Falkner and Boettcher, Appendix A

Percentage Accurate: 90.7% → 99.5%

Time: 15.3s

Alternatives: 16

Speedup: 1.0×

• 22.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.9999999999999999e-154
  2. k = 0.0
  3. m = -2.00000000000000007e154

Specification

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.0002:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{t\_0} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -0.0002)
     (/ (/ a (fma k (+ k 10.0) 1.0)) (pow k (- m)))
     (if (<= m 7.5e-5)
       (/ -1.0 (- (/ -1.0 t_0) (* (/ k a) (/ (+ k 10.0) (pow k m)))))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -0.0002) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / pow(k, -m);
	} else if (m <= 7.5e-5) {
		tmp = -1.0 / ((-1.0 / t_0) - ((k / a) * ((k + 10.0) / pow(k, m))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -0.0002)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / (k ^ Float64(-m)));
	elseif (m <= 7.5e-5)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 / t_0) - Float64(Float64(k / a) * Float64(Float64(k + 10.0) / (k ^ m)))));
	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, -0.0002], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.5e-5], N[(-1.0 / N[(N[(-1.0 / t$95$0), $MachinePrecision] - N[(N[(k / a), $MachinePrecision] * N[(N[(k + 10.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.0000000000000001e-4

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. div-inv 100.0%

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

      5. associate-/r* 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-undefine 100.0%

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

      11. pow-flip 100.0%

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

   6. Applied egg-rr 100.0%

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

   if -2.0000000000000001e-4 < m < 7.49999999999999934e-5

   1. Initial program 92.9%

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

      1. associate-*l/ 92.9%

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

      2. sqr-neg 92.9%

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

      3. associate-+l+ 92.9%

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

      4. sqr-neg 92.9%

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

      5. distribute-rgt-out 92.9%

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

   3. Simplified 92.9%

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

      1. +-commutative 92.9%

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

      2. +-commutative 92.9%

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

      3. fma-undefine 92.9%

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

      4. associate-/r/ 92.9%

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

      5. clear-num 92.8%

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

      6. frac-2neg 92.8%

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

      7. metadata-eval 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. associate-/l/ 92.8%

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

      2. distribute-neg-frac 92.8%

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

      3. neg-sub0 92.8%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 92.8%

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

      5. fma-undefine 92.8%

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

      6. +-commutative 92.8%

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

      7. associate--r+ 92.8%

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

      8. metadata-eval 92.8%

         \[\leadsto \frac{-1}{\frac{\left(\color{blue}{0} - 1\right) - k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}} \]

      9. metadata-eval 92.8%

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

   8. Simplified 92.8%

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

      1. div-sub 92.8%

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

      2. sub-neg 92.8%

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

      3. times-frac 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. sub-neg 99.2%

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

   12. Simplified 99.2%

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

   if 7.49999999999999934e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0002:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\frac{-1}{a \cdot {k}^{m}} - \frac{k}{a} \cdot \frac{k + 10}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-5} \lor \neg \left(m \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -4.2e-5) (not (<= m 9e-9)))
   (* a (pow k m))
   (* a (/ 1.0 (fma k (+ k 10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -4.2e-5) || !(m <= 9e-9)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -4.2e-5) || !(m <= 9e-9))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -4.2e-5], N[Not[LessEqual[m, 9e-9]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -4.19999999999999977e-5 or 8.99999999999999953e-9 < m

   1. Initial program 92.9%

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

      1. associate-*l/ 91.6%

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

      2. sqr-neg 91.6%

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

      3. associate-+l+ 91.6%

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

      4. sqr-neg 91.6%

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

      5. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in k around 0 98.7%

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

   if -4.19999999999999977e-5 < m < 8.99999999999999953e-9

   1. Initial program 92.9%

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

      1. associate-*l/ 92.9%

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

      2. sqr-neg 92.9%

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

      3. associate-+l+ 92.9%

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

      4. sqr-neg 92.9%

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

      5. distribute-rgt-out 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in m around 0 92.1%

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

      1. clear-num 91.9%

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

      2. distribute-lft-in 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-/r/ 92.1%

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

      5. associate-+l+ 92.1%

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

      6. distribute-lft-in 92.1%

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

      7. +-commutative 92.1%

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

      8. +-commutative 92.1%

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

      9. fma-undefine 92.1%

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

   7. Applied egg-rr 92.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-5} \lor \neg \left(m \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e-7)
   (* (pow k m) (/ a (+ 1.0 (* k 10.0))))
   (if (<= m 5e-5) (* a (/ 1.0 (fma k (+ k 10.0) 1.0))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e-7) {
		tmp = pow(k, m) * (a / (1.0 + (k * 10.0)));
	} else if (m <= 5e-5) {
		tmp = a * (1.0 / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e-7)
		tmp = Float64((k ^ m) * Float64(a / Float64(1.0 + Float64(k * 10.0))));
	elseif (m <= 5e-5)
		tmp = Float64(a * Float64(1.0 / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e-7], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-5], N[(a * N[(1.0 / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.69999999999999987e-7

