Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 97.2%

Time: 7.5s

Alternatives: 11

Speedup: 1.1×

• 22.7% 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: 90.6% 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: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+147}:\\ \;\;\;\;\frac{{k}^{m}}{-1 - k \cdot \left(k + 10\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+147)
     (* (/ (pow k m) (- -1.0 (* k (+ k 10.0)))) (- a))
     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))) <= 1e+147) {
		tmp = (pow(k, m) / (-1.0 - (k * (k + 10.0)))) * -a;
	} 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))) <= 1d+147) then
        tmp = ((k ** m) / ((-1.0d0) - (k * (k + 10.0d0)))) * -a
    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))) <= 1e+147) {
		tmp = (Math.pow(k, m) / (-1.0 - (k * (k + 10.0)))) * -a;
	} 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))) <= 1e+147:
		tmp = (math.pow(k, m) / (-1.0 - (k * (k + 10.0)))) * -a
	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))) <= 1e+147)
		tmp = Float64(Float64((k ^ m) / Float64(-1.0 - Float64(k * Float64(k + 10.0)))) * Float64(-a));
	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))) <= 1e+147)
		tmp = ((k ^ m) / (-1.0 - (k * (k + 10.0)))) * -a;
	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], 1e+147], N[(N[(N[Power[k, m], $MachinePrecision] / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $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))) < 9.9999999999999998e146

   1. Initial program 97.6%

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

      1. sqr-neg 97.6%

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

      2. associate-+l+ 97.6%

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

      3. sqr-neg 97.6%

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

      4. distribute-rgt-out 97.6%

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

   3. Simplified 97.6%

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

      1. frac-2neg 97.6%

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

      2. div-inv 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

      4. +-commutative 97.6%

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

      5. fma-def 97.6%

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

      6. +-commutative 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. associate-*l* 97.6%

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

      2. associate-*r/ 97.6%

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

      3. *-rgt-identity 97.6%

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

      4. fma-udef 97.6%

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

      5. +-commutative 97.6%

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

      6. distribute-neg-in 97.6%

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

      7. metadata-eval 97.6%

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

      8. sub-neg 97.6%

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

   7. Simplified 97.6%

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

   if 9.9999999999999998e146 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 70.1%

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

      1. sqr-neg 70.1%

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

      2. associate-+l+ 70.1%

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

      3. sqr-neg 70.1%

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

      4. distribute-rgt-out 70.1%

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

   3. Simplified 70.1%

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

   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 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+147}:\\ \;\;\;\;\frac{{k}^{m}}{-1 - k \cdot \left(k + 10\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 1.95e-20) (/ 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 (m <= 1.95e-20) {
		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 (m <= 1.95d-20) 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 (m <= 1.95e-20) {
		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 m <= 1.95e-20:
		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 (m <= 1.95e-20)
		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 (m <= 1.95e-20)
		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[m, 1.95e-20], 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 m < 1.95000000000000004e-20

   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 1.95000000000000004e-20 < m

   1. Initial program 81.9%

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

      1. sqr-neg 81.9%

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

      2. associate-+l+ 81.9%

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

      3. sqr-neg 81.9%

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

      4. distribute-rgt-out 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in k around 0 98.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-20}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.2 \lor \neg \left(m \leq 1.95 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -4.2) (not (<= m 1.95e-20)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -4.2) || !(m <= 1.95e-20)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-4.2d0)) .or. (.not. (m <= 1.95d-20))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -4.2) || !(m <= 1.95e-20)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -4.2) or not (m <= 1.95e-20):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -4.2) || !(m <= 1.95e-20))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -4.2) || ~((m <= 1.95e-20)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -4.2], N[Not[LessEqual[m, 1.95e-20]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -4.20000000000000018 or 1.95000000000000004e-20 < m

   1. Initial program 89.9%

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

      1. sqr-neg 89.9%

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

      2. associate-+l+ 89.9%

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

      3. sqr-neg 89.9%

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

      4. distribute-rgt-out 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in k around 0 99.4%

