expq3 (problem 3.4.2)

Percentage Accurate: 6.5% → 99.3%

Time: 19.9s

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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(t_0\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}\\ \end{array} \end{array} \]

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 5e-54)))
     (+ (/ 1.0 a) (/ 1.0 b))
     (/ (/ (expm1 t_0) (expm1 (* eps a))) (/ (expm1 (* eps b)) eps)))))

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 <= 5e-54)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = (expm1(t_0) / expm1((eps * a))) / (expm1((eps * b)) / eps);
	}
	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 <= 5e-54)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = (Math.expm1(t_0) / Math.expm1((eps * a))) / (Math.expm1((eps * b)) / eps);
	}
	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 <= 5e-54):
		tmp = (1.0 / a) + (1.0 / b)
	else:
		tmp = (math.expm1(t_0) / math.expm1((eps * a))) / (math.expm1((eps * b)) / eps)
	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 <= 5e-54))
		tmp = Float64(Float64(1.0 / a) + Float64(1.0 / b));
	else
		tmp = Float64(Float64(expm1(t_0) / expm1(Float64(eps * a))) / Float64(expm1(Float64(eps * b)) / eps));
	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, 5e-54]], $MachinePrecision]], N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[t$95$0] - 1), $MachinePrecision] / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision] / eps), $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 5.00000000000000015e-54 < (/.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 0.7%

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

      1. *-commutative 0.7%

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

      2. associate-*l/ 0.7%

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

      3. *-commutative 0.7%

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

      4. expm1-def 2.5%

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

      5. *-commutative 2.5%

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

      6. associate-/r* 2.5%

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

      7. expm1-def 12.5%

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

      8. *-commutative 12.5%

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

      9. expm1-def 47.8%

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

      10. *-commutative 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in eps around 0 82.5%

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

   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))) < 5.00000000000000015e-54

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*l/ 92.4%

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

      3. *-commutative 92.4%

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

      4. expm1-def 92.4%

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

      5. *-commutative 92.4%

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

      6. associate-/r* 92.4%

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

      7. expm1-def 94.7%

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

      8. *-commutative 94.7%

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

      9. expm1-def 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%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.8%

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

      3. clear-num 99.8%

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

      4. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}\\ \end{array} \]

Alternative 2: 85.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= b -1.1e+72)
   (/ 1.0 (/ (expm1 (* eps b)) eps))
   (+ (/ 1.0 a) (+ (/ 1.0 b) (* eps -0.5)))))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= -1.1e+72) {
		tmp = 1.0 / (expm1((eps * b)) / eps);
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= -1.1e+72) {
		tmp = 1.0 / (Math.expm1((eps * b)) / eps);
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= -1.1e+72:
		tmp = 1.0 / (math.expm1((eps * b)) / eps)
	else:
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5))
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= -1.1e+72)
		tmp = Float64(1.0 / Float64(expm1(Float64(eps * b)) / eps));
	else
		tmp = Float64(Float64(1.0 / a) + Float64(Float64(1.0 / b) + Float64(eps * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, -1.1e+72], N[(1.0 / N[(N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] + N[(N[(1.0 / b), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1e72

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. times-frac 26.9%

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

      3. +-commutative 26.9%

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

      4. expm1-def 30.7%

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

      5. *-commutative 30.7%

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

      6. expm1-def 31.6%

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

      7. +-commutative 31.6%

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

      8. *-commutative 31.6%

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

      9. expm1-def 73.8%

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

      10. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in b around 0 30.1%

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

      1. clear-num 30.1%

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

      2. inv-pow 30.1%

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

   6. Applied egg-rr 30.1%

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

      1. unpow-1 30.1%

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

      2. *-commutative 30.1%

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

   8. Simplified 30.1%

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

   if -1.1e72 < b

   1. Initial program 3.0%

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

      1. *-commutative 3.0%

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

      2. associate-*l/ 3.0%

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

      3. *-commutative 3.0%

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

      4. expm1-def 4.9%

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

      5. *-commutative 4.9%

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

      6. associate-/r* 4.9%

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

      7. expm1-def 15.4%

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

      8. *-commutative 15.4%

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

      9. expm1-def 47.4%

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

      10. *-commutative 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in a around 0 9.0%