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   if -1.69999999999999987e-7 < m < 5.00000000000000024e-5

   1. Initial program 92.9%

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

      1. associate-*l/ 92.9%

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

      2. sqr-neg 92.9%

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

      3. associate-+l+ 92.9%

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

      4. sqr-neg 92.9%

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

      5. distribute-rgt-out 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in m around 0 92.1%

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

      1. clear-num 91.9%

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

      2. distribute-lft-in 91.9%

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

      3. associate-+l+ 91.9%

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

      4. associate-/r/ 92.1%

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

      5. associate-+l+ 92.1%

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

      6. distribute-lft-in 92.1%

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

      7. +-commutative 92.1%

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

      8. +-commutative 92.1%

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

      9. fma-undefine 92.1%

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

   7. Applied egg-rr 92.1%

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

   if 5.00000000000000024e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 7.5e-5) (* t_0 (/ -1.0 (- -1.0 (* k (+ k 10.0))))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 7.5e-5) {
		tmp = t_0 * (-1.0 / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 7.5d-5) then
        tmp = t_0 * ((-1.0d0) / ((-1.0d0) - (k * (k + 10.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 7.5e-5) {
		tmp = t_0 * (-1.0 / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 7.5e-5:
		tmp = t_0 * (-1.0 / (-1.0 - (k * (k + 10.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 7.5e-5)
		tmp = Float64(t_0 * Float64(-1.0 / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 7.5e-5)
		tmp = t_0 * (-1.0 / (-1.0 - (k * (k + 10.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 7.5e-5], N[(t$95$0 * N[(-1.0 / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 7.49999999999999934e-5

   1. Initial program 96.3%

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

      1. associate-/l* 96.3%

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

      2. sqr-pow 95.7%

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

      3. associate-/r* 95.7%

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

      4. associate-/r* 95.7%

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

      5. sqr-pow 96.3%

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

      6. sqr-neg 96.3%

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

      7. associate-+l+ 96.3%

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

      8. +-commutative 96.3%

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

      9. sqr-neg 96.3%

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

      10. distribute-rgt-out 96.3%

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

      11. fma-define 96.3%

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

      12. +-commutative 96.3%

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

   3. Simplified 96.3%

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

      1. fma-undefine 96.3%

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

      2. +-commutative 96.3%

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

      3. +-commutative 96.3%

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

      4. distribute-lft-in 96.3%

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

      5. associate-+l+ 96.3%

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

      6. associate-/l* 96.3%

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

      7. clear-num 96.2%

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

      8. associate-/r/ 96.3%

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

      9. associate-+l+ 96.3%

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

      10. distribute-lft-in 96.3%

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

      11. +-commutative 96.3%

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

      12. +-commutative 96.3%

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

      13. fma-undefine 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. frac-2neg 96.3%

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

      2. metadata-eval 96.3%

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

      3. div-inv 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-*r/ 96.3%

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

      2. metadata-eval 96.3%

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

      3. fma-undefine 96.3%

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

      4. +-commutative 96.3%

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

      5. distribute-neg-in 96.3%

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

      6. metadata-eval 96.3%

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

      7. sub-neg 96.3%

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

   10. Simplified 96.3%

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

   if 7.49999999999999934e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{-1}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7.5e-5)
   (* (pow k m) (/ a (+ 1.0 (* k (+ k 10.0)))))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.5e-5) {
		tmp = pow(k, m) * (a / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7.5d-5) then
        tmp = (k ** m) * (a / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.5e-5) {
		tmp = Math.pow(k, m) * (a / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 7.5e-5:
		tmp = math.pow(k, m) * (a / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 7.5e-5)
		tmp = Float64((k ^ m) * Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 7.5e-5)
		tmp = (k ^ m) * (a / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 7.5e-5], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.49999999999999934e-5

   1. Initial program 96.3%

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

      1. associate-*l/ 96.3%

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

      2. sqr-neg 96.3%

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

      3. associate-+l+ 96.3%

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

      4. sqr-neg 96.3%

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

      5. distribute-rgt-out 96.3%

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

   3. Simplified 96.3%

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

   if 7.49999999999999934e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 7.5e-5)
   (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.5e-5) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 7.5d-5) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 7.5e-5) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 7.5e-5:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 7.5e-5)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 7.5e-5)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 7.5e-5], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.49999999999999934e-5

