expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 99.9%

Time: 25.4s

Alternatives: 3

Speedup: 107.0×

Specification

\[\left(\left|a\right| \leq 710 \land \left|b\right| \leq 710\right) \land \left(10^{-27} \cdot \mathsf{min}\left(\left|a\right|, \left|b\right|\right) \leq \varepsilon \land \varepsilon \leq \mathsf{min}\left(\left|a\right|, \left|b\right|\right)\right)\]

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: 0.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

Java

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

Python

def code(a, b, eps):
	return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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 0.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right), \color{blue}{\left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\right) \]

   4. accelerator-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right), \left(\frac{\color{blue}{\varepsilon}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\varepsilon \cdot \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{\color{blue}{e^{b \cdot \varepsilon} - 1}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}\right), \color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \left(e^{a \cdot \varepsilon} - 1\right)\right), \left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right)\right)\right) \]

   11. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(a \cdot \varepsilon\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(\varepsilon \cdot a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   14. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(b \cdot \varepsilon\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(\varepsilon \cdot b\right)\right)\right)\right) \]

   16. *-lowering-*.f64 19.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, b\right)\right)\right)\right) \]

3. Simplified 19.9%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
6. Step-by-step derivation

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{a + b}{a}\right), \color{blue}{b}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a + b\right), a\right), b\right) \]

   4. +-lowering-+.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(a, b\right), a\right), b\right) \]

7. Simplified 99.8%

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

8. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{a}\right), \color{blue}{\left(\frac{1}{b}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, a\right), \left(\frac{\color{blue}{1}}{b}\right)\right) \]

   3. /-lowering-/.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(1, \color{blue}{b}\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}} \]
11. Add Preprocessing

Alternative 2: 60.1% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{-191}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= a -1.65e-191) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -1.65e-191) {
		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 <= (-1.65d-191)) 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 <= -1.65e-191) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if a <= -1.65e-191:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (a <= -1.65e-191)
		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 <= -1.65e-191)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[a, -1.65e-191], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.64999999999999991e-191

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right), \color{blue}{\left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\right) \]

      4. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right), \left(\frac{\color{blue}{\varepsilon}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\varepsilon \cdot \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{\color{blue}{e^{b \cdot \varepsilon} - 1}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}\right), \color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \left(e^{a \cdot \varepsilon} - 1\right)\right), \left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right)\right)\right) \]

      11. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(a \cdot \varepsilon\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(\varepsilon \cdot a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(b \cdot \varepsilon\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(\varepsilon \cdot b\right)\right)\right)\right) \]

      16. *-lowering-*.f64 31.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, b\right)\right)\right)\right) \]

   3. Simplified 31.1%

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

   5. Taylor expanded in b around 0

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

      1. /-lowering-/.f64 68.6%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{b}\right) \]

   7. Simplified 68.6%

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

   if -1.64999999999999991e-191 < a

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right), \color{blue}{\left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\right) \]

      4. accelerator-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right), \left(\frac{\color{blue}{\varepsilon}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\varepsilon \cdot \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{\color{blue}{e^{b \cdot \varepsilon} - 1}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}\right), \color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \left(e^{a \cdot \varepsilon} - 1\right)\right), \left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right)\right)\right) \]

      11. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(a \cdot \varepsilon\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(\varepsilon \cdot a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(b \cdot \varepsilon\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(\varepsilon \cdot b\right)\right)\right)\right) \]

      16. *-lowering-*.f64 13.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, b\right)\right)\right)\right) \]

   3. Simplified 13.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{a}\right) \]

   7. Simplified 59.3%

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

Alternative 3: 49.7% 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 0.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right), \color{blue}{\left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)}\right) \]

   4. accelerator-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right), \left(\frac{\color{blue}{\varepsilon}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\left(\varepsilon \cdot \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(a + b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \left(\frac{\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}}{\color{blue}{e^{b \cdot \varepsilon} - 1}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{\varepsilon}{e^{a \cdot \varepsilon} - 1}\right), \color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \left(e^{a \cdot \varepsilon} - 1\right)\right), \left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right)\right)\right) \]

   11. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(a \cdot \varepsilon\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\left(\varepsilon \cdot a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \]

   14. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(b \cdot \varepsilon\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\left(\varepsilon \cdot b\right)\right)\right)\right) \]

   16. *-lowering-*.f64 19.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{+.f64}\left(a, b\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, a\right)\right)\right), \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(\varepsilon, b\right)\right)\right)\right) \]

3. Simplified 19.9%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{a}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 49.0%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{a}\right) \]

7. Simplified 49.0%

   \[\leadsto \color{blue}{\frac{1}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.9% 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]

Reproduce

herbie shell --seed 2024191 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (and (<= (fabs a) 710.0) (<= (fabs b) 710.0)) (and (<= (* 1e-27 (fmin (fabs a) (fabs b))) eps) (<= eps (fmin (fabs a) (fabs b)))))

  :alt
  (! :herbie-platform default (+ (/ 1 a) (/ 1 b)))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))