Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 98.5%

Time: 19.2s

Alternatives: 16

Speedup: 1.0×

• 21.6% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -8.3 \cdot 10^{-41}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{elif}\;m \leq 2.25 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{\sqrt{a\_m}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -8.3e-41)
    (* a_m (/ (pow k m) (+ 1.0 (* k (+ 10.0 k)))))
    (if (<= m 2.25e-5)
      (pow (/ (sqrt a_m) (hypot 1.0 k)) 2.0)
      (/ a_m (pow k (- m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -8.3e-41) {
		tmp = a_m * (pow(k, m) / (1.0 + (k * (10.0 + k))));
	} else if (m <= 2.25e-5) {
		tmp = pow((sqrt(a_m) / hypot(1.0, k)), 2.0);
	} else {
		tmp = a_m / pow(k, -m);
	}
	return a_s * tmp;
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -8.3e-41) {
		tmp = a_m * (Math.pow(k, m) / (1.0 + (k * (10.0 + k))));
	} else if (m <= 2.25e-5) {
		tmp = Math.pow((Math.sqrt(a_m) / Math.hypot(1.0, k)), 2.0);
	} else {
		tmp = a_m / Math.pow(k, -m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -8.3e-41:
		tmp = a_m * (math.pow(k, m) / (1.0 + (k * (10.0 + k))))
	elif m <= 2.25e-5:
		tmp = math.pow((math.sqrt(a_m) / math.hypot(1.0, k)), 2.0)
	else:
		tmp = a_m / math.pow(k, -m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -8.3e-41)
		tmp = Float64(a_m * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(10.0 + k)))));
	elseif (m <= 2.25e-5)
		tmp = Float64(sqrt(a_m) / hypot(1.0, k)) ^ 2.0;
	else
		tmp = Float64(a_m / (k ^ Float64(-m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -8.3e-41)
		tmp = a_m * ((k ^ m) / (1.0 + (k * (10.0 + k))));
	elseif (m <= 2.25e-5)
		tmp = (sqrt(a_m) / hypot(1.0, k)) ^ 2.0;
	else
		tmp = a_m / (k ^ -m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -8.3e-41], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.25e-5], N[Power[N[(N[Sqrt[a$95$m], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -8.3000000000000004e-41

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   if -8.3000000000000004e-41 < m < 2.25000000000000014e-5

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in k around 0 87.9%

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

      1. add-sqr-sqrt 56.0%

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

      2. pow2 56.0%

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

      3. sqrt-div 43.0%

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

      4. hypot-1-def 47.2%

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

   7. Applied egg-rr 47.2%

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

   8. Taylor expanded in m around 0 47.2%

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

   if 2.25000000000000014e-5 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot {\left(\frac{1}{\mathsf{hypot}\left(1, k\right)} \cdot \sqrt{a\_m \cdot {k}^{m}}\right)}^{2} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (pow (* (/ 1.0 (hypot 1.0 k)) (sqrt (* a_m (pow k m)))) 2.0)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * pow(((1.0 / hypot(1.0, k)) * sqrt((a_m * pow(k, m)))), 2.0);
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * Math.pow(((1.0 / Math.hypot(1.0, k)) * Math.sqrt((a_m * Math.pow(k, m)))), 2.0);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * math.pow(((1.0 / math.hypot(1.0, k)) * math.sqrt((a_m * math.pow(k, m)))), 2.0)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * (Float64(Float64(1.0 / hypot(1.0, k)) * sqrt(Float64(a_m * (k ^ m)))) ^ 2.0))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (((1.0 / hypot(1.0, k)) * sqrt((a_m * (k ^ m)))) ^ 2.0);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[Power[N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. *-commutative 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in k around 0 86.3%

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

   1. add-sqr-sqrt 63.1%

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

   2. pow2 63.1%

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

   3. sqrt-div 57.9%

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

   4. hypot-1-def 62.3%

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

7. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2}} \]
8. Step-by-step derivation