   1. Initial program 96.3%

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

      1. associate-/l* 96.3%

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

      2. sqr-pow 95.7%

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

      3. associate-/r* 95.7%

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

      4. associate-/r* 95.7%

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

      5. sqr-pow 96.3%

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

      6. sqr-neg 96.3%

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

      7. associate-+l+ 96.3%

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

      8. +-commutative 96.3%

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

      9. sqr-neg 96.3%

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

      10. distribute-rgt-out 96.3%

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

      11. fma-define 96.3%

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

      12. +-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in m around inf 96.3%

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

   if 7.49999999999999934e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= m 7.5e-5) (/ t_0 (+ 1.0 (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 7.5e-5) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 7.5d-5) then
        tmp = t_0 / (1.0d0 + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 7.5e-5) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 7.5e-5:
		tmp = t_0 / (1.0 + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 7.5e-5)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 7.5e-5)
		tmp = t_0 / (1.0 + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 7.5e-5], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 7.49999999999999934e-5

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in k around 0 94.9%

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

   if 7.49999999999999934e-5 < m

   1. Initial program 82.5%

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

      1. associate-*l/ 79.4%

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

      2. sqr-neg 79.4%

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

      3. associate-+l+ 79.4%

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

      4. sqr-neg 79.4%

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

      5. distribute-rgt-out 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-8} \lor \neg \left(m \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -8.6e-8) (not (<= m 1.4e-8)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.6e-8) || !(m <= 1.4e-8)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-8.6d-8)) .or. (.not. (m <= 1.4d-8))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.6e-8) || !(m <= 1.4e-8)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -8.6e-8) or not (m <= 1.4e-8):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -8.6e-8) || !(m <= 1.4e-8))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -8.6e-8) || ~((m <= 1.4e-8)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -8.6e-8], N[Not[LessEqual[m, 1.4e-8]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.6000000000000002e-8 or 1.4e-8 < m

   1. Initial program 92.9%

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

      1. associate-*l/ 91.6%

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

      2. sqr-neg 91.6%

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

      3. associate-+l+ 91.6%

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

      4. sqr-neg 91.6%

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

      5. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in k around 0 98.7%

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

   if -8.6000000000000002e-8 < m < 1.4e-8

   1. Initial program 92.9%

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

      1. associate-*l/ 92.9%

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

      2. sqr-neg 92.9%

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

      3. associate-+l+ 92.9%

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

      4. sqr-neg 92.9%

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

      5. distribute-rgt-out 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in m around 0 92.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-8} \lor \neg \left(m \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\left(-1 - k \cdot \left(k + 10\right)\right) \cdot \frac{-1}{-a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.35e+16)
   (/ -1.0 (* (- -1.0 (* k (+ k 10.0))) (/ -1.0 (- a))))
   (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35e+16) {
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) * (-1.0 / -a));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.35d+16) then
        tmp = (-1.0d0) / (((-1.0d0) - (k * (k + 10.0d0))) * ((-1.0d0) / -a))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35e+16) {
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) * (-1.0 / -a));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.35e+16:
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) * (-1.0 / -a))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.35e+16)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 - Float64(k * Float64(k + 10.0))) * Float64(-1.0 / Float64(-a))));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.35e+16)
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) * (-1.0 / -a));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.35e+16], N[(-1.0 / N[(N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35e16

   1. Initial program 95.3%

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

      1. associate-*l/ 95.3%

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

      2. sqr-neg 95.3%

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

      3. associate-+l+ 95.3%

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

      4. sqr-neg 95.3%

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

      5. distribute-rgt-out 95.3%

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

   3. Simplified 95.3%

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

      1. +-commutative 95.3%

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

      2. +-commutative 95.3%

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

      3. fma-undefine 95.3%

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

      4. associate-/r/ 95.3%

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

      5. clear-num 95.2%

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

      6. frac-2neg 95.2%

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

      7. metadata-eval 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. associate-/l/ 95.2%

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

      2. distribute-neg-frac 95.2%

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

      3. neg-sub0 95.2%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 95.2%

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

      5. fma-undefine 95.2%

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

      6. +-commutative 95.2%

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

      7. associate--r+ 95.2%

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

      8. metadata-eval 95.2%

         \[\leadsto \frac{-1}{\frac{\left(\color{blue}{0} - 1\right) - k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}} \]

      9. metadata-eval 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in m around 0 63.7%

