Falkner and Boettcher, Appendix A

Percentage Accurate: 89.8% → 97.6%

Time: 11.1s

Alternatives: 11

Speedup: 1.1×

• 23.2% 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: 89.8% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 5e+269) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+269) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 5d+269) then
        tmp = t_1
    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 t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+269) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 5e+269:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e+269)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 5e+269)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+269], t$95$1, 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))) < 5.0000000000000002e269

   1. Initial program 97.9%

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

   if 5.0000000000000002e269 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 65.1%

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

      1. associate-*r/ 65.1%

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

      2. *-commutative 65.1%

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

      3. sqr-neg 65.1%

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

      4. associate-+l+ 65.1%

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

      5. +-commutative 65.1%

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

      6. sqr-neg 65.1%

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

      7. distribute-rgt-out 65.1%

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

      8. fma-def 65.1%

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

      9. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+269}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.45e-5)
   (/ (pow k m) (/ (+ 1.0 (* k (+ k 10.0))) a))
   (/ a (pow k (- 2.0 m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.45e-5) {
		tmp = pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a / pow(k, (2.0 - m));
	}
	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 <= 1.45d-5) then
        tmp = (k ** m) / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = a / (k ** (2.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.45e-5) {
		tmp = Math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a / Math.pow(k, (2.0 - m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.45e-5:
		tmp = math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = a / math.pow(k, (2.0 - m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.45e-5)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a / (k ^ Float64(2.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.45e-5)
		tmp = (k ^ m) / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.45e-5], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, N[(2.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.45e-5

   1. Initial program 97.4%

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

      1. associate-*r/ 97.3%

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

      2. *-commutative 97.3%

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

      3. sqr-neg 97.3%

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

      4. associate-+l+ 97.3%

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

      5. +-commutative 97.3%

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

      6. sqr-neg 97.3%

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

      7. distribute-rgt-out 97.3%

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

      8. fma-def 97.3%

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

      9. +-commutative 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in m around inf 97.4%

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

      1. *-commutative 97.4%

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

      2. +-commutative 97.4%

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

      3. +-commutative 97.4%

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

      4. fma-udef 97.4%

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

      5. associate-/l* 97.1%

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

   6. Simplified 97.1%

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

   7. Taylor expanded in a around 0 97.1%

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

   if 1.45e-5 < m

   1. Initial program 82.4%

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

   2. Taylor expanded in k around inf 56.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 56.5%

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

   4. Simplified 56.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 42.4%

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

      2. expm1-udef 42.4%

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

      3. associate-/l* 42.4%

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

      4. pow2 42.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      5. pow-div 76.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 76.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\frac{a}{{k}^{\left(2 - m\right)}}} \]

   8. Simplified 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 3: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.25 \cdot 10^{-5} \lor \neg \left(m \leq 0.00046\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.25e-5) (not (<= m 0.00046)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.25e-5) || !(m <= 0.00046)) {
		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 <= (-1.25d-5)) .or. (.not. (m <= 0.00046d0))) 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 <= -1.25e-5) || !(m <= 0.00046)) {
		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 <= -1.25e-5) or not (m <= 0.00046):
		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 <= -1.25e-5) || !(m <= 0.00046))
		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 <= -1.25e-5) || ~((m <= 0.00046)))
		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, -1.25e-5], N[Not[LessEqual[m, 0.00046]], $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 < -1.25000000000000006e-5 or 4.6000000000000001e-4 < m

   1. Initial program 91.2%

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

      1. associate-*r/ 91.2%

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

      2. *-commutative 91.2%

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

      3. sqr-neg 91.2%

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

      4. associate-+l+ 91.2%

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

      5. +-commutative 91.2%

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

      6. sqr-neg 91.2%

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

      7. distribute-rgt-out 91.2%

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

      8. fma-def 91.2%

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

      9. +-commutative 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -1.25000000000000006e-5 < m < 4.6000000000000001e-4

