expq3 (problem 3.4.2)

Percentage Accurate: 6.6% → 98.9%

Time: 14.9s

Alternatives: 4

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: 6.6% 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: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(a + b\right)\\ t_1 := \frac{\varepsilon \cdot \left(e^{t_0} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{-84}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(t_0\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (let* ((t_0 (* eps (+ a b)))
        (t_1
         (/
          (* eps (+ (exp t_0) -1.0))
          (* (+ (exp (* eps a)) -1.0) (+ (exp (* eps b)) -1.0)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e-84)))
     (+ (/ 1.0 a) (/ 1.0 b))
     (* (expm1 t_0) (/ (/ eps (expm1 (* eps a))) (expm1 (* eps b)))))))

C

double code(double a, double b, double eps) {
	double t_0 = eps * (a + b);
	double t_1 = (eps * (exp(t_0) + -1.0)) / ((exp((eps * a)) + -1.0) * (exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e-84)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = expm1(t_0) * ((eps / expm1((eps * a))) / expm1((eps * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double eps) {
	double t_0 = eps * (a + b);
	double t_1 = (eps * (Math.exp(t_0) + -1.0)) / ((Math.exp((eps * a)) + -1.0) * (Math.exp((eps * b)) + -1.0));
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e-84)) {
		tmp = (1.0 / a) + (1.0 / b);
	} else {
		tmp = Math.expm1(t_0) * ((eps / Math.expm1((eps * a))) / Math.expm1((eps * b)));
	}
	return tmp;
}

Python

def code(a, b, eps):
	t_0 = eps * (a + b)
	t_1 = (eps * (math.exp(t_0) + -1.0)) / ((math.exp((eps * a)) + -1.0) * (math.exp((eps * b)) + -1.0))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e-84):
		tmp = (1.0 / a) + (1.0 / b)
	else:
		tmp = math.expm1(t_0) * ((eps / math.expm1((eps * a))) / math.expm1((eps * b)))
	return tmp

Julia

function code(a, b, eps)
	t_0 = Float64(eps * Float64(a + b))
	t_1 = Float64(Float64(eps * Float64(exp(t_0) + -1.0)) / Float64(Float64(exp(Float64(eps * a)) + -1.0) * Float64(exp(Float64(eps * b)) + -1.0)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e-84))
		tmp = Float64(Float64(1.0 / a) + Float64(1.0 / b));
	else
		tmp = Float64(expm1(t_0) * Float64(Float64(eps / expm1(Float64(eps * a))) / expm1(Float64(eps * b))));
	end
	return tmp
end

Wolfram

code[a_, b_, eps_] := Block[{t$95$0 = N[(eps * N[(a + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * N[(N[Exp[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(eps * a), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[Exp[N[(eps * b), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e-84]], $MachinePrecision]], N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[t$95$0] - 1), $MachinePrecision] * N[(N[(eps / N[(Exp[N[(eps * a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(eps * b), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 eps (-.f64 (exp.f64 (*.f64 (+.f64 a b) eps)) 1)) (*.f64 (-.f64 (exp.f64 (*.f64 a eps)) 1) (-.f64 (exp.f64 (*.f64 b eps)) 1))) < -inf.0 or 1e-84 < (/.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. times-frac 0.7%

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

      2. *-commutative 0.7%

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

      3. associate-*l/ 0.7%

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

      4. associate-*r/ 0.7%

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

      5. expm1-def 2.4%

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

      6. *-commutative 2.4%

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

      7. *-commutative 2.4%

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

      8. expm1-def 11.9%

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

      9. *-commutative 11.9%

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

      10. expm1-def 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in eps around 0 83.7%

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

   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))) < 1e-84

   1. Initial program 100.0%

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

      1. times-frac 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. expm1-def 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. expm1-def 100.0%

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

      9. *-commutative 100.0%

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

      10. expm1-def 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)} + -1\right)}{\left(e^{\varepsilon \cdot a} + -1\right) \cdot \left(e^{\varepsilon \cdot b} + -1\right)} \leq 10^{-84}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right) \cdot \frac{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}\\ \end{array} \]

Alternative 2: 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 5.0%

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

   1. times-frac 5.0%

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

   2. *-commutative 5.0%

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

   3. associate-*l/ 5.0%

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

   4. associate-*r/ 5.0%

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

   5. expm1-def 6.6%

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

   6. *-commutative 6.6%

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

   7. *-commutative 6.6%

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

   8. expm1-def 15.7%

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

   9. *-commutative 15.7%

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

   10. expm1-def 49.7%

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

3. Simplified 49.7%

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

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 95.9%

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

6. Final simplification 95.9%

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

Alternative 3: 58.6% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= a -7.6e-114) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -7.6e-114) {
		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 (a <= (-7.6d-114)) 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 (a <= -7.6e-114) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if a <= -7.6e-114:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (a <= -7.6e-114)
		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 (a <= -7.6e-114)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[a, -7.6e-114], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5999999999999997e-114

   1. Initial program 5.8%

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

      1. times-frac 5.8%

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

      2. *-commutative 5.8%

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

      3. associate-*l/ 5.8%

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

      4. associate-*r/ 5.8%

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

      5. expm1-def 7.7%

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

      6. *-commutative 7.7%

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

      7. *-commutative 7.7%

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

      8. expm1-def 20.9%

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

      9. *-commutative 20.9%

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

      10. expm1-def 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in b around 0 66.6%

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

   if -7.5999999999999997e-114 < a

   1. Initial program 4.7%

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

      1. times-frac 4.7%

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

      2. *-commutative 4.7%

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

      3. associate-*l/ 4.7%

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

      4. associate-*r/ 4.7%

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

      5. expm1-def 6.1%

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

      6. *-commutative 6.1%

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

      7. *-commutative 6.1%

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

      8. expm1-def 13.4%

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

      9. *-commutative 13.4%

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

      10. expm1-def 44.6%

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

   3. Simplified 44.6%

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

   4. Taylor expanded in a around 0 62.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4: 48.8% 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 5.0%

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

   1. times-frac 5.0%

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

   2. *-commutative 5.0%

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

   3. associate-*l/ 5.0%

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

   4. associate-*r/ 5.0%

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

   5. expm1-def 6.6%

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

   6. *-commutative 6.6%

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

   7. *-commutative 6.6%

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

   8. expm1-def 15.7%

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

   9. *-commutative 15.7%

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

   10. expm1-def 49.7%

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

3. Simplified 49.7%

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

4. Taylor expanded in a around 0 52.2%

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

5. Final simplification 52.2%

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

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 2023297 
(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))))