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

       1. frac-2neg 63.7%

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

       2. div-inv 63.7%

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

       3. +-commutative 63.7%

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

       4. distribute-neg-in 63.7%

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

       5. metadata-eval 63.7%

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

       6. sub-neg 63.7%

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

   11. Applied egg-rr 63.7%

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

   if 1.35e16 < m

   1. Initial program 85.0%

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

      1. associate-*l/ 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-+l+ 81.7%

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

      4. sqr-neg 81.7%

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

      5. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 11.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\left(-1 - k \cdot \left(k + 10\right)\right) \cdot \frac{-1}{-a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e+103)
   (/ 0.1 (/ k a))
   (if (<= m 1.06e+16) (/ a (+ 1.0 (* k 10.0))) (+ a (* -10.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e+103) {
		tmp = 0.1 / (k / a);
	} else if (m <= 1.06e+16) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.7d+103)) then
        tmp = 0.1d0 / (k / a)
    else if (m <= 1.06d+16) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e+103) {
		tmp = 0.1 / (k / a);
	} else if (m <= 1.06e+16) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.7e+103:
		tmp = 0.1 / (k / a)
	elif m <= 1.06e+16:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e+103)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (m <= 1.06e+16)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.7e+103)
		tmp = 0.1 / (k / a);
	elseif (m <= 1.06e+16)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e+103], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.06e+16], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.6999999999999999e103

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in k around inf 91.2%

      \[\leadsto \color{blue}{\left(0.1 \cdot \frac{a}{k}\right)} \cdot {k}^{m} \]

   9. Taylor expanded in m around 0 23.6%

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

       1. clear-num 24.6%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 24.6%

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

   11. Applied egg-rr 24.6%

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

   if -1.6999999999999999e103 < m < 1.06e16

   1. Initial program 93.4%

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

      1. associate-*l/ 93.4%

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

      2. sqr-neg 93.4%

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

      3. associate-+l+ 93.4%

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

      4. sqr-neg 93.4%

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

      5. distribute-rgt-out 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in m around 0 73.7%

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

   6. Taylor expanded in k around 0 47.0%

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

      1. *-commutative 69.6%

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

   8. Simplified 47.0%

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

   if 1.06e16 < m

   1. Initial program 85.0%

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

      1. associate-*l/ 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-+l+ 81.7%

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

      4. sqr-neg 81.7%

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

      5. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 11.7%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\frac{-1 - k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.7e+16)
   (/ -1.0 (/ (- -1.0 (* k (+ k 10.0))) a))
   (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.7e+16) {
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) / a);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.7d+16) then
        tmp = (-1.0d0) / (((-1.0d0) - (k * (k + 10.0d0))) / a)
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.7e+16) {
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) / a);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.7e+16:
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) / a)
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.7e+16)
		tmp = Float64(-1.0 / Float64(Float64(-1.0 - Float64(k * Float64(k + 10.0))) / a));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.7e+16)
		tmp = -1.0 / ((-1.0 - (k * (k + 10.0))) / a);
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.7e+16], N[(-1.0 / N[(N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.7e16

   1. Initial program 95.3%

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

      1. associate-*l/ 95.3%

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

      2. sqr-neg 95.3%

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

      3. associate-+l+ 95.3%

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

      4. sqr-neg 95.3%

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

      5. distribute-rgt-out 95.3%

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

   3. Simplified 95.3%

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

      1. +-commutative 95.3%

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

      2. +-commutative 95.3%

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

      3. fma-undefine 95.3%

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

      4. associate-/r/ 95.3%

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

      5. clear-num 95.2%

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

      6. frac-2neg 95.2%

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

      7. metadata-eval 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. associate-/l/ 95.2%

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

      2. distribute-neg-frac 95.2%

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

      3. neg-sub0 95.2%

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{fma}\left(k, k + 10, 1\right)}}{a \cdot {k}^{m}}} \]

      4. metadata-eval 95.2%

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

      5. fma-undefine 95.2%

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

      6. +-commutative 95.2%

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

      7. associate--r+ 95.2%

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

      8. metadata-eval 95.2%

         \[\leadsto \frac{-1}{\frac{\left(\color{blue}{0} - 1\right) - k \cdot \left(k + 10\right)}{a \cdot {k}^{m}}} \]

      9. metadata-eval 95.2%

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

   8. Simplified 95.2%

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

   9. Taylor expanded in m around 0 63.7%

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

   if 2.7e16 < m

   1. Initial program 85.0%

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

      1. associate-*l/ 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-+l+ 81.7%

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

      4. sqr-neg 81.7%

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

      5. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 11.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\frac{-1 - k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{+17}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1e+17) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1e+17) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1d+17) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1e+17) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1e+17:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1e+17)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1e+17)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1e+17], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1e17

