Falkner and Boettcher, Appendix A

Percentage Accurate: 89.6% → 99.4%

Time: 9.0s

Alternatives: 11

Speedup: 1.1×

• 24.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;m \leq -8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{elif}\;m \leq 0.00052:\\ \;\;\;\;\frac{\frac{a}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= m -8e-25)
     (/ (/ a (fma k (+ k 10.0) 1.0)) (pow (/ 1.0 k) m))
     (if (<= m 0.00052) (/ (/ a t_0) t_0) (* a (pow k m))))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (m <= -8e-25) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / pow((1.0 / k), m);
	} else if (m <= 0.00052) {
		tmp = (a / t_0) / t_0;
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (m <= -8e-25)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / (Float64(1.0 / k) ^ m));
	elseif (m <= 0.00052)
		tmp = Float64(Float64(a / t_0) / t_0);
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[m, -8e-25], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 / k), $MachinePrecision], m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00052], N[(N[(a / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.00000000000000031e-25

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 57.3%

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

      1. expm1-log1p-u 42.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)\right)} \]

      2. expm1-udef 40.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)} - 1} \]

   6. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 77.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

   8. Simplified 100.0%

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

   if -8.00000000000000031e-25 < m < 5.19999999999999954e-4

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. associate-+l+ 93.0%

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

      3. sqr-neg 93.0%

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

      4. distribute-rgt-out 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in m around 0 93.0%

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

      1. *-un-lft-identity 93.0%

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

      2. add-sqr-sqrt 92.9%

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

      3. times-frac 92.9%

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

      4. +-commutative 92.9%

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

      5. +-commutative 92.9%

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

      6. fma-udef 92.9%

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

      7. +-commutative 92.9%

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

      8. +-commutative 92.9%

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

      9. fma-udef 92.9%

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

   6. Applied egg-rr 92.9%

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

      1. associate-*l/ 92.9%

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

      2. *-lft-identity 92.9%

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

      3. fma-def 92.9%

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

      4. +-commutative 92.9%

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

      5. distribute-rgt-in 92.9%

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

      6. unpow2 92.9%

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

      7. +-commutative 92.9%

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

      8. associate-+r+ 92.9%

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

      9. +-commutative 92.9%

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

      10. *-commutative 92.9%

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

      11. fma-udef 92.9%

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

      12. +-commutative 92.9%

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

      13. unpow2 92.9%

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

      14. rem-square-sqrt 92.9%

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

      15. hypot-def 92.9%

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

      16. fma-def 92.9%

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

      17. +-commutative 92.9%

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

      18. distribute-rgt-in 92.9%

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

      19. unpow2 92.9%

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

      20. +-commutative 92.9%

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

   8. Simplified 99.8%

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

   if 5.19999999999999954e-4 < m

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. associate-+l+ 82.6%

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

      3. sqr-neg 82.6%

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

      4. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{elif}\;m \leq 0.00052:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.7)
   (/ (/ a (fma k (+ k 10.0) 1.0)) (pow (/ 1.0 k) m))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.7) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / pow((1.0 / k), m);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.7)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / (Float64(1.0 / k) ^ m));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.7], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 / k), $MachinePrecision], m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7000000000000002

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in k around inf 75.1%

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

      1. expm1-log1p-u 61.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)\right)} \]

      2. expm1-udef 42.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)} - 1} \]

   6. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 78.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)\right)} \]

      2. expm1-log1p 96.5%

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

      3. associate-/r* 96.5%

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

   8. Simplified 96.5%

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

   if 3.7000000000000002 < m

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. associate-+l+ 82.6%

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

      3. sqr-neg 82.6%

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

      4. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.5)
   (* a (/ (- (pow k m)) (- -1.0 (* k (+ k 10.0)))))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.5) {
		tmp = a * (-pow(k, m) / (-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 <= 4.5d0) then
        tmp = a * (-(k ** m) / ((-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 <= 4.5) {
		tmp = a * (-Math.pow(k, m) / (-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 <= 4.5:
		tmp = a * (-math.pow(k, m) / (-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 <= 4.5)
		tmp = Float64(a * Float64(Float64(-(k ^ m)) / 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 <= 4.5)
		tmp = a * (-(k ^ m) / (-1.0 - (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. frac-2neg 96.5%