   1. Initial program 94.7%

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

      1. associate-*r/ 94.6%

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

      2. *-commutative 94.6%

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

      3. sqr-neg 94.6%

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

      4. associate-+l+ 94.6%

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

      5. +-commutative 94.6%

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

      6. sqr-neg 94.6%

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

      7. distribute-rgt-out 94.6%

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

      8. fma-def 94.6%

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

      9. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in m around 0 93.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.25 \cdot 10^{-5} \lor \neg \left(m \leq 0.00046\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.0132:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.0132) (* a (pow k m)) (/ a (pow k (- 2.0 m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.0132) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / pow(k, (2.0 - m));
	}
	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 <= 0.0132d0) then
        tmp = a * (k ** m)
    else
        tmp = a / (k ** (2.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.0132) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / Math.pow(k, (2.0 - m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.0132:
		tmp = a * math.pow(k, m)
	else:
		tmp = a / math.pow(k, (2.0 - m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.0132)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / (k ^ Float64(2.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.0132)
		tmp = a * (k ^ m);
	else
		tmp = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.0132], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, N[(2.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0132

   1. Initial program 96.2%

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

      1. associate-*r/ 96.2%

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

      2. *-commutative 96.2%

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

      3. sqr-neg 96.2%

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

      4. associate-+l+ 96.2%

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

      5. +-commutative 96.2%

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

      6. sqr-neg 96.2%

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

      7. distribute-rgt-out 96.2%

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

      8. fma-def 96.2%

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

      9. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in k around 0 99.4%

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

   if 0.0132 < k

   1. Initial program 86.1%

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

   2. Taylor expanded in k around inf 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 85.5%

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

   4. Simplified 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 75.8%

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

      2. expm1-udef 65.5%

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

      3. associate-/l* 65.5%

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

      4. pow2 65.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      5. pow-div 71.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 82.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 94.8%

         \[\leadsto \color{blue}{\frac{a}{{k}^{\left(2 - m\right)}}} \]

   8. Simplified 94.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0132:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 5: 45.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-299} \lor \neg \left(k \leq 0.0132\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 9.8e-299) (not (<= k 0.0132)))
   (/ a (* k k))
   (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 9.8e-299) || !(k <= 0.0132)) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (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 ((k <= 9.8d-299) .or. (.not. (k <= 0.0132d0))) then
        tmp = a / (k * k)
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 9.8e-299) || !(k <= 0.0132)) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 9.8e-299) or not (k <= 0.0132):
		tmp = a / (k * k)
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 9.8e-299) || !(k <= 0.0132))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 9.8e-299) || ~((k <= 0.0132)))
		tmp = a / (k * k);
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 9.8e-299], N[Not[LessEqual[k, 0.0132]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.7999999999999999e-299 or 0.0132 < k

   1. Initial program 89.0%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. sqr-neg 88.9%

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

      4. associate-+l+ 88.9%

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

      5. +-commutative 88.9%

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

      6. sqr-neg 88.9%

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

      7. distribute-rgt-out 88.9%

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

      8. fma-def 88.9%

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

      9. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in m around 0 49.7%

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

   5. Taylor expanded in k around inf 52.6%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 52.6%

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

   7. Simplified 52.6%

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

   if 9.7999999999999999e-299 < k < 0.0132

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. distribute-lft1-in 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-def 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in m around 0 40.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.8 \cdot 10^{-299} \lor \neg \left(k \leq 0.0132\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 6: 47.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.0132:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.2e-298)
   (/ a (* k k))
   (if (<= k 0.0132) (* a (+ 1.0 (* k -10.0))) (* (/ 1.0 k) (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.2e-298) {
		tmp = a / (k * k);
	} else if (k <= 0.0132) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (1.0 / k) * (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 <= 4.2d-298) then
        tmp = a / (k * k)
    else if (k <= 0.0132d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (1.0d0 / k) * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.2e-298) {
		tmp = a / (k * k);
	} else if (k <= 0.0132) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.2e-298:
		tmp = a / (k * k)
	elif k <= 0.0132:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (1.0 / k) * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.2e-298)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.0132)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.2e-298)
		tmp = a / (k * k);
	elseif (k <= 0.0132)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (1.0 / k) * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.2e-298], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0132], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.2000000000000001e-298