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

   if -4.20000000000000018 < m < 1.95000000000000004e-20

   1. Initial program 94.6%

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

      1. sqr-neg 94.6%

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

      2. associate-+l+ 94.6%

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

      3. sqr-neg 94.6%

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

      4. distribute-rgt-out 94.6%

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

   3. Simplified 94.6%

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

      1. frac-2neg 94.6%

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

      2. div-inv 94.6%

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

      3. distribute-rgt-neg-in 94.6%

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

      4. +-commutative 94.6%

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

      5. fma-def 94.6%

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

      6. +-commutative 94.6%

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

   5. Applied egg-rr 94.6%

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

      1. associate-*l* 94.6%

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

      2. associate-*r/ 94.6%

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

      3. *-rgt-identity 94.6%

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

      4. fma-udef 94.6%

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

      5. +-commutative 94.6%

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

      6. distribute-neg-in 94.6%

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

      7. metadata-eval 94.6%

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

      8. sub-neg 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in m around 0 94.5%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.2 \lor \neg \left(m \leq 1.95 \cdot 10^{-20}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 51.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.3:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.3) (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0))))) (* a (* k 10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.3) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (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 <= 0.3d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.3) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.3:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.3)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.3)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.3], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.299999999999999989

   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%

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

      1. frac-2neg 97.1%

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

      2. div-inv 97.1%

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

      3. distribute-rgt-neg-in 97.1%

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

      4. +-commutative 97.1%

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

      5. fma-def 97.1%

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

      6. +-commutative 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. associate-*l* 97.1%

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

      2. associate-*r/ 97.1%

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

      3. *-rgt-identity 97.1%

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

      4. fma-udef 97.1%

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

      5. +-commutative 97.1%

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

      6. distribute-neg-in 97.1%

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

      7. metadata-eval 97.1%

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

      8. sub-neg 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in m around 0 69.4%

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

   if 0.299999999999999989 < m

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. associate-+l+ 81.7%

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

      3. sqr-neg 81.7%

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

      4. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 6.1%

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

   6. Taylor expanded in k around inf 23.0%

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

      1. associate-*r* 23.0%

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

      2. *-commutative 23.0%

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

      3. associate-*r* 23.0%

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

   8. Simplified 23.0%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)\right)} \]

      2. expm1-udef 42.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)} - 1} \]

      3. add-sqr-sqrt 23.9%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{-10 \cdot k} \cdot \sqrt{-10 \cdot k}\right)}\right)} - 1 \]

      4. sqrt-unprod 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{\left(-10 \cdot k\right) \cdot \left(-10 \cdot k\right)}}\right)} - 1 \]

      5. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot -10\right)} \cdot \left(-10 \cdot k\right)}\right)} - 1 \]

      6. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot -10\right) \cdot \color{blue}{\left(k \cdot -10\right)}}\right)} - 1 \]

      7. swap-sqr 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(-10 \cdot -10\right)}}\right)} - 1 \]

      8. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{100}}\right)} - 1 \]

      9. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{\left(10 \cdot 10\right)}}\right)} - 1 \]

      10. swap-sqr 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot 10\right) \cdot \left(k \cdot 10\right)}}\right)} - 1 \]

      11. sqrt-unprod 21.2%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{k \cdot 10} \cdot \sqrt{k \cdot 10}\right)}\right)} - 1 \]

      12. add-sqr-sqrt 44.0%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(k \cdot 10\right)}\right)} - 1 \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)\right)} \]

       2. expm1-log1p 27.7%

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

       3. *-commutative 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.3:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \]

Alternative 5: 37.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.96:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.5e-5)
   (/ 0.1 (/ k a))
   (if (<= m 0.96) (/ a (+ 1.0 (* k 10.0))) (* a (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-5) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.96) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (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 <= (-3.5d-5)) then
        tmp = 0.1d0 / (k / a)
    else if (m <= 0.96d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a * (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.5e-5) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.96) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a * (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.5e-5:
		tmp = 0.1 / (k / a)
	elif m <= 0.96:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a * (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.5e-5)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (m <= 0.96)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a * Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.5e-5)
		tmp = 0.1 / (k / a);
	elseif (m <= 0.96)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a * (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.5e-5], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.96], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.4999999999999997e-5

   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 41.4%

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

   5. Taylor expanded in k around 0 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in k around inf 30.0%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.0%

         \[\leadsto \color{blue}{\frac{0.1 \cdot a}{k}} \]

      2. associate-/l* 32.0%

         \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]