   1. clear-num 62.3%

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

   2. associate-/r/ 62.3%

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

9. Applied egg-rr 62.3%

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

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot {\left(\frac{\sqrt{a\_m \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (pow (/ (sqrt (* a_m (pow k m))) (hypot 1.0 k)) 2.0)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * pow((sqrt((a_m * pow(k, m))) / hypot(1.0, k)), 2.0);
}

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * Math.pow((Math.sqrt((a_m * Math.pow(k, m))) / Math.hypot(1.0, k)), 2.0);
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * math.pow((math.sqrt((a_m * math.pow(k, m))) / math.hypot(1.0, k)), 2.0)

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * (Float64(sqrt(Float64(a_m * (k ^ m))) / hypot(1.0, k)) ^ 2.0))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * ((sqrt((a_m * (k ^ m))) / hypot(1.0, k)) ^ 2.0);
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[Power[N[(N[Sqrt[N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. *-commutative 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in k around 0 86.3%

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

   1. add-sqr-sqrt 63.1%

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

   2. pow2 63.1%

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

   3. sqrt-div 57.9%

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

   4. hypot-1-def 62.3%

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

7. Applied egg-rr 62.3%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2}} \]
8. Add Preprocessing

Alternative 4: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.65e-5)
    (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
    (/ a_m (pow k (- m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = a_m / pow(k, -m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.65d-5) then
        tmp = (a_m * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    else
        tmp = a_m / (k ** -m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = a_m / Math.pow(k, -m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 2.65e-5:
		tmp = (a_m * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	else:
		tmp = a_m / math.pow(k, -m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.65e-5)
		tmp = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)));
	else
		tmp = Float64(a_m / (k ^ Float64(-m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 2.65e-5)
		tmp = (a_m * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	else
		tmp = a_m / (k ^ -m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.65e-5], N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.65e-5

   1. Initial program 94.6%

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

   if 2.65e-5 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{\left(-m\right)}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{a\_m}{k \cdot \left(k + 10\right) + 1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (pow k (- m))))
   (*
    a_s
    (if (<= m 2.65e-5) (/ (/ a_m (+ (* k (+ k 10.0)) 1.0)) t_0) (/ a_m t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, -m);
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m / ((k * (k + 10.0)) + 1.0)) / t_0;
	} else {
		tmp = a_m / t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k ** -m
    if (m <= 2.65d-5) then
        tmp = (a_m / ((k * (k + 10.0d0)) + 1.0d0)) / t_0
    else
        tmp = a_m / t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = Math.pow(k, -m);
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m / ((k * (k + 10.0)) + 1.0)) / t_0;
	} else {
		tmp = a_m / t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = math.pow(k, -m)
	tmp = 0
	if m <= 2.65e-5:
		tmp = (a_m / ((k * (k + 10.0)) + 1.0)) / t_0
	else:
		tmp = a_m / t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = k ^ Float64(-m)
	tmp = 0.0
	if (m <= 2.65e-5)
		tmp = Float64(Float64(a_m / Float64(Float64(k * Float64(k + 10.0)) + 1.0)) / t_0);
	else
		tmp = Float64(a_m / t_0);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = k ^ -m;
	tmp = 0.0;
	if (m <= 2.65e-5)
		tmp = (a_m / ((k * (k + 10.0)) + 1.0)) / t_0;
	else
		tmp = a_m / t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[Power[k, (-m)], $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, 2.65e-5], N[(N[(a$95$m / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(a$95$m / t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.65e-5

