Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 97.5%

Time: 10.0s

Alternatives: 10

Speedup: 1.1×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

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: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 5e+137)
     (/ t_0 (fma k (+ k 10.0) 1.0))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 5e+137) {
		tmp = t_0 / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 5e+137)
		tmp = Float64(t_0 / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+137], N[(t$95$0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.0000000000000002e137

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   5. Simplified 96.5%

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

   if 5.0000000000000002e137 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 64.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.00011:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -5.8e-8)
     t_0
     (if (<= m 0.00011) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -5.8e-8) {
		tmp = t_0;
	} else if (m <= 0.00011) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -5.8e-8)
		tmp = t_0;
	elseif (m <= 0.00011)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.8e-8], t$95$0, If[LessEqual[m, 0.00011], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.8000000000000003e-8 or 1.10000000000000004e-4 < m

   1. Initial program 88.8%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Simplified 100.0%

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

   if -5.8000000000000003e-8 < m < 1.10000000000000004e-4

   1. Initial program 93.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 91.1

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

   5. Simplified 91.1%

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

Alternative 3: 76.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.18:\\ \;\;\;\;\frac{a \cdot \frac{100}{k \cdot k}}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.18)
   (/ (* a (/ 100.0 (* k k))) (* k k))
   (if (<= m 2.65e+14) (/ a (fma k (+ k 10.0) 1.0)) (* 99.0 (* a (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.18) {
		tmp = (a * (100.0 / (k * k))) / (k * k);
	} else if (m <= 2.65e+14) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.18)
		tmp = Float64(Float64(a * Float64(100.0 / Float64(k * k))) / Float64(k * k));
	elseif (m <= 2.65e+14)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.18], N[(N[(a * N[(100.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e+14], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.17999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 40.3

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

   5. Simplified 40.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in k around -inf

      \[\leadsto \color{blue}{\frac{a + -1 \cdot \frac{-100 \cdot \frac{a}{k} + 10 \cdot a}{k}}{{k}^{2}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + -1 \cdot \frac{-100 \cdot \frac{a}{k} + 10 \cdot a}{k}}{{k}^{2}}} \]

   11. Simplified 69.7%

       \[\leadsto \color{blue}{\frac{a + \frac{a \cdot \left(\frac{100}{k} + -10\right)}{k}}{k \cdot k}} \]

   12. Taylor expanded in k around 0

       \[\leadsto \frac{\color{blue}{100 \cdot \frac{a}{{k}^{2}}}}{k \cdot k} \]
   13. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{\frac{100 \cdot a}{{k}^{2}}}}{k \cdot k} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{a \cdot 100}}{{k}^{2}}}{k \cdot k} \]

       3. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{a \cdot \frac{100}{{k}^{2}}}}{k \cdot k} \]

       4. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{100 \cdot 1}}{{k}^{2}}}{k \cdot k} \]

       5. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\left(100 \cdot \frac{1}{{k}^{2}}\right)}}{k \cdot k} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(100 \cdot \frac{1}{{k}^{2}}\right)}}{k \cdot k} \]

       7. associate-*r/ N/A

          \[\leadsto \frac{a \cdot \color{blue}{\frac{100 \cdot 1}{{k}^{2}}}}{k \cdot k} \]

       8. metadata-eval N/A

          \[\leadsto \frac{a \cdot \frac{\color{blue}{100}}{{k}^{2}}}{k \cdot k} \]

       9. lower-/.f64 N/A

          \[\leadsto \frac{a \cdot \color{blue}{\frac{100}{{k}^{2}}}}{k \cdot k} \]

       10. unpow2 N/A

           \[\leadsto \frac{a \cdot \frac{100}{\color{blue}{k \cdot k}}}{k \cdot k} \]

       11. lower-*.f64 76.6

           \[\leadsto \frac{a \cdot \frac{100}{\color{blue}{k \cdot k}}}{k \cdot k} \]

   14. Simplified 76.6%

       \[\leadsto \frac{\color{blue}{a \cdot \frac{100}{k \cdot k}}}{k \cdot k} \]

   if -0.17999999999999999 < m < 2.65e14

   1. Initial program 92.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 88.3

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

   5. Simplified 88.3%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.4e+21)
   (/ a (* k k))
   (if (<= m 2.65e+14) (/ a (fma k (+ k 10.0) 1.0)) (* 99.0 (* a (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.4e+21) {
		tmp = a / (k * k);
	} else if (m <= 2.65e+14) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.4e+21)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.65e+14)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.4e+21], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e+14], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.4e21

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 39.5

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

   5. Simplified 39.5%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

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

   8. Simplified 67.7%

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

   if -1.4e21 < m < 2.65e14

   1. Initial program 92.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 87.1

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

   5. Simplified 87.1%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 61.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.7e-20)
   (/ a (* k k))
   (if (<= m 2.65e+14) (/ a (fma k 10.0 1.0)) (* 99.0 (* a (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.7e-20) {
		tmp = a / (k * k);
	} else if (m <= 2.65e+14) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.7e-20)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.65e+14)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.7e-20], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e+14], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.7e-20

