expq3 (problem 3.4.2)

Percentage Accurate: 6.6% → 96.9%

Time: 17.5s

Alternatives: 6

Speedup: 63.4×

• 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: 6.6% 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: 96.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

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 / b) + ((1.0d0 / a) - eps)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. associate-*l/ 7.2%

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

   3. *-commutative 7.2%

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

   4. expm1-def 8.7%

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

   5. *-commutative 8.7%

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

   6. associate-/r* 8.7%

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

   7. expm1-def 14.3%

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

   8. *-commutative 14.3%

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

   9. expm1-def 49.6%

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

   10. *-commutative 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in a around 0 16.8%

   \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{\varepsilon \cdot e^{b \cdot \varepsilon}}{e^{b \cdot \varepsilon} - 1}\right) - 0.5 \cdot \varepsilon} \]
5. Step-by-step derivation

   1. associate--l+ 16.8%

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

   2. cancel-sign-sub-inv 16.8%

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

   3. metadata-eval 16.8%

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

   4. associate-/l* 16.8%

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

   5. expm1-def 65.4%

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

   6. *-commutative 65.4%

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

6. Simplified 65.4%

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

7. Taylor expanded in b around 0 95.4%

   \[\leadsto \color{blue}{\left(\varepsilon + \left(-0.5 \cdot \varepsilon + \left(\frac{1}{a} + \frac{1}{b}\right)\right)\right) - 0.5 \cdot \varepsilon} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 95.4%

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

   2. associate-+r+ 95.4%

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

   3. distribute-rgt1-in 95.4%

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

   4. metadata-eval 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-+l+ 95.4%

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

   7. add-sqr-sqrt 44.7%

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

   8. sqrt-unprod 95.5%

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

   9. swap-sqr 95.5%

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

   10. metadata-eval 95.5%

       \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\varepsilon \cdot \varepsilon\right)} + \left(\left(\frac{1}{a} + \frac{1}{b}\right) + -0.5 \cdot \varepsilon\right) \]

   11. metadata-eval 95.5%

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

   12. swap-sqr 95.5%

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

   13. sqrt-unprod 51.4%

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

   14. add-sqr-sqrt 97.6%

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

   15. *-commutative 97.6%

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

   16. +-commutative 97.6%

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

   17. associate-+r+ 97.6%

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

   18. *-commutative 97.6%

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

   19. fma-def 97.6%

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

9. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\varepsilon \cdot -0.5 + \left(\mathsf{fma}\left(\varepsilon, -0.5, \frac{1}{a}\right) + \frac{1}{b}\right)} \]
10. Step-by-step derivation

    1. +-commutative 97.6%

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

    2. +-commutative 97.6%

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

    3. associate-+l+ 97.6%

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

    4. fma-udef 97.6%

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

    5. associate-+r+ 97.6%

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

    6. *-commutative 97.6%

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

    7. +-commutative 97.6%

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

    8. *-commutative 97.6%

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

    9. associate-+r+ 97.6%

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

    10. distribute-lft-out 97.6%

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

    11. metadata-eval 97.6%

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

    12. metadata-eval 97.6%

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

    13. distribute-rgt-neg-in 97.6%

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

    14. *-rgt-identity 97.6%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 2: 58.4% accurate, 45.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} - \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= b 5e-152) (/ 1.0 b) (- (/ 1.0 a) eps)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 5e-152) {
		tmp = 1.0 / b;
	} else {
		tmp = (1.0 / a) - eps;
	}
	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 <= 5d-152) then
        tmp = 1.0d0 / b
    else
        tmp = (1.0d0 / a) - eps
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 5e-152) {
		tmp = 1.0 / b;
	} else {
		tmp = (1.0 / a) - eps;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= 5e-152:
		tmp = 1.0 / b
	else:
		tmp = (1.0 / a) - eps
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= 5e-152)
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(Float64(1.0 / a) - eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 5e-152)
		tmp = 1.0 / b;
	else
		tmp = (1.0 / a) - eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, 5e-152], N[(1.0 / b), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] - eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999997e-152

   1. Initial program 4.9%

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

      1. *-commutative 4.9%

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

      2. associate-*l/ 4.9%

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

      3. *-commutative 4.9%

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

      4. expm1-def 6.6%

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

      5. *-commutative 6.6%

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

      6. associate-/r* 6.6%

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

      7. expm1-def 12.2%

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

      8. *-commutative 12.2%

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

      9. expm1-def 37.9%

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

      10. *-commutative 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around 0 60.4%