   1. Initial program 92.4%

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

      1. associate-*r/ 92.4%

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

      2. *-commutative 92.4%

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

      3. sqr-neg 92.4%

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

      4. associate-+l+ 92.4%

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

      5. +-commutative 92.4%

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

      6. sqr-neg 92.4%

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

      7. distribute-rgt-out 92.4%

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

      8. fma-def 92.4%

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

      9. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in m around 0 32.9%

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

   5. Taylor expanded in k around inf 39.8%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 39.8%

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

   7. Simplified 39.8%

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

   if 4.2000000000000001e-298 < k < 0.0132

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 69.3%

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

      1. associate-*r* 69.3%

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

      2. distribute-lft1-in 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-def 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in m around 0 40.0%

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

   if 0.0132 < k

   1. Initial program 86.1%

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

   2. Taylor expanded in k around inf 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 85.5%

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

   4. Simplified 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   5. Step-by-step derivation

      1. *-commutative 85.5%

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

      2. times-frac 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in m around 0 66.6%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.0132:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 7: 47.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-297}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.0132:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.15e-297)
   (/ a (* k k))
   (if (<= k 0.0132) (/ a (+ 1.0 (* k 10.0))) (* (/ 1.0 k) (/ a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.15e-297) {
		tmp = a / (k * k);
	} else if (k <= 0.0132) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (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 <= 1.15d-297) then
        tmp = a / (k * k)
    else if (k <= 0.0132d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (1.0d0 / k) * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.15e-297) {
		tmp = a / (k * k);
	} else if (k <= 0.0132) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.15e-297:
		tmp = a / (k * k)
	elif k <= 0.0132:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (1.0 / k) * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.15e-297)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.0132)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.15e-297)
		tmp = a / (k * k);
	elseif (k <= 0.0132)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (1.0 / k) * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.15e-297], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0132], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.15e-297

   1. Initial program 92.4%

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

      1. associate-*r/ 92.4%

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

      2. *-commutative 92.4%

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

      3. sqr-neg 92.4%

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

      4. associate-+l+ 92.4%

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

      5. +-commutative 92.4%

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

      6. sqr-neg 92.4%

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

      7. distribute-rgt-out 92.4%

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

      8. fma-def 92.4%

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

      9. +-commutative 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in m around 0 32.9%

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

   5. Taylor expanded in k around inf 39.8%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 39.8%

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

   7. Simplified 39.8%

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

   if 1.15e-297 < k < 0.0132

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.6%

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

   5. Taylor expanded in k around 0 40.0%

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

      1. *-commutative 40.0%

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

   7. Simplified 40.0%

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

   if 0.0132 < k

   1. Initial program 86.1%

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

   2. Taylor expanded in k around inf 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 85.5%

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

   4. Simplified 85.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   5. Step-by-step derivation

      1. *-commutative 85.5%

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

      2. times-frac 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in m around 0 66.6%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-297}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.0132:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 8: 52.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.24) (* a (/ 1.0 (* k k))) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.24) {
		tmp = a * (1.0 / (k * k));
	} 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 <= (-0.24d0)) then
        tmp = a * (1.0d0 / (k * k))
    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 <= -0.24) {
		tmp = a * (1.0 / (k * k));
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.24:
		tmp = a * (1.0 / (k * k))
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.24)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	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 <= -0.24)
		tmp = a * (1.0 / (k * k));
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.24], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $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 < -0.23999999999999999

   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. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 43.6%

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

   5. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 68.2%

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

   7. Simplified 68.2%

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

      1. associate-/r* 62.8%

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

      2. div-inv 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. times-frac 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. /-rgt-identity 68.2%

