Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 99.8%

Time: 12.9s

Alternatives: 13

Speedup: 1.0×

• 21.4% 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: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ t_1 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)) (t_1 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= k -0.1) t_0 (/ (/ t_0 t_1) t_1))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double t_1 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (k <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	t_1 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (k <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, -0.1], t$95$0, N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < -0.10000000000000001

   1. Initial program 73.9%

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

      1. associate-/l* 73.9%

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

      2. sqr-neg 73.9%

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

      3. associate-+l+ 73.9%

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

      4. +-commutative 73.9%

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

      5. sqr-neg 73.9%

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

      6. distribute-rgt-out 73.9%

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

      7. fma-define 73.9%

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

      8. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -0.10000000000000001 < k

   1. Initial program 94.3%

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

      1. associate-/l* 94.3%

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

      2. sqr-neg 94.3%

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

      3. associate-+l+ 94.3%

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

      4. +-commutative 94.3%

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

      5. sqr-neg 94.3%

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

      6. distribute-rgt-out 94.3%

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

      7. fma-define 94.3%

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

      8. +-commutative 94.3%

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

   3. Simplified 94.3%

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

      1. associate-*r/ 94.3%

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

      2. add-sqr-sqrt 94.3%

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

      3. associate-/r* 94.3%

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

      4. *-commutative 94.3%

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

      5. fma-undefine 94.3%

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

      6. distribute-rgt-in 94.3%

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

      7. associate-+r+ 94.3%

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

      8. +-commutative 94.3%

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

      9. add-sqr-sqrt 94.3%

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

      10. hypot-define 94.3%

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

      11. +-commutative 94.3%

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

      12. *-commutative 94.3%

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

      13. fma-define 94.3%

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

      14. fma-undefine 94.3%

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

      15. distribute-rgt-in 94.3%

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

      16. associate-+r+ 94.3%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 10^{-254}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{t\_0} \cdot \left(k + 10\right) + \frac{1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 1e-254) t_0 (/ 1.0 (+ (* (/ k t_0) (+ k 10.0)) (/ 1.0 t_0))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 1e-254) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((k / t_0) * (k + 10.0)) + (1.0 / 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 = (k ** m) * a
    if (k <= 1d-254) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((k / t_0) * (k + 10.0d0)) + (1.0d0 / t_0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (k <= 1e-254) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((k / t_0) * (k + 10.0)) + (1.0 / t_0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if k <= 1e-254:
		tmp = t_0
	else:
		tmp = 1.0 / (((k / t_0) * (k + 10.0)) + (1.0 / t_0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 1e-254)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(k / t_0) * Float64(k + 10.0)) + Float64(1.0 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (k <= 1e-254)
		tmp = t_0;
	else
		tmp = 1.0 / (((k / t_0) * (k + 10.0)) + (1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 1e-254], t$95$0, N[(1.0 / N[(N[(N[(k / t$95$0), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999991e-255

   1. Initial program 88.6%

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

      1. associate-/l* 88.6%

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

      2. sqr-neg 88.6%

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

      3. associate-+l+ 88.6%

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

      4. +-commutative 88.6%

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

      5. sqr-neg 88.6%

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

      6. distribute-rgt-out 88.6%

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

      7. fma-define 88.6%

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

      8. +-commutative 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 9.9999999999999991e-255 < k

   1. Initial program 92.1%

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

      1. associate-/l* 92.1%

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

      2. sqr-neg 92.1%

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

      3. associate-+l+ 92.1%

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

      4. +-commutative 92.1%

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

      5. sqr-neg 92.1%

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

      6. distribute-rgt-out 92.1%

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

      7. fma-define 92.1%

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

      8. +-commutative 92.1%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 92.1%

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

      2. clear-num 92.1%

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

      3. *-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

   7. Taylor expanded in k around 0 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. *-commutative 99.9%

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

       2. associate-*l/ 99.9%

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

       3. associate-*r/ 99.9%

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

       4. distribute-rgt-out 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-254}:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{{k}^{m} \cdot a} \cdot \left(k + 10\right) + \frac{1}{{k}^{m} \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 3.0) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 3.0) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} 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 = (k ** m) * a
    if (m <= 3.0d0) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (m <= 3.0) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if m <= 3.0:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 3.0)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (m <= 3.0)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, 3.0], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3

   1. Initial program 98.2%

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

   if 3 < m

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-+l+ 77.4%

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

      4. +-commutative 77.4%

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

      5. sqr-neg 77.4%

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

      6. distribute-rgt-out 77.4%

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

      7. fma-define 77.4%

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

      8. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 0.0011:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.02e-5)
   (* a (/ (pow k m) (+ 1.0 (* k 10.0))))
   (if (<= m 0.0011) (/ a (+ 1.0 (* k (+ k 10.0)))) (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02e-5) {
		tmp = a * (pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.0011) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = pow(k, m) * 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 (m <= (-1.02d-5)) then
        tmp = a * ((k ** m) / (1.0d0 + (k * 10.0d0)))
    else if (m <= 0.0011d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02e-5) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * 10.0)));
	} else if (m <= 0.0011) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.02e-5:
		tmp = a * (math.pow(k, m) / (1.0 + (k * 10.0)))
	elif m <= 0.0011:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.02e-5)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * 10.0))));
	elseif (m <= 0.0011)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.02e-5)
		tmp = a * ((k ^ m) / (1.0 + (k * 10.0)));
	elseif (m <= 0.0011)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.02e-5], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0011], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.0200000000000001e-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. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. sqr-neg 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 98.7%

