Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.5%

Time: 8.5s

Alternatives: 8

Speedup: 1.1×

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.9999999999999999e-154
  2. k = 0.0
  3. m = -2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 5e+178)
     (/ t_0 (+ 1.0 (* k (+ k 10.0))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 5e+178) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 5d+178) then
        tmp = t_0 / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 5e+178) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 5e+178:
		tmp = t_0 / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 5e+178)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 5e+178)
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+178], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 4.9999999999999999e178

   1. Initial program 97.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 97.1%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 97.1%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 97.1%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 97.1%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   if 4.9999999999999999e178 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 55.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 55.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 55.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 55.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 55.6%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in k around 0 100.0%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+178}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-18} \lor \neg \left(m \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -8.6e-18) (not (<= m 1.85e-6)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.6e-18) || !(m <= 1.85e-6)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-8.6d-18)) .or. (.not. (m <= 1.85d-6))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.6e-18) || !(m <= 1.85e-6)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -8.6e-18) or not (m <= 1.85e-6):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -8.6e-18) || !(m <= 1.85e-6))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -8.6e-18) || ~((m <= 1.85e-6)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -8.6e-18], N[Not[LessEqual[m, 1.85e-6]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.6000000000000005e-18 or 1.8500000000000001e-6 < m

   1. Initial program 87.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in k around 0 99.4%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   if -8.6000000000000005e-18 < m < 1.8500000000000001e-6

   1. Initial program 93.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 93.3%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 93.3%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 93.3%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 93.3%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 93.0%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-18} \lor \neg \left(m \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3: 50.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 880000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 880000.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 880000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 880000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 880000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 880000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 880000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 880000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 880000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.8e5

   1. Initial program 95.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 67.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   if 8.8e5 < m

   1. Initial program 77.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 7.8%

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 7.8%

         \[\leadsto a + -10 \cdot \color{blue}{\left(k \cdot a\right)} \]

   7. Simplified 7.8%

      \[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 7.8%

         \[\leadsto a + \color{blue}{\left(-10 \cdot k\right) \cdot a} \]

      2. distribute-rgt1-in 7.8%

         \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   9. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   10. Taylor expanded in k around inf 18.6%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 880000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 4: 29.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 3e-309) (* -10.0 (* a k)) (if (<= k 0.21) a (* 0.1 (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 3e-309) {
		tmp = -10.0 * (a * k);
	} else if (k <= 0.21) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 3d-309) then
        tmp = (-10.0d0) * (a * k)
    else if (k <= 0.21d0) then
        tmp = a
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 3e-309) {
		tmp = -10.0 * (a * k);
	} else if (k <= 0.21) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 3e-309:
		tmp = -10.0 * (a * k)
	elif k <= 0.21:
		tmp = a
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 3e-309)
		tmp = Float64(-10.0 * Float64(a * k));
	elseif (k <= 0.21)
		tmp = a;
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 3e-309)
		tmp = -10.0 * (a * k);
	elseif (k <= 0.21)
		tmp = a;
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 3e-309], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.21], a, N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.000000000000001e-309

   1. Initial program 85.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 85.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 85.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 85.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 85.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 17.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 8.5%

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto a + -10 \cdot \color{blue}{\left(k \cdot a\right)} \]

   7. Simplified 8.5%

      \[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 8.5%

         \[\leadsto a + \color{blue}{\left(-10 \cdot k\right) \cdot a} \]

      2. distribute-rgt1-in 8.5%

         \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   9. Applied egg-rr 8.5%

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   10. Taylor expanded in k around inf 12.8%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   if 3.000000000000001e-309 < k < 0.209999999999999992

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 58.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 57.8%

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

   if 0.209999999999999992 < k

   1. Initial program 83.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 83.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 83.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 83.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 83.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   4. Step-by-step derivation

      1. frac-2neg 83.8%

         \[\leadsto \color{blue}{\frac{-a \cdot {k}^{m}}{-\left(1 + k \cdot \left(10 + k\right)\right)}} \]

      2. div-inv 83.8%

         \[\leadsto \color{blue}{\left(-a \cdot {k}^{m}\right) \cdot \frac{1}{-\left(1 + k \cdot \left(10 + k\right)\right)}} \]

      3. distribute-rgt-neg-in 83.8%

         \[\leadsto \color{blue}{\left(a \cdot \left(-{k}^{m}\right)\right)} \cdot \frac{1}{-\left(1 + k \cdot \left(10 + k\right)\right)} \]

      4. +-commutative 83.8%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\color{blue}{\left(k \cdot \left(10 + k\right) + 1\right)}} \]

