1-def 99.8%

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

      10. *-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 10^{-5}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \end{array} \]

Alternative 2: 93.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.32 \cdot 10^{-43}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= eps 1.32e-43)
   (- (+ (/ 1.0 a) (/ 1.0 b)) (* eps 0.5))
   (/ eps (expm1 (* eps b)))))

C

double code(double a, double b, double eps) {
	double tmp;
	if (eps <= 1.32e-43) {
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
	} else {
		tmp = eps / expm1((eps * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (eps <= 1.32e-43) {
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
	} else {
		tmp = eps / Math.expm1((eps * b));
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if eps <= 1.32e-43:
		tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5)
	else:
		tmp = eps / math.expm1((eps * b))
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (eps <= 1.32e-43)
		tmp = Float64(Float64(Float64(1.0 / a) + Float64(1.0 / b)) - Float64(eps * 0.5));
	else
		tmp = Float64(eps / expm1(Float64(eps * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[eps, 1.32e-43], N[(N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision], N[(eps / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.32000000000000002e-43

   1. Initial program 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l/ 5.7%

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

      3. *-commutative 5.7%

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

      4. expm1-def 7.2%

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

      5. *-commutative 7.2%

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

      6. associate-/r* 7.2%

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

      7. expm1-def 17.8%

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

      8. *-commutative 17.8%

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

      9. expm1-def 55.4%

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

      10. *-commutative 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in b around 0 15.7%

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

      1. associate-/r* 15.7%

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

      2. expm1-def 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in eps around 0 46.6%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(a + b\right)\right)} \cdot \frac{\frac{1}{b}}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \]
   8. Step-by-step derivation

      1. *-commutative 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in a around 0 96.0%

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

   if 1.32000000000000002e-43 < eps

   1. Initial program 50.9%

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

      1. *-commutative 50.9%

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

      2. times-frac 50.9%

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

      3. +-commutative 50.9%

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

      4. expm1-def 58.6%

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

      5. *-commutative 58.6%

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

      6. expm1-def 59.8%

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

      7. +-commutative 59.8%

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

      8. *-commutative 59.8%

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

      9. expm1-def 99.3%

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

      10. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in b around 0 67.4%

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.32 \cdot 10^{-43}:\\ \;\;\;\;\left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \]

Alternative 3: 95.1% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{a} + \frac{1}{b}\right) - \varepsilon \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (- (+ (/ 1.0 a) (/ 1.0 b)) (* eps 0.5)))

C

double code(double a, double b, double eps) {
	return ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
}

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)) - (eps * 0.5d0)
end function

Java

public static double code(double a, double b, double eps) {
	return ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
}

Python

def code(a, b, eps):
	return ((1.0 / a) + (1.0 / b)) - (eps * 0.5)

Julia

function code(a, b, eps)
	return Float64(Float64(Float64(1.0 / a) + Float64(1.0 / b)) - Float64(eps * 0.5))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = ((1.0 / a) + (1.0 / b)) - (eps * 0.5);
end

Wolfram

code[a_, b_, eps_] := N[(N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. *-commutative 7.8%

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

   2. associate-*l/ 7.8%

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

   3. *-commutative 7.8%

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

   4. expm1-def 9.3%

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

   5. *-commutative 9.3%

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

   6. associate-/r* 9.3%

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

   7. expm1-def 19.8%

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

   8. *-commutative 19.8%

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

   9. expm1-def 57.1%

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

   10. *-commutative 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in b around 0 15.9%

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

   1. associate-/r* 15.9%

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

   2. expm1-def 50.7%

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

6. Simplified 50.7%

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

7. Taylor expanded in eps around 0 47.6%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(a + b\right)\right)} \cdot \frac{\frac{1}{b}}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \]
8. Step-by-step derivation

   1. *-commutative 47.6%

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

9. Simplified 47.6%

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

10. Taylor expanded in a around 0 94.1%

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

11. Final simplification 94.1%

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

Alternative 4: 94.7% 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]

Derivation

1. Initial program 7.8%

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

   1. *-commutative 7.8%

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

   2. associate-*l/ 7.8%

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

   3. *-commutative 7.8%

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

   4. expm1-def 9.3%

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

   5. *-commutative 9.3%

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

   6. associate-/r* 9.3%

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

   7. expm1-def 19.8%

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

   8. *-commutative 19.8%

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

   9. expm1-def 57.1%

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

   10. *-commutative 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in eps around 0 79.0%