   10. Simplified 32.0%

       \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]

   if -3.4999999999999997e-5 < m < 0.95999999999999996

   1. Initial program 94.5%

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

      1. sqr-neg 94.5%

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

      2. associate-+l+ 94.5%

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

      3. sqr-neg 94.5%

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

      4. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in m around 0 94.4%

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

   5. Taylor expanded in k around 0 60.6%

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

      1. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   if 0.95999999999999996 < m

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. associate-+l+ 81.7%

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

      3. sqr-neg 81.7%

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

      4. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 6.1%

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

   6. Taylor expanded in k around inf 23.0%

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

      1. associate-*r* 23.0%

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

      2. *-commutative 23.0%

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

      3. associate-*r* 23.0%

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

   8. Simplified 23.0%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)\right)} \]

      2. expm1-udef 42.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)} - 1} \]

      3. add-sqr-sqrt 23.9%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{-10 \cdot k} \cdot \sqrt{-10 \cdot k}\right)}\right)} - 1 \]

      4. sqrt-unprod 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{\left(-10 \cdot k\right) \cdot \left(-10 \cdot k\right)}}\right)} - 1 \]

      5. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot -10\right)} \cdot \left(-10 \cdot k\right)}\right)} - 1 \]

      6. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot -10\right) \cdot \color{blue}{\left(k \cdot -10\right)}}\right)} - 1 \]

      7. swap-sqr 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(-10 \cdot -10\right)}}\right)} - 1 \]

      8. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{100}}\right)} - 1 \]

      9. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{\left(10 \cdot 10\right)}}\right)} - 1 \]

      10. swap-sqr 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot 10\right) \cdot \left(k \cdot 10\right)}}\right)} - 1 \]

      11. sqrt-unprod 21.2%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{k \cdot 10} \cdot \sqrt{k \cdot 10}\right)}\right)} - 1 \]

      12. add-sqr-sqrt 44.0%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(k \cdot 10\right)}\right)} - 1 \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)\right)} \]

       2. expm1-log1p 27.7%

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

       3. *-commutative 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.96:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \]

Alternative 6: 51.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.3) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (* k 10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (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 <= 1.3d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.3) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.3:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.3)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.3)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.3], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.30000000000000004

   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%

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

   4. Taylor expanded in m around 0 69.4%

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

   if 1.30000000000000004 < m

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. associate-+l+ 81.7%

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

      3. sqr-neg 81.7%

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

      4. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 6.1%

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

   6. Taylor expanded in k around inf 23.0%

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

      1. associate-*r* 23.0%

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

      2. *-commutative 23.0%

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

      3. associate-*r* 23.0%

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

   8. Simplified 23.0%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)\right)} \]

      2. expm1-udef 42.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)} - 1} \]

      3. add-sqr-sqrt 23.9%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{-10 \cdot k} \cdot \sqrt{-10 \cdot k}\right)}\right)} - 1 \]

      4. sqrt-unprod 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{\left(-10 \cdot k\right) \cdot \left(-10 \cdot k\right)}}\right)} - 1 \]

      5. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot -10\right)} \cdot \left(-10 \cdot k\right)}\right)} - 1 \]

      6. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot -10\right) \cdot \color{blue}{\left(k \cdot -10\right)}}\right)} - 1 \]

      7. swap-sqr 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(-10 \cdot -10\right)}}\right)} - 1 \]

      8. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{100}}\right)} - 1 \]

      9. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{\left(10 \cdot 10\right)}}\right)} - 1 \]

      10. swap-sqr 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot 10\right) \cdot \left(k \cdot 10\right)}}\right)} - 1 \]

      11. sqrt-unprod 21.2%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{k \cdot 10} \cdot \sqrt{k \cdot 10}\right)}\right)} - 1 \]

      12. add-sqr-sqrt 44.0%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(k \cdot 10\right)}\right)} - 1 \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)\right)} \]

       2. expm1-log1p 27.7%

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

       3. *-commutative 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \]

Alternative 7: 32.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.45 \cdot 10^{-12}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.45e-12) (* 0.1 (/ a k)) (if (<= m 5e+24) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.45e-12) {
		tmp = 0.1 * (a / k);
	} else if (m <= 5e+24) {
		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 <= (-3.45d-12)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 5d+24) 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 <= -3.45e-12) {
		tmp = 0.1 * (a / k);
	} else if (m <= 5e+24) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.45e-12:
		tmp = 0.1 * (a / k)
	elif m <= 5e+24:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.45e-12)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 5e+24)
		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 <= -3.45e-12)
		tmp = 0.1 * (a / k);
	elseif (m <= 5e+24)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.45e-12], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e+24], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.45e-12