      5. fma-def 83.8%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      6. +-commutative 83.8%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   5. Applied egg-rr 83.8%

      \[\leadsto \color{blue}{\left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(k, k + 10, 1\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 83.8%

         \[\leadsto \color{blue}{a \cdot \left(\left(-{k}^{m}\right) \cdot \frac{1}{-\mathsf{fma}\left(k, k + 10, 1\right)}\right)} \]

      2. associate-*r/ 83.8%

         \[\leadsto a \cdot \color{blue}{\frac{\left(-{k}^{m}\right) \cdot 1}{-\mathsf{fma}\left(k, k + 10, 1\right)}} \]

      3. *-rgt-identity 83.8%

         \[\leadsto a \cdot \frac{\color{blue}{-{k}^{m}}}{-\mathsf{fma}\left(k, k + 10, 1\right)} \]

      4. fma-udef 83.8%

         \[\leadsto a \cdot \frac{-{k}^{m}}{-\color{blue}{\left(k \cdot \left(k + 10\right) + 1\right)}} \]

      5. +-commutative 83.8%

         \[\leadsto a \cdot \frac{-{k}^{m}}{-\color{blue}{\left(1 + k \cdot \left(k + 10\right)\right)}} \]

      6. distribute-neg-in 83.8%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{\left(-1\right) + \left(-k \cdot \left(k + 10\right)\right)}} \]

      7. metadata-eval 83.8%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{-1} + \left(-k \cdot \left(k + 10\right)\right)} \]

      8. sub-neg 83.8%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{-1 - k \cdot \left(k + 10\right)}} \]

   7. Simplified 83.8%

      \[\leadsto \color{blue}{a \cdot \frac{-{k}^{m}}{-1 - k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in m around 0 62.7%

      \[\leadsto a \cdot \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \]

   9. Taylor expanded in k around 0 23.9%

      \[\leadsto a \cdot \frac{1}{1 + \color{blue}{10 \cdot k}} \]
   10. Step-by-step derivation

       1. *-commutative 23.9%

          \[\leadsto a \cdot \frac{1}{1 + \color{blue}{k \cdot 10}} \]

   11. Simplified 23.9%

       \[\leadsto a \cdot \frac{1}{1 + \color{blue}{k \cdot 10}} \]

   12. Taylor expanded in k around inf 23.9%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-309}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 5: 28.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (* a (+ 1.0 (* k -10.0))) (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.075d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   1. Initial program 92.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 92.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 92.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 92.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 92.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 38.4%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 34.0%

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 34.0%

         \[\leadsto a + -10 \cdot \color{blue}{\left(k \cdot a\right)} \]

   7. Simplified 34.0%

      \[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 34.0%

         \[\leadsto a + \color{blue}{\left(-10 \cdot k\right) \cdot a} \]

      2. distribute-rgt1-in 34.0%

         \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   9. Applied egg-rr 34.0%

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   if 0.0749999999999999972 < k

   1. Initial program 84.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 84.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 84.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 84.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 84.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   4. Step-by-step derivation

      1. frac-2neg 84.0%

         \[\leadsto \color{blue}{\frac{-a \cdot {k}^{m}}{-\left(1 + k \cdot \left(10 + k\right)\right)}} \]

      2. div-inv 84.0%

         \[\leadsto \color{blue}{\left(-a \cdot {k}^{m}\right) \cdot \frac{1}{-\left(1 + k \cdot \left(10 + k\right)\right)}} \]

      3. distribute-rgt-neg-in 84.0%

         \[\leadsto \color{blue}{\left(a \cdot \left(-{k}^{m}\right)\right)} \cdot \frac{1}{-\left(1 + k \cdot \left(10 + k\right)\right)} \]

      4. +-commutative 84.0%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\color{blue}{\left(k \cdot \left(10 + k\right) + 1\right)}} \]

      5. fma-def 84.0%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

      6. +-commutative 84.0%

         \[\leadsto \left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   5. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\left(a \cdot \left(-{k}^{m}\right)\right) \cdot \frac{1}{-\mathsf{fma}\left(k, k + 10, 1\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* 84.0%

         \[\leadsto \color{blue}{a \cdot \left(\left(-{k}^{m}\right) \cdot \frac{1}{-\mathsf{fma}\left(k, k + 10, 1\right)}\right)} \]

      2. associate-*r/ 84.0%

         \[\leadsto a \cdot \color{blue}{\frac{\left(-{k}^{m}\right) \cdot 1}{-\mathsf{fma}\left(k, k + 10, 1\right)}} \]

