Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 98.7%

Time: 7.9s

Alternatives: 9

Speedup: 1.1×

• 23.5% 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: 90.0% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a \cdot {k}^{\left(m + -1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.5e-7) (* a (pow k m)) (/ 1.0 (/ k (* a (pow k (+ m -1.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.5e-7) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / (k / (a * pow(k, (m + -1.0))));
	}
	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 <= 4.5d-7) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / (k / (a * (k ** (m + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.5e-7) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / (k / (a * Math.pow(k, (m + -1.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.5e-7:
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / (k / (a * math.pow(k, (m + -1.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.5e-7)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(k / Float64(a * (k ^ Float64(m + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.5e-7)
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / (k / (a * (k ^ (m + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.5e-7], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k / N[(a * N[Power[k, N[(m + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4999999999999998e-7

   1. Initial program 95.6%

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

      1. sqr-neg 95.6%

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

      2. associate-+l+ 95.6%

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

      3. sqr-neg 95.6%

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

      4. distribute-rgt-out 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 99.5%

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

   if 4.4999999999999998e-7 < k

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. associate-+l+ 85.9%

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

      3. sqr-neg 85.9%

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

      4. distribute-rgt-out 85.9%

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

   3. Simplified 85.9%

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

      1. add-sqr-sqrt 85.9%

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

      2. times-frac 84.9%

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

      3. +-commutative 84.9%

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

      4. fma-def 84.9%

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

      5. +-commutative 84.9%

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

      6. +-commutative 84.9%

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

      7. fma-def 84.9%

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

      8. +-commutative 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*r/ 85.9%

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

      3. associate-/l* 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in k around inf 84.9%

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

   9. Taylor expanded in k around inf 96.8%

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

       1. clear-num 95.4%

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

       2. inv-pow 95.4%

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

       3. pow1 95.4%

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

       4. pow-div 95.4%

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

   11. Applied egg-rr 95.4%

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

       1. unpow-1 95.4%

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

       2. associate-/r* 98.5%

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

       3. sub-neg 98.5%

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

       4. metadata-eval 98.5%

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

   13. Simplified 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{a \cdot {k}^{\left(m + -1\right)}}}\\ \end{array} \]

Alternative 2: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 4.5e-7) (* a (pow k m)) (* (pow k (+ m -1.0)) (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.5e-7) {
		tmp = a * pow(k, m);
	} else {
		tmp = pow(k, (m + -1.0)) * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 4.5d-7) then
        tmp = a * (k ** m)
    else
        tmp = (k ** (m + (-1.0d0))) * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 4.5e-7) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = Math.pow(k, (m + -1.0)) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 4.5e-7:
		tmp = a * math.pow(k, m)
	else:
		tmp = math.pow(k, (m + -1.0)) * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 4.5e-7)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64((k ^ Float64(m + -1.0)) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 4.5e-7)
		tmp = a * (k ^ m);
	else
		tmp = (k ^ (m + -1.0)) * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 4.5e-7], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[Power[k, N[(m + -1.0), $MachinePrecision]], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.4999999999999998e-7

   1. Initial program 95.6%

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

      1. sqr-neg 95.6%

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

      2. associate-+l+ 95.6%

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

      3. sqr-neg 95.6%

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

      4. distribute-rgt-out 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 99.5%

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

   if 4.4999999999999998e-7 < k

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. associate-+l+ 85.9%

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

      3. sqr-neg 85.9%

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

      4. distribute-rgt-out 85.9%

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

   3. Simplified 85.9%

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

      1. add-sqr-sqrt 85.9%

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

      2. times-frac 84.9%

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

      3. +-commutative 84.9%

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

      4. fma-def 84.9%

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

      5. +-commutative 84.9%

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

      6. +-commutative 84.9%

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

      7. fma-def 84.9%

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

      8. +-commutative 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*r/ 85.9%

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

      3. associate-/l* 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in k around inf 84.9%

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

   9. Taylor expanded in k around inf 96.8%

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

       1. div-inv 96.8%

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

       2. clear-num 96.8%

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

       3. pow1 96.8%

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

       4. pow-div 96.8%

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

   11. Applied egg-rr 96.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{\left(m + -1\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 3: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.15 \cdot 10^{-6} \lor \neg \left(m \leq 3.15 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.15e-6) (not (<= m 3.15e-6)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.15e-6) || !(m <= 3.15e-6)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-1.15d-6)) .or. (.not. (m <= 3.15d-6))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.15e-6) || !(m <= 3.15e-6)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.15e-6) or not (m <= 3.15e-6):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.15e-6) || !(m <= 3.15e-6))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -1.15e-6) || ~((m <= 3.15e-6)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.15e-6], N[Not[LessEqual[m, 3.15e-6]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.15e-6 or 3.14999999999999991e-6 < m

   1. Initial program 91.1%

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

      1. sqr-neg 91.1%

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

      2. associate-+l+ 91.1%

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

      3. sqr-neg 91.1%

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

      4. distribute-rgt-out 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -1.15e-6 < m < 3.14999999999999991e-6

   1. Initial program 94.1%

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

      1. sqr-neg 94.1%

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

      2. associate-+l+ 94.1%

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

      3. sqr-neg 94.1%

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

      4. distribute-rgt-out 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 94.1%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.15 \cdot 10^{-6} \lor \neg \left(m \leq 3.15 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 49.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -23.5:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -23.5) (/ (/ 1.0 k) (/ k a)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -23.5) {
		tmp = (1.0 / k) / (k / 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 <= (-23.5d0)) then
        tmp = (1.0d0 / k) / (k / 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 <= -23.5) {
		tmp = (1.0 / k) / (k / a);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -23.5:
		tmp = (1.0 / k) / (k / a)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -23.5)
		tmp = Float64(Float64(1.0 / k) / Float64(k / 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 <= -23.5)
		tmp = (1.0 / k) / (k / a);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -23.5], N[(N[(1.0 / k), $MachinePrecision] / N[(k / a), $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 < -23.5