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

   if 4.9999999999999997e-152 < b

   1. Initial program 11.6%

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

      1. *-commutative 11.6%

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

      2. associate-*l/ 11.6%

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

      3. *-commutative 11.6%

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

      4. expm1-def 12.7%

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

      5. *-commutative 12.7%

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

      6. associate-/r* 12.7%

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

      7. expm1-def 18.4%

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

      8. *-commutative 18.4%

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

      9. expm1-def 71.8%

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

      10. *-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in a around 0 25.4%

      \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{\varepsilon \cdot e^{b \cdot \varepsilon}}{e^{b \cdot \varepsilon} - 1}\right) - 0.5 \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. associate--l+ 25.4%

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

      2. cancel-sign-sub-inv 25.4%

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

      3. metadata-eval 25.4%

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

      4. associate-/l* 25.4%

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

      5. expm1-def 83.9%

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

      6. *-commutative 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in b around 0 93.2%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(-0.5 \cdot \varepsilon + \left(\frac{1}{a} + \frac{1}{b}\right)\right)\right) - 0.5 \cdot \varepsilon} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 93.2%

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

      2. associate-+r+ 93.2%

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

      3. distribute-rgt1-in 93.2%

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

      4. metadata-eval 93.2%

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

      5. metadata-eval 93.2%

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

      6. associate-+l+ 93.2%

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

      7. add-sqr-sqrt 36.7%

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

      8. sqrt-unprod 93.5%

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

      9. swap-sqr 93.5%

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

      10. metadata-eval 93.5%

          \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\varepsilon \cdot \varepsilon\right)} + \left(\left(\frac{1}{a} + \frac{1}{b}\right) + -0.5 \cdot \varepsilon\right) \]

      11. metadata-eval 93.5%

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

      12. swap-sqr 93.5%

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

      13. sqrt-unprod 58.6%

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

      14. add-sqr-sqrt 94.2%

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

      15. *-commutative 94.2%

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

      16. +-commutative 94.2%

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

      17. associate-+r+ 94.2%

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

      18. *-commutative 94.2%

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

      19. fma-def 94.2%

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

   9. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\varepsilon \cdot -0.5 + \left(\mathsf{fma}\left(\varepsilon, -0.5, \frac{1}{a}\right) + \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 94.2%

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

       2. +-commutative 94.2%

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

       3. associate-+l+ 94.2%

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

       4. fma-udef 94.2%

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

       5. associate-+r+ 94.2%

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

       6. *-commutative 94.2%

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

       7. +-commutative 94.2%

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

       8. *-commutative 94.2%

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

       9. associate-+r+ 94.2%

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

       10. distribute-lft-out 94.2%

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

       11. metadata-eval 94.2%

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

       12. metadata-eval 94.2%

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

       13. distribute-rgt-neg-in 94.2%

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

       14. *-rgt-identity 94.2%

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

   11. Simplified 94.2%

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

   12. Taylor expanded in b around inf 69.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} - \varepsilon\\ \end{array} \]

Alternative 3: 94.6% accurate, 45.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double eps) {
	return (1.0 / b) + (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 / b) + (1.0d0 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. associate-*l/ 7.2%

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

   3. *-commutative 7.2%

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

   4. expm1-def 8.7%

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

   5. *-commutative 8.7%

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

   6. associate-/r* 8.7%

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

   7. expm1-def 14.3%

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

   8. *-commutative 14.3%

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

   9. expm1-def 49.6%

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

   10. *-commutative 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in eps around 0 75.9%

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

5. Taylor expanded in a around 0 94.0%

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

6. Final simplification 94.0%

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

Alternative 4: 37.1% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.56 \cdot 10^{+60}:\\ \;\;\;\;-\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (if (<= b -1.56e+60) (- eps) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= -1.56e+60) {
		tmp = -eps;
	} 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 <= (-1.56d+60)) then
        tmp = -eps
    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 <= -1.56e+60) {
		tmp = -eps;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= -1.56e+60:
		tmp = -eps
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= -1.56e+60)
		tmp = Float64(-eps);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= -1.56e+60)
		tmp = -eps;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, -1.56e+60], (-eps), N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.56000000000000009e60