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

   if -1.0200000000000001e-5 < m < 0.00110000000000000007

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. sqr-neg 96.8%

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

      3. associate-+l+ 96.8%

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

      4. +-commutative 96.8%

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

      5. sqr-neg 96.8%

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

      6. distribute-rgt-out 96.8%

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

      7. fma-define 96.8%

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

      8. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in m around 0 96.7%

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

   if 0.00110000000000000007 < m

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-+l+ 77.4%

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

      4. +-commutative 77.4%

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

      5. sqr-neg 77.4%

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

      6. distribute-rgt-out 77.4%

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

      7. fma-define 77.4%

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

      8. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 0.0011:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5200 \lor \neg \left(m \leq 0.0012\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5200.0) (not (<= m 0.0012)))
   (* (pow k m) a)
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5200.0) || !(m <= 0.0012)) {
		tmp = pow(k, m) * a;
	} 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 <= (-5200.0d0)) .or. (.not. (m <= 0.0012d0))) then
        tmp = (k ** m) * a
    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 <= -5200.0) || !(m <= 0.0012)) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -5200.0) or not (m <= 0.0012):
		tmp = math.pow(k, m) * a
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5200.0) || !(m <= 0.0012))
		tmp = Float64((k ^ m) * a);
	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 <= -5200.0) || ~((m <= 0.0012)))
		tmp = (k ^ m) * a;
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5200.0], N[Not[LessEqual[m, 0.0012]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $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 < -5200 or 0.00119999999999999989 < m

   1. Initial program 87.2%

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

      1. associate-/l* 87.2%

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

      2. sqr-neg 87.2%

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

      3. associate-+l+ 87.2%

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

      4. +-commutative 87.2%

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

      5. sqr-neg 87.2%

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

      6. distribute-rgt-out 87.2%

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

      7. fma-define 87.2%

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

      8. +-commutative 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -5200 < m < 0.00119999999999999989

   1. Initial program 96.9%

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

      1. associate-/l* 96.9%

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

      2. sqr-neg 96.9%

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

      3. associate-+l+ 96.9%

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

      4. +-commutative 96.9%

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

      5. sqr-neg 96.9%

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

      6. distribute-rgt-out 96.9%

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

      7. fma-define 96.9%

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

      8. +-commutative 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in m around 0 95.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5200 \lor \neg \left(m \leq 0.0012\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(100 \cdot \left(k \cdot a\right) - a \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.1)
   (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a))
   (+ a (* k (- (* 100.0 (* k a)) (* a 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a + (k * ((100.0 * (k * a)) - (a * 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 <= 2.1d0) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = a + (k * ((100.0d0 * (k * a)) - (a * 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.1) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = a + (k * ((100.0 * (k * a)) - (a * 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.1:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = a + (k * ((100.0 * (k * a)) - (a * 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.1)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a + Float64(k * Float64(Float64(100.0 * Float64(k * a)) - Float64(a * 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.1)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = a + (k * ((100.0 * (k * a)) - (a * 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.1], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a + N[(k * N[(N[(100.0 * N[(k * a), $MachinePrecision]), $MachinePrecision] - N[(a * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 98.2%

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

      1. associate-/l* 98.2%

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

      2. sqr-neg 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. sqr-neg 98.2%

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

      6. distribute-rgt-out 98.2%

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

      7. fma-define 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. associate-*r/ 98.2%

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

      2. clear-num 98.2%

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

      3. *-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in m around 0 72.6%

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

   if 2.10000000000000009 < m

   1. Initial program 77.4%

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

      1. associate-/l* 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-+l+ 77.4%

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

      4. +-commutative 77.4%

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

      5. sqr-neg 77.4%

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

      6. distribute-rgt-out 77.4%

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

      7. fma-define 77.4%

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

      8. +-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 2.9%

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

      1. *-commutative 2.9%

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

   8. Simplified 2.9%

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

   9. Taylor expanded in k around 0 22.7%

      \[\leadsto \color{blue}{a + k \cdot \left(100 \cdot \left(a \cdot k\right) - 10 \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(100 \cdot \left(k \cdot a\right) - a \cdot 10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.6) (/ 1.0 (/ (+ 1.0 (* k (+ k 10.0))) a)) (* (* k a) 0.001)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.6) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= 1.6d0) then
        tmp = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.6) {
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.6:
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a)
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.6)
		tmp = Float64(1.0 / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.6)
		tmp = 1.0 / ((1.0 + (k * (k + 10.0))) / a);
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.6], N[(1.0 / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6000000000000001