   1. Initial program 94.6%

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

      1. associate-/l* 94.6%

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

      2. remove-double-neg 94.6%

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

      3. distribute-frac-neg2 94.6%

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

      4. distribute-neg-frac2 94.6%

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

      5. remove-double-neg 94.6%

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

      6. sqr-neg 94.6%

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

      7. associate-+l+ 94.6%

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

      8. sqr-neg 94.6%

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

      9. distribute-rgt-out 94.6%

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

   3. Simplified 94.6%

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

      1. flip-+ 62.4%

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

      2. associate-/r/ 62.4%

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

      3. metadata-eval 62.4%

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

      4. pow2 62.4%

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

      5. +-commutative 62.4%

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

      6. +-commutative 62.4%

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

   6. Applied egg-rr 62.4%

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

      1. associate-/r/ 62.4%

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

      2. metadata-eval 62.4%

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

      3. unpow2 62.4%

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

      4. flip-+ 94.6%

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

      5. +-commutative 94.6%

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

      6. fma-undefine 94.6%

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

      7. clear-num 94.6%

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

      8. div-inv 94.6%

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

      9. div-inv 94.6%

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

      10. associate-/r* 94.6%

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

      11. pow-flip 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. fma-undefine 94.6%

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

   10. Applied egg-rr 94.6%

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

   if 2.65e-5 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.65e-5)
    (* a_m (/ (pow k m) (+ 1.0 (* k (+ 10.0 k)))))
    (/ a_m (pow k (- m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = a_m * (pow(k, m) / (1.0 + (k * (10.0 + k))));
	} else {
		tmp = a_m / pow(k, -m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.65d-5) then
        tmp = a_m * ((k ** m) / (1.0d0 + (k * (10.0d0 + k))))
    else
        tmp = a_m / (k ** -m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = a_m * (Math.pow(k, m) / (1.0 + (k * (10.0 + k))));
	} else {
		tmp = a_m / Math.pow(k, -m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 2.65e-5:
		tmp = a_m * (math.pow(k, m) / (1.0 + (k * (10.0 + k))))
	else:
		tmp = a_m / math.pow(k, -m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.65e-5)
		tmp = Float64(a_m * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(10.0 + k)))));
	else
		tmp = Float64(a_m / (k ^ Float64(-m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 2.65e-5)
		tmp = a_m * ((k ^ m) / (1.0 + (k * (10.0 + k))));
	else
		tmp = a_m / (k ^ -m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.65e-5], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.65e-5

   1. Initial program 94.6%

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

      1. associate-/l* 94.6%

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

      2. remove-double-neg 94.6%

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

      3. distribute-frac-neg2 94.6%

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

      4. distribute-neg-frac2 94.6%

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

      5. remove-double-neg 94.6%

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

      6. sqr-neg 94.6%

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

      7. associate-+l+ 94.6%

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

      8. sqr-neg 94.6%

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

      9. distribute-rgt-out 94.6%

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

   3. Simplified 94.6%

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

   if 2.65e-5 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 7: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 3.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{a\_m}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -2.7e-5)
    (* a_m (pow k m))
    (if (<= m 3.7e-6)
      (/ a_m (+ (+ 1.0 (* k 10.0)) (* k k)))
      (/ a_m (pow k (- m)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -2.7e-5) {
		tmp = a_m * pow(k, m);
	} else if (m <= 3.7e-6) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a_m / pow(k, -m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-2.7d-5)) then
        tmp = a_m * (k ** m)
    else if (m <= 3.7d-6) then
        tmp = a_m / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = a_m / (k ** -m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -2.7e-5) {
		tmp = a_m * Math.pow(k, m);
	} else if (m <= 3.7e-6) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a_m / Math.pow(k, -m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -2.7e-5:
		tmp = a_m * math.pow(k, m)
	elif m <= 3.7e-6:
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = a_m / math.pow(k, -m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -2.7e-5)
		tmp = Float64(a_m * (k ^ m));
	elseif (m <= 3.7e-6)
		tmp = Float64(a_m / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(a_m / (k ^ Float64(-m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -2.7e-5)
		tmp = a_m * (k ^ m);
	elseif (m <= 3.7e-6)
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = a_m / (k ^ -m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -2.7e-5], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.7e-6], N[(a$95$m / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.6999999999999999e-5

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -2.6999999999999999e-5 < m < 3.7000000000000002e-6

   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. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in m around 0 89.7%