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

5. Taylor expanded in a around 0 93.5%

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

6. Final simplification 93.5%

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

Alternative 5: 59.5% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (if (<= b 4.2e-101) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 4.2e-101) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (b <= 4.2d-101) then
        tmp = 1.0d0 / b
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 4.2e-101) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= 4.2e-101:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= 4.2e-101)
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 4.2e-101)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, 4.2e-101], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.20000000000000031e-101

   1. Initial program 6.9%

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

      1. *-commutative 6.9%

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

      2. associate-*l/ 6.9%

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

      3. *-commutative 6.9%

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

      4. expm1-def 8.5%

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

      5. *-commutative 8.5%

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

      6. associate-/r* 8.5%

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

      7. expm1-def 17.7%

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

      8. *-commutative 17.7%

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

      9. expm1-def 50.6%

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

      10. *-commutative 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in b around 0 54.7%

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

   if 4.20000000000000031e-101 < b

   1. Initial program 9.7%

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

      1. *-commutative 9.7%

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

      2. associate-*l/ 9.7%

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

      3. *-commutative 9.7%

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

      4. expm1-def 11.0%

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

      5. *-commutative 11.0%

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

      6. associate-/r* 11.0%

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

      7. expm1-def 24.1%

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

      8. *-commutative 24.1%

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

      9. expm1-def 70.6%

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

      10. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   4. Taylor expanded in a around 0 65.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 6: 3.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (* eps -0.5))

C

double code(double a, double b, double eps) {
	return eps * -0.5;
}

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 * (-0.5d0)
end function

Java

public static double code(double a, double b, double eps) {
	return eps * -0.5;
}

Python

def code(a, b, eps):
	return eps * -0.5

Julia

function code(a, b, eps)
	return Float64(eps * -0.5)
end

MATLAB

function tmp = code(a, b, eps)
	tmp = eps * -0.5;
end

Wolfram

code[a_, b_, eps_] := N[(eps * -0.5), $MachinePrecision]

Derivation

1. Initial program 7.8%

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

   1. *-commutative 7.8%

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

   2. associate-*l/ 7.8%

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

   3. *-commutative 7.8%

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

   4. expm1-def 9.3%

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

   5. *-commutative 9.3%

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

   6. associate-/r* 9.3%

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

   7. expm1-def 19.8%

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

   8. *-commutative 19.8%

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

   9. expm1-def 57.1%

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

   10. *-commutative 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in b around 0 15.9%

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

   1. associate-/r* 15.9%

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

   2. expm1-def 50.7%

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

6. Simplified 50.7%

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

7. Taylor expanded in eps around 0 47.6%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(a + b\right)\right)} \cdot \frac{\frac{1}{b}}{\mathsf{expm1}\left(a \cdot \varepsilon\right)} \]
8. Step-by-step derivation

   1. *-commutative 47.6%

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

9. Simplified 47.6%

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

10. Taylor expanded in a around 0 94.1%

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

11. Taylor expanded in eps around inf 3.3%

    \[\leadsto \color{blue}{-0.5 \cdot \varepsilon} \]
12. Step-by-step derivation

    1. *-commutative 3.3%

       \[\leadsto \color{blue}{\varepsilon \cdot -0.5} \]

13. Simplified 3.3%

    \[\leadsto \color{blue}{\varepsilon \cdot -0.5} \]

14. Final simplification 3.3%

    \[\leadsto \varepsilon \cdot -0.5 \]

Alternative 7: 47.9% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

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

   1. *-commutative 7.8%

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

   2. associate-*l/ 7.8%

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

   3. *-commutative 7.8%

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

   4. expm1-def 9.3%

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

   5. *-commutative 9.3%

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

   6. associate-/r* 9.3%

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

   7. expm1-def 19.8%

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

   8. *-commutative 19.8%

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

   9. expm1-def 57.1%

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

   10. *-commutative 57.1%

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

3. Simplified 57.1%

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

4. Taylor expanded in a around 0 49.2%

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

5. Final simplification 49.2%

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

Developer target: 78.0% 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 2023293 
(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))))