   1. Initial program 98.2%

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

      1. associate-/l* 98.2%

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

      2. sqr-neg 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. sqr-neg 98.2%

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

      6. distribute-rgt-out 98.2%

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

      7. fma-define 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

      1. associate-*r/ 98.2%

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

      2. clear-num 98.2%

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

      3. *-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in m around 0 72.6%

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

   if 1.6000000000000001 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-18}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.62:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e-18) (* 0.1 (/ a k)) (if (<= m 0.62) a (* (* k a) 0.001))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e-18) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.62) {
		tmp = a;
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= (-1.05d-18)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 0.62d0) then
        tmp = a
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e-18) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.62) {
		tmp = a;
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.05e-18:
		tmp = 0.1 * (a / k)
	elif m <= 0.62:
		tmp = a
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e-18)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 0.62)
		tmp = a;
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.05e-18)
		tmp = 0.1 * (a / k);
	elseif (m <= 0.62)
		tmp = a;
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e-18], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.62], a, N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e-18

   1. Initial program 98.8%

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

   3. Taylor expanded in m around 0 43.9%

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

   4. Taylor expanded in k around inf 46.1%

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

      1. *-commutative 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in k around 0 25.0%

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

   if -1.05e-18 < m < 0.619999999999999996

   1. Initial program 97.7%

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

   3. Taylor expanded in m around 0 97.6%

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

   4. Taylor expanded in k around inf 97.5%

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

   5. Taylor expanded in k around 0 57.7%

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

   if 0.619999999999999996 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-18}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.62:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* (* k a) 0.001)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= 1.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 98.2%

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

      1. associate-/l* 98.2%

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

      2. sqr-neg 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. sqr-neg 98.2%

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

      6. distribute-rgt-out 98.2%

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

      7. fma-define 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in m around 0 72.2%

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

   if 1 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.9) (/ a (+ 1.0 (* k k))) (* (* k a) 0.001)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= 0.9d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.9) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.9:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.9)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.9)
		tmp = a / (1.0 + (k * k));
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.9], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.900000000000000022

   1. Initial program 98.2%

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

   3. Taylor expanded in m around 0 72.2%

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

   4. Taylor expanded in k around inf 72.1%

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

   5. Taylor expanded in k around 0 72.2%

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

   if 0.900000000000000022 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.4) (/ a (+ 1.0 (* k 10.0))) (* (* k a) 0.001)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.4) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= 1.4d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.4) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.4)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.4)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3999999999999999

   1. Initial program 98.2%

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

      1. associate-/l* 98.2%

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

      2. sqr-neg 98.2%

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

      3. associate-+l+ 98.2%

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

      4. +-commutative 98.2%

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

      5. sqr-neg 98.2%

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

      6. distribute-rgt-out 98.2%

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

      7. fma-define 98.2%

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

      8. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in m around 0 72.2%

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

   6. Taylor expanded in k around 0 46.6%

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

      1. *-commutative 46.6%

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

   8. Simplified 46.6%

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

   if 1.3999999999999999 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.55:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.55) a (* (* k a) 0.001)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.55) {
		tmp = a;
	} else {
		tmp = (k * a) * 0.001;
	}
	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 <= 0.55d0) then
        tmp = a
    else
        tmp = (k * a) * 0.001d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.55) {
		tmp = a;
	} else {
		tmp = (k * a) * 0.001;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.55:
		tmp = a
	else:
		tmp = (k * a) * 0.001
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.55)
		tmp = a;
	else
		tmp = Float64(Float64(k * a) * 0.001);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.55)
		tmp = a;
	else
		tmp = (k * a) * 0.001;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.55], a, N[(N[(k * a), $MachinePrecision] * 0.001), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.55000000000000004

   1. Initial program 98.2%

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

   3. Taylor expanded in m around 0 72.2%

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

   4. Taylor expanded in k around inf 72.1%

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

   5. Taylor expanded in k around 0 32.3%

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

   if 0.55000000000000004 < m

   1. Initial program 77.4%

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

   3. Taylor expanded in m around 0 3.0%

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

   4. Taylor expanded in k around inf 2.3%

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

      1. *-commutative 2.3%

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

   6. Simplified 2.3%

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

   7. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{\frac{0.1 \cdot a + k \cdot \left(0.001 \cdot \left(a \cdot k\right) - 0.01 \cdot a\right)}{k}} \]

   8. Taylor expanded in k around inf 19.5%

      \[\leadsto \color{blue}{0.001 \cdot \left(a \cdot k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.55:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot 0.001\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 20.4% 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 90.7%

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

3. Taylor expanded in m around 0 47.1%

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

4. Taylor expanded in k around inf 47.0%

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

5. Taylor expanded in k around 0 21.9%

   \[\leadsto \color{blue}{a} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))