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

      1. associate--l+ 9.0%

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

      2. cancel-sign-sub-inv 9.0%

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

      3. metadata-eval 9.0%

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

      4. associate-/l* 9.0%

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

      5. expm1-def 66.8%

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

      6. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in eps around inf 9.0%

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

      1. expm1-def 66.8%

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

      2. associate-/l* 66.8%

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

      3. expm1-def 9.0%

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

      4. div-sub 6.0%

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

      5. *-commutative 6.0%

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

      6. exp-prod 2.8%

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

      7. *-commutative 2.8%

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

      8. exp-prod 8.6%

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

      9. *-inverses 9.5%

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

      10. rec-exp 9.5%

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

      11. distribute-rgt-neg-in 9.5%

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

   9. Simplified 9.5%

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

   10. Taylor expanded in eps around 0 98.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{expm1}\left(\varepsilon \cdot b\right)}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 3: 85.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= b -1.1e+72)
   (/ eps (expm1 (* eps b)))
   (+ (/ 1.0 a) (+ (/ 1.0 b) (* eps -0.5)))))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= -1.1e+72) {
		tmp = eps / expm1((eps * b));
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= -1.1e+72) {
		tmp = eps / Math.expm1((eps * b));
	} else {
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5));
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= -1.1e+72:
		tmp = eps / math.expm1((eps * b))
	else:
		tmp = (1.0 / a) + ((1.0 / b) + (eps * -0.5))
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= -1.1e+72)
		tmp = Float64(eps / expm1(Float64(eps * b)));
	else
		tmp = Float64(Float64(1.0 / a) + Float64(Float64(1.0 / b) + Float64(eps * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, -1.1e+72], N[(eps / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] + N[(N[(1.0 / b), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1e72

   1. Initial program 26.9%

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

      1. *-commutative 26.9%

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

      2. times-frac 26.9%

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

      3. +-commutative 26.9%

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

      4. expm1-def 30.7%

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

      5. *-commutative 30.7%

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

      6. expm1-def 31.6%

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

      7. +-commutative 31.6%

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

      8. *-commutative 31.6%

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

      9. expm1-def 73.8%

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

      10. *-commutative 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in b around 0 30.1%

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

   if -1.1e72 < b

   1. Initial program 3.0%

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

      1. *-commutative 3.0%

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

      2. associate-*l/ 3.0%

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

      3. *-commutative 3.0%

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

      4. expm1-def 4.9%

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

      5. *-commutative 4.9%

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

      6. associate-/r* 4.9%

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

      7. expm1-def 15.4%

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

      8. *-commutative 15.4%

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

      9. expm1-def 47.4%

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

      10. *-commutative 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in a around 0 9.0%

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

      1. associate--l+ 9.0%

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

      2. cancel-sign-sub-inv 9.0%

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

      3. metadata-eval 9.0%

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

      4. associate-/l* 9.0%

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

      5. expm1-def 66.8%

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

      6. *-commutative 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in eps around inf 9.0%

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

      1. expm1-def 66.8%

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

      2. associate-/l* 66.8%

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

      3. expm1-def 9.0%

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

      4. div-sub 6.0%

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

      5. *-commutative 6.0%

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

      6. exp-prod 2.8%

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

      7. *-commutative 2.8%

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

      8. exp-prod 8.6%

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

      9. *-inverses 9.5%

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

      10. rec-exp 9.5%

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

      11. distribute-rgt-neg-in 9.5%

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

   9. Simplified 9.5%

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

   10. Taylor expanded in eps around 0 98.1%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \left(\frac{1}{b} + \varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 4: 95.0% accurate, 29.2× speedup