   1. Initial program 95.3%

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

      1. associate-*l/ 95.3%

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

      2. sqr-neg 95.3%

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

      3. associate-+l+ 95.3%

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

      4. sqr-neg 95.3%

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

      5. distribute-rgt-out 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in m around 0 63.6%

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

   if 1e17 < m

   1. Initial program 85.0%

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

      1. associate-*l/ 81.7%

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

      2. sqr-neg 81.7%

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

      3. associate-+l+ 81.7%

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

      4. sqr-neg 81.7%

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

      5. distribute-rgt-out 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 11.7%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{+17}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (+ a (* -10.0 (* a k))) (/ 0.1 (/ k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = 0.1d0 / (k / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = 0.1 / (k / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(0.1 / Float64(k / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = 0.1 / (k / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   1. Initial program 97.5%

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

      1. associate-*l/ 97.5%

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

      2. sqr-neg 97.5%

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

      3. associate-+l+ 97.5%

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

      4. sqr-neg 97.5%

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

      5. distribute-rgt-out 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in m around 0 39.2%

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

   6. Taylor expanded in k around 0 33.2%

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

   if 0.0749999999999999972 < k

   1. Initial program 84.9%

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

      1. associate-*l/ 82.8%

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

      2. sqr-neg 82.8%

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

      3. associate-+l+ 82.8%

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

      4. sqr-neg 82.8%

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

      5. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in k around 0 51.5%

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

      1. *-commutative 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in k around inf 51.5%

      \[\leadsto \color{blue}{\left(0.1 \cdot \frac{a}{k}\right)} \cdot {k}^{m} \]

   9. Taylor expanded in m around 0 23.4%

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

       1. clear-num 24.2%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 24.2%

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

   11. Applied egg-rr 24.2%

       \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k 0.1) a (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = a;
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.1d0) then
        tmp = a
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 97.5%

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

      1. associate-*l/ 97.5%

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

      2. sqr-neg 97.5%

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

      3. associate-+l+ 97.5%

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

      4. sqr-neg 97.5%

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

      5. distribute-rgt-out 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in m around 0 39.2%

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

   6. Taylor expanded in k around 0 30.3%

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

   if 0.10000000000000001 < k

   1. Initial program 84.9%

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

      1. associate-*l/ 82.8%

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

      2. sqr-neg 82.8%

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

      3. associate-+l+ 82.8%

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

      4. sqr-neg 82.8%

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

      5. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in k around 0 51.5%

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

      1. *-commutative 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in k around inf 51.5%

      \[\leadsto \color{blue}{\left(0.1 \cdot \frac{a}{k}\right)} \cdot {k}^{m} \]

   9. Taylor expanded in m around 0 23.4%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k 0.1) a (/ 0.1 (/ k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = a;
	} else {
		tmp = 0.1 / (k / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.1d0) then
        tmp = a
    else
        tmp = 0.1d0 / (k / a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.1:
		tmp = a
	else:
		tmp = 0.1 / (k / a)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.1)
		tmp = a;
	else
		tmp = 0.1 / (k / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.1], a, N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 97.5%

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

      1. associate-*l/ 97.5%

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

      2. sqr-neg 97.5%

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

      3. associate-+l+ 97.5%

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

      4. sqr-neg 97.5%

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

      5. distribute-rgt-out 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in m around 0 39.2%

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

   6. Taylor expanded in k around 0 30.3%

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

   if 0.10000000000000001 < k

   1. Initial program 84.9%

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

      1. associate-*l/ 82.8%

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

      2. sqr-neg 82.8%

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

      3. associate-+l+ 82.8%

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

      4. sqr-neg 82.8%

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

      5. distribute-rgt-out 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in k around 0 51.5%

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

      1. *-commutative 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in k around inf 51.5%

      \[\leadsto \color{blue}{\left(0.1 \cdot \frac{a}{k}\right)} \cdot {k}^{m} \]

   9. Taylor expanded in m around 0 23.4%

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

       1. clear-num 24.2%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       2. un-div-inv 24.2%

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

   11. Applied egg-rr 24.2%

       \[\leadsto \color{blue}{\frac{0.1}{\frac{k}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.7% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 92.9%

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

   1. associate-*l/ 92.1%

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

   2. sqr-neg 92.1%

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

   3. associate-+l+ 92.1%

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

   4. sqr-neg 92.1%

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

   5. distribute-rgt-out 92.1%

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

3. Simplified 92.1%

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

5. Taylor expanded in m around 0 49.6%

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

6. Taylor expanded in k around 0 20.7%

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

7. Final simplification 20.7%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))