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

      2. div-inv 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

      4. +-commutative 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. associate-*l* 96.5%

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

      2. associate-*r/ 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. fma-udef 96.5%

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

      5. +-commutative 96.5%

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

      6. distribute-neg-in 96.5%

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

      7. metadata-eval 96.5%

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

      8. sub-neg 96.5%

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

   7. Simplified 96.5%

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

   if 4.5 < m

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. associate-+l+ 82.6%

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

      3. sqr-neg 82.6%

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

      4. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 3.6:\\ \;\;\;\;\frac{t_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 3.6) (/ t_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 <= 3.6) {
		tmp = t_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 <= 3.6d0) then
        tmp = t_0 / (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 <= 3.6) {
		tmp = t_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 <= 3.6:
		tmp = t_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 <= 3.6)
		tmp = Float64(t_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 <= 3.6)
		tmp = t_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, 3.6], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.60000000000000009

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   if 3.60000000000000009 < m

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. associate-+l+ 82.6%

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

      3. sqr-neg 82.6%

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

      4. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-12} \lor \neg \left(m \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -6e-12) (not (<= m 1.05e-10)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -6e-12) || !(m <= 1.05e-10)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-6d-12)) .or. (.not. (m <= 1.05d-10))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -6e-12) || !(m <= 1.05e-10)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -6e-12) or not (m <= 1.05e-10):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -6e-12) || !(m <= 1.05e-10))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -6e-12) || ~((m <= 1.05e-10)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -6e-12], N[Not[LessEqual[m, 1.05e-10]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.0000000000000003e-12 or 1.05e-10 < m

   1. Initial program 90.7%

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

      1. sqr-neg 90.7%

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

      2. associate-+l+ 90.7%

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

      3. sqr-neg 90.7%

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

      4. distribute-rgt-out 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -6.0000000000000003e-12 < m < 1.05e-10

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. associate-+l+ 93.0%

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

      3. sqr-neg 93.0%

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

      4. distribute-rgt-out 93.0%

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

   3. Simplified 93.0%

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

      1. frac-2neg 93.0%

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

      2. div-inv 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

      4. +-commutative 93.1%

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

      5. fma-def 93.1%

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

      6. +-commutative 93.1%

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

   5. Applied egg-rr 93.1%

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

      1. associate-*l* 93.1%

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

      2. associate-*r/ 93.1%

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

      3. *-rgt-identity 93.1%

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

      4. fma-udef 93.1%

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

      5. +-commutative 93.1%

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

      6. distribute-neg-in 93.1%

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

      7. metadata-eval 93.1%

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

      8. sub-neg 93.1%

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

   7. Simplified 93.1%

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

   8. Taylor expanded in m around 0 93.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-12} \lor \neg \left(m \leq 1.05 \cdot 10^{-10}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-12)
   (/ a (pow (/ 1.0 k) m))
   (if (<= m 1.05e-8) (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0))))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-12) {
		tmp = a / pow((1.0 / k), m);
	} else if (m <= 1.05e-8) {
		tmp = a * (1.0 / (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 <= (-5.4d-12)) then
        tmp = a / ((1.0d0 / k) ** m)
    else if (m <= 1.05d-8) then
        tmp = a * (1.0d0 / (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 <= -5.4e-12) {
		tmp = a / Math.pow((1.0 / k), m);
	} else if (m <= 1.05e-8) {
		tmp = a * (1.0 / (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 <= -5.4e-12:
		tmp = a / math.pow((1.0 / k), m)
	elif m <= 1.05e-8:
		tmp = a * (1.0 / (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 <= -5.4e-12)
		tmp = Float64(a / (Float64(1.0 / k) ^ m));
	elseif (m <= 1.05e-8)
		tmp = Float64(a * Float64(1.0 / 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 <= -5.4e-12)
		tmp = a / ((1.0 / k) ^ m);
	elseif (m <= 1.05e-8)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e-12], N[(a / N[Power[N[(1.0 / k), $MachinePrecision], m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05e-8], N[(a * N[(1.0 / 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 3 regimes