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-def 100.0%

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

      8. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in k around inf 85.9%

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

   9. Taylor expanded in k around inf 100.0%

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

   10. Taylor expanded in m around 0 52.0%

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

   if -23.5 < m

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. associate-+l+ 86.9%

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

      3. sqr-neg 86.9%

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

      4. distribute-rgt-out 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in m around 0 47.1%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -23.5:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5: 26.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.1e-19) (* (/ a k) 0.1) (+ a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e-19) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a + (-10.0 * (k * 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 <= (-4.1d-19)) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e-19) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.1e-19:
		tmp = (a / k) * 0.1
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.1e-19)
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.1e-19)
		tmp = (a / k) * 0.1;
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.1e-19], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -4.09999999999999985e-19

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.1%

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

   5. Taylor expanded in k around 0 15.9%

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

      1. *-commutative 15.9%

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

   7. Simplified 15.9%

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

   8. Taylor expanded in k around inf 24.7%

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

   if -4.09999999999999985e-19 < m

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. associate-+l+ 86.7%

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

      3. sqr-neg 86.7%

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

      4. distribute-rgt-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in m around 0 46.7%

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

   5. Taylor expanded in k around 0 27.3%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 6: 30.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -23.5:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -23.5) (* (/ a k) 0.1) (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -23.5) {
		tmp = (a / k) * 0.1;
	} 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 (m <= (-23.5d0)) then
        tmp = (a / k) * 0.1d0
    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 (m <= -23.5) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -23.5:
		tmp = (a / k) * 0.1
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -23.5)
		tmp = Float64(Float64(a / k) * 0.1);
	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 (m <= -23.5)
		tmp = (a / k) * 0.1;
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -23.5], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -23.5

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 39.2%

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

   5. Taylor expanded in k around 0 15.2%

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

      1. *-commutative 15.2%

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

   7. Simplified 15.2%

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

   8. Taylor expanded in k around inf 24.3%

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

   if -23.5 < m

   1. Initial program 86.9%

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

      1. sqr-neg 86.9%

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

      2. associate-+l+ 86.9%

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

      3. sqr-neg 86.9%

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

      4. distribute-rgt-out 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in m around 0 47.1%

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

   5. Taylor expanded in k around 0 33.2%

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

      1. *-commutative 33.2%

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

   7. Simplified 33.2%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -23.5:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 7: 37.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2e-204) (/ (/ 1.0 k) (/ k a)) (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-204) {
		tmp = (1.0 / k) / (k / a);
	} 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 (m <= (-2d-204)) then
        tmp = (1.0d0 / k) / (k / a)
    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 (m <= -2e-204) {
		tmp = (1.0 / k) / (k / a);
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2e-204:
		tmp = (1.0 / k) / (k / a)
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2e-204)
		tmp = Float64(Float64(1.0 / k) / Float64(k / a));
	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 (m <= -2e-204)
		tmp = (1.0 / k) / (k / a);
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2e-204], N[(N[(1.0 / k), $MachinePrecision] / N[(k / a), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2e-204

   1. Initial program 98.0%

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

      1. sqr-neg 98.0%

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

      2. associate-+l+ 98.0%

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

      3. sqr-neg 98.0%

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

      4. distribute-rgt-out 98.0%

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

   3. Simplified 98.0%

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

      1. add-sqr-sqrt 98.0%

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

      2. times-frac 98.0%

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

      3. +-commutative 98.0%

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

      4. fma-def 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. fma-def 98.0%

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

      8. +-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 98.0%

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

      3. associate-/l* 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in k around inf 79.8%

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

   9. Taylor expanded in k around inf 92.6%

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

   10. Taylor expanded in m around 0 55.1%

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

   if -2e-204 < m

   1. Initial program 85.9%

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

      1. sqr-neg 85.9%

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

      2. associate-+l+ 85.9%

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

      3. sqr-neg 85.9%

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

      4. distribute-rgt-out 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in m around 0 37.4%

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

   5. Taylor expanded in k around 0 30.0%

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

      1. *-commutative 30.0%

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

   7. Simplified 30.0%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 8: 25.8% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m -4.1e-19) (* (/ a k) 0.1) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e-19) {
		tmp = (a / k) * 0.1;
	} 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 (m <= (-4.1d-19)) then
        tmp = (a / k) * 0.1d0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e-19) {
		tmp = (a / k) * 0.1;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.1e-19:
		tmp = (a / k) * 0.1
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.1e-19)
		tmp = Float64(Float64(a / k) * 0.1);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.1e-19)
		tmp = (a / k) * 0.1;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.1e-19], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -4.09999999999999985e-19

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.1%

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

   5. Taylor expanded in k around 0 15.9%

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

      1. *-commutative 15.9%

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

   7. Simplified 15.9%

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

   8. Taylor expanded in k around inf 24.7%

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

   if -4.09999999999999985e-19 < m

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. associate-+l+ 86.7%

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

      3. sqr-neg 86.7%

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

      4. distribute-rgt-out 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in m around 0 46.7%

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

   5. Taylor expanded in k around 0 26.1%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 9: 19.6% 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 92.0%

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

   1. sqr-neg 92.0%

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

   2. associate-+l+ 92.0%

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

   3. sqr-neg 92.0%

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

   4. distribute-rgt-out 92.0%

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

3. Simplified 92.0%

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

4. Taylor expanded in m around 0 44.1%

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

5. Taylor expanded in k around 0 17.4%

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

6. Final simplification 17.4%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023308 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))