      3. *-rgt-identity 84.0%

         \[\leadsto a \cdot \frac{\color{blue}{-{k}^{m}}}{-\mathsf{fma}\left(k, k + 10, 1\right)} \]

      4. fma-udef 84.0%

         \[\leadsto a \cdot \frac{-{k}^{m}}{-\color{blue}{\left(k \cdot \left(k + 10\right) + 1\right)}} \]

      5. +-commutative 84.0%

         \[\leadsto a \cdot \frac{-{k}^{m}}{-\color{blue}{\left(1 + k \cdot \left(k + 10\right)\right)}} \]

      6. distribute-neg-in 84.0%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{\left(-1\right) + \left(-k \cdot \left(k + 10\right)\right)}} \]

      7. metadata-eval 84.0%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{-1} + \left(-k \cdot \left(k + 10\right)\right)} \]

      8. sub-neg 84.0%

         \[\leadsto a \cdot \frac{-{k}^{m}}{\color{blue}{-1 - k \cdot \left(k + 10\right)}} \]

   7. Simplified 84.0%

      \[\leadsto \color{blue}{a \cdot \frac{-{k}^{m}}{-1 - k \cdot \left(k + 10\right)}} \]

   8. Taylor expanded in m around 0 62.1%

      \[\leadsto a \cdot \color{blue}{\frac{1}{1 + k \cdot \left(10 + k\right)}} \]

   9. Taylor expanded in k around 0 23.7%

      \[\leadsto a \cdot \frac{1}{1 + \color{blue}{10 \cdot k}} \]
   10. Step-by-step derivation

       1. *-commutative 23.7%

          \[\leadsto a \cdot \frac{1}{1 + \color{blue}{k \cdot 10}} \]

   11. Simplified 23.7%

       \[\leadsto a \cdot \frac{1}{1 + \color{blue}{k \cdot 10}} \]

   12. Taylor expanded in k around inf 23.7%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 6: 33.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 750000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 750000.0) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 750000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 750000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 750000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 750000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 750000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 750000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 750000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.5e5

   1. Initial program 95.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 95.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 67.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 43.8%

      \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 43.8%

         \[\leadsto a \cdot \frac{1}{1 + \color{blue}{k \cdot 10}} \]

   7. Simplified 43.8%

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   if 7.5e5 < m

   1. Initial program 77.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 77.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 7.8%

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 7.8%

         \[\leadsto a + -10 \cdot \color{blue}{\left(k \cdot a\right)} \]

   7. Simplified 7.8%

      \[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 7.8%

         \[\leadsto a + \color{blue}{\left(-10 \cdot k\right) \cdot a} \]

      2. distribute-rgt1-in 7.8%

         \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   9. Applied egg-rr 7.8%

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   10. Taylor expanded in k around inf 18.6%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 750000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 7: 24.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{+40}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 6.8e+40) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.8e+40) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 6.8d+40) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 6.8e+40) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 6.8e+40:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 6.8e+40)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 6.8e+40)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 6.8e+40], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.79999999999999977e40

   1. Initial program 95.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 95.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 95.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 95.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 95.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 64.3%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 29.7%

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

   if 6.79999999999999977e40 < m

   1. Initial program 77.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. sqr-neg 77.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

      2. associate-+l+ 77.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

      3. sqr-neg 77.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      4. distribute-rgt-out 77.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

   4. Taylor expanded in m around 0 2.9%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

   5. Taylor expanded in k around 0 8.6%

      \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto a + -10 \cdot \color{blue}{\left(k \cdot a\right)} \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{a + -10 \cdot \left(k \cdot a\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 8.6%

         \[\leadsto a + \color{blue}{\left(-10 \cdot k\right) \cdot a} \]

      2. distribute-rgt1-in 8.6%

         \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   9. Applied egg-rr 8.6%

      \[\leadsto \color{blue}{\left(-10 \cdot k + 1\right) \cdot a} \]

   10. Taylor expanded in k around inf 20.7%

       \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{+40}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 19.7% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.8%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Step-by-step derivation

   1. sqr-neg 89.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + \color{blue}{\left(-k\right) \cdot \left(-k\right)}} \]

   2. associate-+l+ 89.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{1 + \left(10 \cdot k + \left(-k\right) \cdot \left(-k\right)\right)}} \]

   3. sqr-neg 89.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   4. distribute-rgt-out 89.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

3. Simplified 89.8%

   \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]

4. Taylor expanded in m around 0 46.5%

   \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(10 + k\right)}} \]

5. Taylor expanded in k around 0 22.1%

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

6. Final simplification 22.1%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2024024 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))