   1. Initial program 15.5%

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

      1. *-commutative 15.5%

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

      2. associate-*l/ 15.5%

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

      3. *-commutative 15.5%

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

      4. expm1-def 16.7%

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

      5. *-commutative 16.7%

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

      6. associate-/r* 16.7%

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

      7. expm1-def 22.0%

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

      8. *-commutative 22.0%

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

      9. expm1-def 63.8%

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

      10. *-commutative 63.8%

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

   3. Simplified 63.8%

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

   4. Taylor expanded in a around 0 31.4%

      \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{\varepsilon \cdot e^{b \cdot \varepsilon}}{e^{b \cdot \varepsilon} - 1}\right) - 0.5 \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. associate--l+ 31.4%

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

      2. cancel-sign-sub-inv 31.4%

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

      3. metadata-eval 31.4%

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

      4. associate-/l* 31.4%

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

      5. expm1-def 85.3%

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

      6. *-commutative 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in b around 0 87.7%

      \[\leadsto \color{blue}{\left(\varepsilon + \left(-0.5 \cdot \varepsilon + \left(\frac{1}{a} + \frac{1}{b}\right)\right)\right) - 0.5 \cdot \varepsilon} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 87.7%

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

      2. associate-+r+ 87.7%

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

      3. distribute-rgt1-in 87.7%

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

      4. metadata-eval 87.7%

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

      5. metadata-eval 87.7%

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

      6. associate-+l+ 87.7%

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

      7. add-sqr-sqrt 45.2%

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

      8. sqrt-unprod 88.0%

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

      9. swap-sqr 88.0%

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

      10. metadata-eval 88.0%

          \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\varepsilon \cdot \varepsilon\right)} + \left(\left(\frac{1}{a} + \frac{1}{b}\right) + -0.5 \cdot \varepsilon\right) \]

      11. metadata-eval 88.0%

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

      12. swap-sqr 88.0%

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

      13. sqrt-unprod 42.5%

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

      14. add-sqr-sqrt 97.9%

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

      15. *-commutative 97.9%

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

      16. +-commutative 97.9%

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

      17. associate-+r+ 97.9%

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

      18. *-commutative 97.9%

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

      19. fma-def 97.9%

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

   9. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\varepsilon \cdot -0.5 + \left(\mathsf{fma}\left(\varepsilon, -0.5, \frac{1}{a}\right) + \frac{1}{b}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 97.9%

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

       2. +-commutative 97.9%

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

       3. associate-+l+ 97.9%

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

       4. fma-udef 97.9%

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

       5. associate-+r+ 97.9%

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

       6. *-commutative 97.9%

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

       7. +-commutative 97.9%

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

       8. *-commutative 97.9%

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

       9. associate-+r+ 97.9%

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

       10. distribute-lft-out 97.9%

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

       11. metadata-eval 97.9%

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

       12. metadata-eval 97.9%

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

       13. distribute-rgt-neg-in 97.9%

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

       14. *-rgt-identity 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in eps around inf 15.5%

       \[\leadsto \color{blue}{-1 \cdot \varepsilon} \]
   13. Step-by-step derivation

       1. mul-1-neg 15.5%

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

   14. Simplified 15.5%

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

   if -1.56000000000000009e60 < b

   1. Initial program 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*l/ 5.3%

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

      3. *-commutative 5.3%

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

      4. expm1-def 6.9%

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

      5. *-commutative 6.9%

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

      6. associate-/r* 6.9%

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

      7. expm1-def 12.6%

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

      8. *-commutative 12.6%

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

      9. expm1-def 46.4%

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

      10. *-commutative 46.4%

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

   3. Simplified 46.4%

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

   4. Taylor expanded in a around 0 40.6%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.56 \cdot 10^{+60}:\\ \;\;\;\;-\varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 5: 57.2% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (if (<= b 5e-152) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 5e-152) {
		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 <= 5d-152) 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 <= 5e-152) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= 5e-152:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= 5e-152)
		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 <= 5e-152)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, 5e-152], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999997e-152

   1. Initial program 4.9%

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

      1. *-commutative 4.9%

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

      2. associate-*l/ 4.9%

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

      3. *-commutative 4.9%

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

      4. expm1-def 6.6%

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

      5. *-commutative 6.6%

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

      6. associate-/r* 6.6%

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

      7. expm1-def 12.2%

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

      8. *-commutative 12.2%

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

      9. expm1-def 37.9%

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

      10. *-commutative 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around 0 60.4%