   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 41.4%

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

   5. Taylor expanded in k around 0 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in k around inf 30.0%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]

   if -3.45e-12 < m < 5.00000000000000045e24

   1. Initial program 91.8%

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

      1. sqr-neg 91.8%

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

      2. associate-+l+ 91.8%

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

      3. sqr-neg 91.8%

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

      4. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in m around 0 86.6%

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

   5. Taylor expanded in k around 0 45.4%

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

   if 5.00000000000000045e24 < m

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. associate-+l+ 83.5%

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

      3. sqr-neg 83.5%

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

      4. distribute-rgt-out 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 6.6%

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

   6. Taylor expanded in k around inf 25.1%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.45 \cdot 10^{-12}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 33.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-6}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3e-6) (* 0.1 (/ a k)) (if (<= m 0.27) a (* a (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3e-6) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.27) {
		tmp = a;
	} else {
		tmp = a * (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 <= (-3d-6)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 0.27d0) then
        tmp = a
    else
        tmp = a * (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3e-6) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.27) {
		tmp = a;
	} else {
		tmp = a * (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3e-6:
		tmp = 0.1 * (a / k)
	elif m <= 0.27:
		tmp = a
	else:
		tmp = a * (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3e-6)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 0.27)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3e-6)
		tmp = 0.1 * (a / k);
	elseif (m <= 0.27)
		tmp = a;
	else
		tmp = a * (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3e-6], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.27], a, N[(a * N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.0000000000000001e-6

   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 41.4%

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

   5. Taylor expanded in k around 0 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in k around inf 30.0%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]

   if -3.0000000000000001e-6 < m < 0.27000000000000002

   1. Initial program 94.5%

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

      1. sqr-neg 94.5%

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

      2. associate-+l+ 94.5%

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

      3. sqr-neg 94.5%

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

      4. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in m around 0 94.4%

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

   5. Taylor expanded in k around 0 49.3%

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

   if 0.27000000000000002 < m

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. associate-+l+ 81.7%

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

      3. sqr-neg 81.7%

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

      4. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 6.1%

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

   6. Taylor expanded in k around inf 23.0%

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

      1. associate-*r* 23.0%

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

      2. *-commutative 23.0%

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

      3. associate-*r* 23.0%

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

   8. Simplified 23.0%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)\right)} \]

      2. expm1-udef 42.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)} - 1} \]

      3. add-sqr-sqrt 23.9%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{-10 \cdot k} \cdot \sqrt{-10 \cdot k}\right)}\right)} - 1 \]

      4. sqrt-unprod 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{\left(-10 \cdot k\right) \cdot \left(-10 \cdot k\right)}}\right)} - 1 \]

      5. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot -10\right)} \cdot \left(-10 \cdot k\right)}\right)} - 1 \]

      6. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot -10\right) \cdot \color{blue}{\left(k \cdot -10\right)}}\right)} - 1 \]

      7. swap-sqr 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(-10 \cdot -10\right)}}\right)} - 1 \]

      8. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{100}}\right)} - 1 \]

      9. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{\left(10 \cdot 10\right)}}\right)} - 1 \]

      10. swap-sqr 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot 10\right) \cdot \left(k \cdot 10\right)}}\right)} - 1 \]

      11. sqrt-unprod 21.2%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{k \cdot 10} \cdot \sqrt{k \cdot 10}\right)}\right)} - 1 \]

      12. add-sqr-sqrt 44.0%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(k \cdot 10\right)}\right)} - 1 \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)\right)} \]

       2. expm1-log1p 27.7%

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

       3. *-commutative 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{-6}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \]

Alternative 9: 33.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.63:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.1e-7) (/ 0.1 (/ k a)) (if (<= m 0.63) a (* a (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-7) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.63) {
		tmp = a;
	} else {
		tmp = a * (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 <= (-3.1d-7)) then
        tmp = 0.1d0 / (k / a)
    else if (m <= 0.63d0) then
        tmp = a
    else
        tmp = a * (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-7) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.63) {
		tmp = a;
	} else {
		tmp = a * (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.1e-7:
		tmp = 0.1 / (k / a)
	elif m <= 0.63:
		tmp = a
	else:
		tmp = a * (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.1e-7)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (m <= 0.63)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.1e-7)
		tmp = 0.1 / (k / a);
	elseif (m <= 0.63)
		tmp = a;
	else
		tmp = a * (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.1e-7], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.63], a, N[(a * N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.1e-7