Math

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

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(1.0 / a) + Float64(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[(1.0 / a), $MachinePrecision] + N[(N[(1.0 / b), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

   1. *-commutative 6.4%

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

   2. associate-*l/ 6.4%

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

   3. *-commutative 6.4%

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

   4. expm1-def 8.1%

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

   5. *-commutative 8.1%

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

   6. associate-/r* 8.1%

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

   7. expm1-def 17.6%

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

   8. *-commutative 17.6%

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

   9. expm1-def 51.1%

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

   10. *-commutative 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in a around 0 14.7%

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

   1. associate--l+ 14.7%

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

   2. cancel-sign-sub-inv 14.7%

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

   3. metadata-eval 14.7%

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

   4. associate-/l* 14.7%

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

   5. expm1-def 68.5%

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

   6. *-commutative 68.5%

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

6. Simplified 68.5%

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

7. Taylor expanded in eps around inf 14.7%

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

   1. expm1-def 68.5%

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

   2. associate-/l* 68.5%

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

   3. expm1-def 14.7%

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

   4. div-sub 6.3%

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

   5. *-commutative 6.3%

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

   6. exp-prod 2.6%

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

   7. *-commutative 2.6%

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

   8. exp-prod 14.3%

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

   9. *-inverses 15.1%

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

   10. rec-exp 15.1%

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

   11. distribute-rgt-neg-in 15.1%

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

9. Simplified 15.1%

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

10. Taylor expanded in eps around 0 95.6%

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

11. Final simplification 95.6%

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

Alternative 5: 94.6% 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 6.4%

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

   1. *-commutative 6.4%

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

   2. associate-*l/ 6.4%

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

   3. *-commutative 6.4%

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

   4. expm1-def 8.1%

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

   5. *-commutative 8.1%

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

   6. associate-/r* 8.1%

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

   7. expm1-def 17.6%

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

   8. *-commutative 17.6%

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

   9. expm1-def 51.1%

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

   10. *-commutative 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in eps around 0 78.9%

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

5. Taylor expanded in a around 0 94.8%

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

6. Final simplification 94.8%

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

Alternative 6: 57.6% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (if (<= b 5.5e-160) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 5.5e-160) {
		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 <= 5.5d-160) 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 <= 5.5e-160) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= 5.5e-160:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= 5.5e-160)
		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 <= 5.5e-160)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, 5.5e-160], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e-160

   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 8.6%

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

      5. *-commutative 8.6%

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

      6. associate-/r* 8.6%

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

      7. expm1-def 19.2%

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

      8. *-commutative 19.2%

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

      9. expm1-def 47.8%

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

      10. *-commutative 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in b around 0 56.9%

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

   if 5.5e-160 < b

   1. Initial program 5.5%

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

      1. *-commutative 5.5%

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

      2. associate-*l/ 5.5%

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

      3. *-commutative 5.5%

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

      4. expm1-def 7.1%

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

      5. *-commutative 7.1%

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

      6. associate-/r* 7.1%

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

      7. expm1-def 14.6%

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

      8. *-commutative 14.6%

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

      9. expm1-def 57.3%

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

      10. *-commutative 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in a around 0 58.9%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

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 6.4%

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

   1. *-commutative 6.4%

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

   2. associate-*l/ 6.4%

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

   3. *-commutative 6.4%

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

   4. expm1-def 8.1%

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

   5. *-commutative 8.1%

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

   6. associate-/r* 8.1%

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

   7. expm1-def 17.6%

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

   8. *-commutative 17.6%

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

   9. expm1-def 51.1%

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

   10. *-commutative 51.1%

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

3. Simplified 51.1%

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

4. Taylor expanded in a around 0 46.5%

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

5. Final simplification 46.5%

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

Developer target: 77.9% 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 2023285 
(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))))