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

       2. associate-/l/ 69.3%

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

   11. Simplified 69.3%

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

   if -0.23999999999999999 < m

   1. Initial program 88.5%

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

      1. associate-*r/ 88.5%

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

      2. *-commutative 88.5%

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

      3. sqr-neg 88.5%

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

      4. associate-+l+ 88.5%

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

      5. +-commutative 88.5%

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

      6. sqr-neg 88.5%

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

      7. distribute-rgt-out 88.5%

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

      8. fma-def 88.5%

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

      9. +-commutative 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in m around 0 48.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.24:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9: 45.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-299} \lor \neg \left(k \leq 0.0132\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 2.05e-299) (not (<= k 0.0132))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.05e-299) || !(k <= 0.0132)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	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 <= 2.05d-299) .or. (.not. (k <= 0.0132d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.05e-299) || !(k <= 0.0132)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 2.05e-299) or not (k <= 0.0132):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 2.05e-299) || !(k <= 0.0132))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 2.05e-299) || ~((k <= 0.0132)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 2.05e-299], N[Not[LessEqual[k, 0.0132]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 2.05e-299 or 0.0132 < k

   1. Initial program 89.0%

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

      1. associate-*r/ 88.9%

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

      2. *-commutative 88.9%

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

      3. sqr-neg 88.9%

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

      4. associate-+l+ 88.9%

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

      5. +-commutative 88.9%

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

      6. sqr-neg 88.9%

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

      7. distribute-rgt-out 88.9%

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

      8. fma-def 88.9%

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

      9. +-commutative 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in m around 0 49.7%

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

   5. Taylor expanded in k around inf 52.6%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 52.6%

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

   7. Simplified 52.6%

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

   if 2.05e-299 < k < 0.0132

   1. Initial program 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.6%

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

   5. Taylor expanded in k around 0 39.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{-299} \lor \neg \left(k \leq 0.0132\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10: 51.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.86:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.86) (* a (/ 1.0 (* k k))) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.86) {
		tmp = a * (1.0 / (k * k));
	} else {
		tmp = a / (1.0 + (k * 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 <= (-0.86d0)) then
        tmp = a * (1.0d0 / (k * k))
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.86) {
		tmp = a * (1.0 / (k * k));
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.86:
		tmp = a * (1.0 / (k * k))
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.86)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.86)
		tmp = a * (1.0 / (k * k));
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.86], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.859999999999999987

   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. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 43.6%

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

   5. Taylor expanded in k around inf 68.2%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 68.2%

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

   7. Simplified 68.2%

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

      1. associate-/r* 62.8%

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

      2. div-inv 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. times-frac 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. /-rgt-identity 68.2%

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

       2. associate-/l/ 69.3%

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

   11. Simplified 69.3%

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

   if -0.859999999999999987 < m

   1. Initial program 88.5%

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

      1. associate-*r/ 88.5%

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

      2. *-commutative 88.5%

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

      3. sqr-neg 88.5%

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

      4. associate-+l+ 88.5%

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

      5. +-commutative 88.5%

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

      6. sqr-neg 88.5%

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

      7. distribute-rgt-out 88.5%

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

      8. fma-def 88.5%

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

      9. +-commutative 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in m around 0 48.5%

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

   5. Taylor expanded in k around inf 47.8%

      \[\leadsto \frac{a}{1 + \color{blue}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 47.8%

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

   7. Simplified 47.8%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.86:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 11: 19.8% 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 92.4%

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

   1. associate-*r/ 92.4%

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

   2. *-commutative 92.4%

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

   3. sqr-neg 92.4%

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

   4. associate-+l+ 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. distribute-rgt-out 92.4%

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

   8. fma-def 92.4%

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

   9. +-commutative 92.4%

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

3. Simplified 92.4%

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

4. Taylor expanded in m around 0 46.9%

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

5. Taylor expanded in k around 0 15.7%

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

6. Final simplification 15.7%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023278 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))