   1. Initial program 99.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 41.3

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

   5. Simplified 41.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.6

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

   8. Simplified 65.6%

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

   if -2.7e-20 < m < 2.65e14

   1. Initial program 92.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 89.5

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

   5. Simplified 89.5%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 58.6

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

   8. Simplified 58.6%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 8.8e-158)
   (/ a (* k k))
   (if (<= m 2.65e+14) a (* 99.0 (* a (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.8e-158) {
		tmp = a / (k * k);
	} else if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8.8d-158) then
        tmp = a / (k * k)
    else if (m <= 2.65d+14) then
        tmp = a
    else
        tmp = 99.0d0 * (a * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.8e-158) {
		tmp = a / (k * k);
	} else if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.8e-158:
		tmp = a / (k * k)
	elif m <= 2.65e+14:
		tmp = a
	else:
		tmp = 99.0 * (a * (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.8e-158)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.65e+14)
		tmp = a;
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8.8e-158)
		tmp = a / (k * k);
	elseif (m <= 2.65e+14)
		tmp = a;
	else
		tmp = 99.0 * (a * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.8e-158], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e+14], a, N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 8.8000000000000004e-158

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 67.5

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

   5. Simplified 67.5%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.0

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

   8. Simplified 59.0%

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

   if 8.8000000000000004e-158 < m < 2.65e14

   1. Initial program 89.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 61.7

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

   5. Simplified 61.7%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 51.2%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00013:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.00013)
   (/ a (* k 10.0))
   (if (<= m 2.65e+14) a (* 99.0 (* a (* k k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.00013) {
		tmp = a / (k * 10.0);
	} else if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.00013d0)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 2.65d+14) then
        tmp = a
    else
        tmp = 99.0d0 * (a * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.00013) {
		tmp = a / (k * 10.0);
	} else if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.00013:
		tmp = a / (k * 10.0)
	elif m <= 2.65e+14:
		tmp = a
	else:
		tmp = 99.0 * (a * (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.00013)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 2.65e+14)
		tmp = a;
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.00013)
		tmp = a / (k * 10.0);
	elseif (m <= 2.65e+14)
		tmp = a;
	else
		tmp = 99.0 * (a * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.00013], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e+14], a, N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.29999999999999989e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 40.3

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

   5. Simplified 40.3%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 44.6

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

   8. Simplified 44.6%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 20.0

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

   11. Simplified 20.0%

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

   if -1.29999999999999989e-4 < m < 2.65e14

   1. Initial program 92.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.2

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

   5. Simplified 51.2%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 47.2%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00013:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.65e+14) a (* 99.0 (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.65d+14) then
        tmp = a
    else
        tmp = 99.0d0 * (a * (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.65e+14) {
		tmp = a;
	} else {
		tmp = 99.0 * (a * (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.65e+14:
		tmp = a
	else:
		tmp = 99.0 * (a * (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.65e+14)
		tmp = a;
	else
		tmp = Float64(99.0 * Float64(a * Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.65e+14)
		tmp = a;
	else
		tmp = 99.0 * (a * (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.65e+14], a, N[(99.0 * N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.65e14

   1. Initial program 95.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 30.6%

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

   if 2.65e14 < m

   1. Initial program 80.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10 \cdot a\right)\right)}, a\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right)} + \left(\mathsf{neg}\left(10 \cdot a\right)\right), a\right) \]

      5. distribute-neg-out N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{neg}\left(\left(k \cdot \left(a + -100 \cdot a\right) + 10 \cdot a\right)\right)}, a\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(k \cdot \color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} + 10 \cdot a\right)\right), a\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\left(\color{blue}{\left(k \cdot \left(-100 + 1\right)\right) \cdot a} + 10 \cdot a\right)\right), a\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(\color{blue}{a \cdot \left(k \cdot \left(-100 + 1\right) + 10\right)}\right), a\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{neg}\left(a \cdot \color{blue}{\mathsf{fma}\left(k, -100 + 1, 10\right)}\right), a\right) \]

      12. metadata-eval 28.9

          \[\leadsto \mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, \color{blue}{-99}, 10\right), a\right) \]

   8. Simplified 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -a \cdot \mathsf{fma}\left(k, -99, 10\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{99 \cdot \left(\left(a \cdot k\right) \cdot k\right)} \]

       4. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot \left(k \cdot k\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto 99 \cdot \color{blue}{\left(a \cdot {k}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       8. lower-*.f64 60.0

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

   11. Simplified 60.0%

       \[\leadsto \color{blue}{99 \cdot \left(a \cdot \left(k \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;99 \cdot \left(a \cdot \left(k \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 25.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{+37}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 9.2e+37) a (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 9.2e+37) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 9.2d+37) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 9.2e+37) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 9.2e+37:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 9.2e+37)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 9.2e+37)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 9.2e+37], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.2000000000000001e37

   1. Initial program 95.4%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 70.7

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

   5. Simplified 70.7%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 29.7%

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

   if 9.2000000000000001e37 < m

   1. Initial program 78.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. lower-+.f64 3.2

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

   5. Simplified 3.2%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 16.0

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

   8. Simplified 16.0%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. distribute-lft-neg-in N/A

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

       6. lower-*.f64 N/A

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

       7. distribute-lft-neg-in N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 29.1

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

   11. Simplified 29.1%

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

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{+37}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 20.3% accurate, 134.0× 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 90.5%

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

3. Taylor expanded in k around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 79.4

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

5. Simplified 79.4%

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

6. Taylor expanded in m around 0

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

8. Simplified 22.2%

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

9. Final simplification 22.2%

   \[\leadsto a \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))