   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 41.4%

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

   5. Taylor expanded in k around 0 23.1%

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

      1. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in k around inf 30.0%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.0%

         \[\leadsto \color{blue}{\frac{0.1 \cdot a}{k}} \]

      2. associate-/l* 32.0%

         \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]

   10. Simplified 32.0%

       \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]

   if -3.1e-7 < m < 0.630000000000000004

   1. Initial program 94.5%

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

      1. sqr-neg 94.5%

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

      2. associate-+l+ 94.5%

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

      3. sqr-neg 94.5%

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

      4. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in m around 0 94.4%

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

   5. Taylor expanded in k around 0 49.3%

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

   if 0.630000000000000004 < m

   1. Initial program 81.7%

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

      1. sqr-neg 81.7%

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

      2. associate-+l+ 81.7%

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

      3. sqr-neg 81.7%

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

      4. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in m around 0 3.2%

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

   5. Taylor expanded in k around 0 6.1%

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

   6. Taylor expanded in k around inf 23.0%

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

      1. associate-*r* 23.0%

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

      2. *-commutative 23.0%

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

      3. associate-*r* 23.0%

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

   8. Simplified 23.0%

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

      1. expm1-log1p-u 20.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)\right)} \]

      2. expm1-udef 42.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(-10 \cdot k\right)\right)} - 1} \]

      3. add-sqr-sqrt 23.9%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{-10 \cdot k} \cdot \sqrt{-10 \cdot k}\right)}\right)} - 1 \]

      4. sqrt-unprod 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\sqrt{\left(-10 \cdot k\right) \cdot \left(-10 \cdot k\right)}}\right)} - 1 \]

      5. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot -10\right)} \cdot \left(-10 \cdot k\right)}\right)} - 1 \]

      6. *-commutative 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot -10\right) \cdot \color{blue}{\left(k \cdot -10\right)}}\right)} - 1 \]

      7. swap-sqr 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(-10 \cdot -10\right)}}\right)} - 1 \]

      8. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{100}}\right)} - 1 \]

      9. metadata-eval 53.3%

         \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\left(k \cdot k\right) \cdot \color{blue}{\left(10 \cdot 10\right)}}\right)} - 1 \]

      10. swap-sqr 53.3%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \sqrt{\color{blue}{\left(k \cdot 10\right) \cdot \left(k \cdot 10\right)}}\right)} - 1 \]

      11. sqrt-unprod 21.2%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(\sqrt{k \cdot 10} \cdot \sqrt{k \cdot 10}\right)}\right)} - 1 \]

      12. add-sqr-sqrt 44.0%

          \[\leadsto e^{\mathsf{log1p}\left(a \cdot \color{blue}{\left(k \cdot 10\right)}\right)} - 1 \]

   10. Applied egg-rr 44.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 21.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \left(k \cdot 10\right)\right)\right)} \]

       2. expm1-log1p 27.7%

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

       3. *-commutative 27.7%

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

   12. Simplified 27.7%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.63:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot 10\right)\\ \end{array} \]

Alternative 10: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 5e+24) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5e+24) {
		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 <= 5d+24) 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 <= 5e+24) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5e+24:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5e+24)
		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 <= 5e+24)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5e+24], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.00000000000000045e24

   1. Initial program 95.5%

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

      1. sqr-neg 95.5%

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

      2. associate-+l+ 95.5%

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

      3. sqr-neg 95.5%

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

      4. distribute-rgt-out 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in m around 0 66.2%

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

   5. Taylor expanded in k around 0 26.7%

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

   if 5.00000000000000045e24 < m

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. associate-+l+ 83.5%

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

      3. sqr-neg 83.5%

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

      4. distribute-rgt-out 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 6.6%

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

   6. Taylor expanded in k around inf 25.1%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 20.3% 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 91.5%

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

   1. sqr-neg 91.5%

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

   2. associate-+l+ 91.5%

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

   3. sqr-neg 91.5%

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

   4. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

4. Taylor expanded in m around 0 45.3%

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

5. Taylor expanded in k around 0 19.1%

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

6. Final simplification 19.1%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023302 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))