2. if m < -5.39999999999999961e-12

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-neg 100.0%

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

      4. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around inf 56.8%

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

      1. expm1-log1p-u 41.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)\right)} \]

      2. expm1-udef 40.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{1 + k \cdot \left(10 + k\right)}\right)} - 1} \]

   6. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 77.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right) \cdot {\left(\frac{1}{k}\right)}^{m}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in k around 0 100.0%

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

   if -5.39999999999999961e-12 < m < 1.04999999999999997e-8

   1. Initial program 93.0%

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

      1. sqr-neg 93.0%

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

      2. associate-+l+ 93.0%

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

      3. sqr-neg 93.0%

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

      4. distribute-rgt-out 93.0%

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

   3. Simplified 93.0%

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

      1. frac-2neg 93.0%

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

      2. div-inv 93.1%

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

      3. distribute-rgt-neg-in 93.1%

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

      4. +-commutative 93.1%

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

      5. fma-def 93.1%

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

      6. +-commutative 93.1%

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

   5. Applied egg-rr 93.1%

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

      1. associate-*l* 93.1%

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

      2. associate-*r/ 93.1%

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

      3. *-rgt-identity 93.1%

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

      4. fma-udef 93.1%

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

      5. +-commutative 93.1%

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

      6. distribute-neg-in 93.1%

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

      7. metadata-eval 93.1%

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

      8. sub-neg 93.1%

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

   7. Simplified 93.1%

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

   8. Taylor expanded in m around 0 93.0%

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

   if 1.04999999999999997e-8 < m

   1. Initial program 82.6%

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

      1. sqr-neg 82.6%

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

      2. associate-+l+ 82.6%

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

      3. sqr-neg 82.6%

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

      4. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{{\left(\frac{1}{k}\right)}^{m}}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 7: 49.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 31000000000:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 31000000000.0)
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))
   (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 31000000000.0) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = -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 <= 31000000000.0d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = (-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 <= 31000000000.0) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 31000000000.0:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 31000000000.0)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 31000000000.0)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 31000000000.0], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.1e10

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. frac-2neg 96.5%

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

      2. div-inv 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

      4. +-commutative 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. associate-*l* 96.5%

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

      2. associate-*r/ 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. fma-udef 96.5%

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

      5. +-commutative 96.5%

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

      6. distribute-neg-in 96.5%

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

      7. metadata-eval 96.5%

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

      8. sub-neg 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in m around 0 65.0%

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

   if 3.1e10 < m

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. associate-+l+ 82.4%

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

      3. sqr-neg 82.4%

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

      4. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 31000000000:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 49.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 600000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 600000000.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 600000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -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 <= 600000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-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 <= 600000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 600000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 600000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 600000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 600000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6e8

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 65.0%

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

   if 6e8 < m

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. associate-+l+ 82.4%

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

      3. sqr-neg 82.4%

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

      4. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 600000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 9: 33.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 680000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 680000000.0) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 680000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -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 <= 680000000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-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 <= 680000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 680000000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 680000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 680000000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 680000000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.8e8

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 65.0%

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

   5. Taylor expanded in k around 0 41.7%

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

      1. *-commutative 41.7%

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

   7. Simplified 41.7%

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

   if 6.8e8 < m

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. associate-+l+ 82.4%

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

      3. sqr-neg 82.4%

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

      4. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 680000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 10: 25.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1250000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1250000000.0) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1250000000.0) {
		tmp = a;
	} else {
		tmp = -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 <= 1250000000.0d0) then
        tmp = a
    else
        tmp = (-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 <= 1250000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1250000000.0:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1250000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1250000000.0)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1250000000.0], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.25e9

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 65.0%

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

   5. Taylor expanded in k around 0 26.6%

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

   if 1.25e9 < m

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. associate-+l+ 82.4%

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

      3. sqr-neg 82.4%

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

      4. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1250000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 19.8% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.5%

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

   1. sqr-neg 91.5%

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

   2. associate-+l+ 91.5%

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

   3. sqr-neg 91.5%

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

   4. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

4. Taylor expanded in m around 0 42.9%

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

5. Taylor expanded in k around 0 18.3%

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

6. Final simplification 18.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023333 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))