expq3 (problem 3.4.2)

Percentage Accurate: 6.6% → 95.0%

Time: 11.7s

Alternatives: 5

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: 95.0% accurate, 21.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double eps) {
	return (-0.5 * ((eps * a) / a)) + ((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 = ((-0.5d0) * ((eps * a) / a)) + ((1.0d0 / a) + (1.0d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.8%

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

   1. *-commutative 3.8%

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

   2. times-frac 3.8%

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

   3. +-commutative 3.8%

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

   4. expm1-def 12.4%

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

   5. *-commutative 12.4%

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

   6. expm1-def 13.5%

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

   7. +-commutative 13.5%

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

   8. *-commutative 13.5%

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

   9. expm1-def 52.6%

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

   10. *-commutative 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in eps around 0 57.8%

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

5. Taylor expanded in eps around 0 87.6%

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

6. Taylor expanded in a around inf 97.3%

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

7. Final simplification 97.3%

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

Alternative 2: 57.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-161} \lor \neg \left(b \leq 1.55 \cdot 10^{-56}\right) \land \left(b \leq 4.6 \cdot 10^{-39} \lor \neg \left(b \leq 6.5 \cdot 10^{-14}\right) \land b \leq 14\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (or (<= b 5.2e-161)
         (and (not (<= b 1.55e-56))
              (or (<= b 4.6e-39) (and (not (<= b 6.5e-14)) (<= b 14.0)))))
   (/ 1.0 b)
   (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if ((b <= 5.2e-161) || (!(b <= 1.55e-56) && ((b <= 4.6e-39) || (!(b <= 6.5e-14) && (b <= 14.0))))) {
		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 <= 5.2d-161) .or. (.not. (b <= 1.55d-56)) .and. (b <= 4.6d-39) .or. (.not. (b <= 6.5d-14)) .and. (b <= 14.0d0)) 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 <= 5.2e-161) || (!(b <= 1.55e-56) && ((b <= 4.6e-39) || (!(b <= 6.5e-14) && (b <= 14.0))))) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if (b <= 5.2e-161) or (not (b <= 1.55e-56) and ((b <= 4.6e-39) or (not (b <= 6.5e-14) and (b <= 14.0)))):
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if ((b <= 5.2e-161) || (!(b <= 1.55e-56) && ((b <= 4.6e-39) || (!(b <= 6.5e-14) && (b <= 14.0)))))
		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 <= 5.2e-161) || (~((b <= 1.55e-56)) && ((b <= 4.6e-39) || (~((b <= 6.5e-14)) && (b <= 14.0)))))
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[Or[LessEqual[b, 5.2e-161], And[N[Not[LessEqual[b, 1.55e-56]], $MachinePrecision], Or[LessEqual[b, 4.6e-39], And[N[Not[LessEqual[b, 6.5e-14]], $MachinePrecision], LessEqual[b, 14.0]]]]], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.19999999999999991e-161 or 1.54999999999999994e-56 < b < 4.60000000000000016e-39 or 6.5000000000000001e-14 < b < 14

   1. Initial program 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-*l/ 2.6%

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

      3. *-commutative 2.6%

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

      4. expm1-def 4.3%

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

      5. *-commutative 4.3%

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

      6. associate-/r* 4.3%

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

      7. expm1-def 12.2%

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

      8. *-commutative 12.2%

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

      9. expm1-def 37.3%

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

      10. *-commutative 37.3%

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

   3. Simplified 37.3%

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

   4. Taylor expanded in b around 0 59.1%

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

   if 5.19999999999999991e-161 < b < 1.54999999999999994e-56 or 4.60000000000000016e-39 < b < 6.5000000000000001e-14 or 14 < b

   1. Initial program 5.7%

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

      1. *-commutative 5.7%

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

      2. associate-*l/ 5.7%

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

      3. *-commutative 5.7%

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

      4. expm1-def 7.6%

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

      5. *-commutative 7.6%

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

      6. associate-/r* 7.6%

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

      7. expm1-def 15.7%

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

      8. *-commutative 15.7%

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

      9. expm1-def 62.0%

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

      10. *-commutative 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in a around 0 63.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-161} \lor \neg \left(b \leq 1.55 \cdot 10^{-56}\right) \land \left(b \leq 4.6 \cdot 10^{-39} \lor \neg \left(b \leq 6.5 \cdot 10^{-14}\right) \land b \leq 14\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 3: 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 3.8%

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

   1. *-commutative 3.8%

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

   2. associate-*l/ 3.8%

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

   3. *-commutative 3.8%

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

   4. expm1-def 5.6%

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

   5. *-commutative 5.6%

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

   6. associate-/r* 5.6%

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

   7. expm1-def 13.5%

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

   8. *-commutative 13.5%

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

   9. expm1-def 46.7%

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

   10. *-commutative 46.7%

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

3. Simplified 46.7%

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

4. Taylor expanded in eps around 0 78.2%

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

5. Taylor expanded in a around 0 97.0%

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

6. Final simplification 97.0%

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

Alternative 4: 3.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (-0.5d0) * eps
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.8%

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

   1. *-commutative 3.8%

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

   2. times-frac 3.8%

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

   3. +-commutative 3.8%

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

   4. expm1-def 12.4%

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

   5. *-commutative 12.4%

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

   6. expm1-def 13.5%

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

   7. +-commutative 13.5%

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

   8. *-commutative 13.5%

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

   9. expm1-def 52.6%

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

   10. *-commutative 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in eps around 0 57.8%

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

5. Taylor expanded in eps around 0 83.1%

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

6. Taylor expanded in eps around inf 3.7%

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

7. Taylor expanded in a around inf 2.9%

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

8. Final simplification 2.9%

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

Alternative 5: 48.4% 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 3.8%

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

   1. *-commutative 3.8%

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

   2. associate-*l/ 3.8%

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

   3. *-commutative 3.8%

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

   4. expm1-def 5.6%

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

   5. *-commutative 5.6%

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

   6. associate-/r* 5.6%

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

   7. expm1-def 13.5%

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

   8. *-commutative 13.5%

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

   9. expm1-def 46.7%

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

   10. *-commutative 46.7%

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

3. Simplified 46.7%

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

4. Taylor expanded in a around 0 49.3%

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

5. Final simplification 49.3%

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

Developer target: 78.0% accurate, 45.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023263 
(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))))