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

   if 4.9999999999999997e-152 < b

   1. Initial program 11.6%

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

      1. *-commutative 11.6%

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

      2. associate-*l/ 11.6%

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

      3. *-commutative 11.6%

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

      4. expm1-def 12.7%

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

      5. *-commutative 12.7%

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

      6. associate-/r* 12.7%

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

      7. expm1-def 18.4%

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

      8. *-commutative 18.4%

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

      9. expm1-def 71.8%

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

      10. *-commutative 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in a around 0 65.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 6: 5.4% accurate, 160.5× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, eps_] := (-eps)

Derivation

1. Initial program 7.2%

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

   1. *-commutative 7.2%

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

   2. associate-*l/ 7.2%

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

   3. *-commutative 7.2%

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

   4. expm1-def 8.7%

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

   5. *-commutative 8.7%

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

   6. associate-/r* 8.7%

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

   7. expm1-def 14.3%

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

   8. *-commutative 14.3%

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

   9. expm1-def 49.6%

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

   10. *-commutative 49.6%

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

3. Simplified 49.6%

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

4. Taylor expanded in a around 0 16.8%

   \[\leadsto \color{blue}{\left(\frac{1}{a} + \frac{\varepsilon \cdot e^{b \cdot \varepsilon}}{e^{b \cdot \varepsilon} - 1}\right) - 0.5 \cdot \varepsilon} \]
5. Step-by-step derivation

   1. associate--l+ 16.8%

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

   2. cancel-sign-sub-inv 16.8%

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

   3. metadata-eval 16.8%

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

   4. associate-/l* 16.8%

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

   5. expm1-def 65.4%

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

   6. *-commutative 65.4%

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

6. Simplified 65.4%

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

7. Taylor expanded in b around 0 95.4%

   \[\leadsto \color{blue}{\left(\varepsilon + \left(-0.5 \cdot \varepsilon + \left(\frac{1}{a} + \frac{1}{b}\right)\right)\right) - 0.5 \cdot \varepsilon} \]
8. Step-by-step derivation

   1. cancel-sign-sub-inv 95.4%

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

   2. associate-+r+ 95.4%

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

   3. distribute-rgt1-in 95.4%

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

   4. metadata-eval 95.4%

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

   5. metadata-eval 95.4%

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

   6. associate-+l+ 95.4%

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

   7. add-sqr-sqrt 44.7%

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

   8. sqrt-unprod 95.5%

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

   9. swap-sqr 95.5%

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

   10. metadata-eval 95.5%

       \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\varepsilon \cdot \varepsilon\right)} + \left(\left(\frac{1}{a} + \frac{1}{b}\right) + -0.5 \cdot \varepsilon\right) \]

   11. metadata-eval 95.5%

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

   12. swap-sqr 95.5%

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

   13. sqrt-unprod 51.4%

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

   14. add-sqr-sqrt 97.6%

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

   15. *-commutative 97.6%

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

   16. +-commutative 97.6%

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

   17. associate-+r+ 97.6%

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

   18. *-commutative 97.6%

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

   19. fma-def 97.6%

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

9. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\varepsilon \cdot -0.5 + \left(\mathsf{fma}\left(\varepsilon, -0.5, \frac{1}{a}\right) + \frac{1}{b}\right)} \]
10. Step-by-step derivation

    1. +-commutative 97.6%

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

    2. +-commutative 97.6%

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

    3. associate-+l+ 97.6%

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

    4. fma-udef 97.6%

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

    5. associate-+r+ 97.6%

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

    6. *-commutative 97.6%

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

    7. +-commutative 97.6%

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

    8. *-commutative 97.6%

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

    9. associate-+r+ 97.6%

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

    10. distribute-lft-out 97.6%

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

    11. metadata-eval 97.6%

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

    12. metadata-eval 97.6%

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

    13. distribute-rgt-neg-in 97.6%

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

    14. *-rgt-identity 97.6%

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

11. Simplified 97.6%

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

12. Taylor expanded in eps around inf 6.3%

    \[\leadsto \color{blue}{-1 \cdot \varepsilon} \]
13. Step-by-step derivation

    1. mul-1-neg 6.3%

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

14. Simplified 6.3%

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

15. Final simplification 6.3%

    \[\leadsto -\varepsilon \]

Developer target: 77.1% 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 2023275 
(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))))