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

   if 3.7000000000000002e-6 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 8: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\frac{a\_m \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.65e-5)
    (/ (* a_m (pow k m)) (+ 1.0 (* k k)))
    (/ a_m (pow k (- m))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m * pow(k, m)) / (1.0 + (k * k));
	} else {
		tmp = a_m / pow(k, -m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.65d-5) then
        tmp = (a_m * (k ** m)) / (1.0d0 + (k * k))
    else
        tmp = a_m / (k ** -m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.65e-5) {
		tmp = (a_m * Math.pow(k, m)) / (1.0 + (k * k));
	} else {
		tmp = a_m / Math.pow(k, -m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 2.65e-5:
		tmp = (a_m * math.pow(k, m)) / (1.0 + (k * k))
	else:
		tmp = a_m / math.pow(k, -m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.65e-5)
		tmp = Float64(Float64(a_m * (k ^ m)) / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a_m / (k ^ Float64(-m)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 2.65e-5)
		tmp = (a_m * (k ^ m)) / (1.0 + (k * k));
	else
		tmp = a_m / (k ^ -m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.65e-5], N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.65e-5

   1. Initial program 94.6%

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

      1. *-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in k around 0 93.1%

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

   if 2.65e-5 < m

   1. Initial program 67.1%

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

      1. associate-/l* 67.1%

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

      2. remove-double-neg 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-neg-frac2 67.1%

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

      5. remove-double-neg 67.1%

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

      6. sqr-neg 67.1%

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

      7. associate-+l+ 67.1%

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

      8. sqr-neg 67.1%

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

      9. distribute-rgt-out 67.1%

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

   3. Simplified 67.1%

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

      1. flip-+ 59.7%

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

      2. associate-/r/ 59.7%

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

      3. metadata-eval 59.7%

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

      4. pow2 59.7%

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

      5. +-commutative 59.7%

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

      6. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. associate-/r/ 59.7%

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

      2. metadata-eval 59.7%

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

      3. unpow2 59.7%

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

      4. flip-+ 67.1%

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

      5. +-commutative 67.1%

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

      6. fma-undefine 67.1%

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

      7. clear-num 67.1%

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

      8. div-inv 67.2%

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

      9. div-inv 67.2%

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

      10. associate-/r* 59.7%

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

      11. pow-flip 59.7%

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

   8. Applied egg-rr 59.7%

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

   9. Taylor expanded in k around 0 100.0%

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

Alternative 9: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{a\_m}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= m -0.9)
      t_0
      (if (<= m 2.55e-6) (/ a_m (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= -0.9) {
		tmp = t_0;
	} else if (m <= 2.55e-6) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if (m <= (-0.9d0)) then
        tmp = t_0
    else if (m <= 2.55d-6) then
        tmp = a_m / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if (m <= -0.9) {
		tmp = t_0;
	} else if (m <= 2.55e-6) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if m <= -0.9:
		tmp = t_0
	elif m <= 2.55e-6:
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= -0.9)
		tmp = t_0;
	elseif (m <= 2.55e-6)
		tmp = Float64(a_m / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if (m <= -0.9)
		tmp = t_0;
	elseif (m <= 2.55e-6)
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -0.9], t$95$0, If[LessEqual[m, 2.55e-6], N[(a$95$m / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.900000000000000022 or 2.5500000000000001e-6 < 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* 85.0%

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

      2. remove-double-neg 85.0%

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

      3. distribute-frac-neg2 85.0%

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

      4. distribute-neg-frac2 85.0%

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

      5. remove-double-neg 85.0%

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

      6. sqr-neg 85.0%

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

      7. associate-+l+ 85.0%

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

      8. sqr-neg 85.0%

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

      9. distribute-rgt-out 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -0.900000000000000022 < m < 2.5500000000000001e-6

   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. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in m around 0 89.7%

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

Alternative 10: 56.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.7 \cdot 10^{+76}:\\ \;\;\;\;\frac{a\_m}{10 \cdot k + k \cdot k}\\ \mathbf{elif}\;m \leq 1.75:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(10 + k\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot \left(99 \cdot k - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -3.7e+76)
    (/ a_m (+ (* 10.0 k) (* k k)))
    (if (<= m 1.75)
      (/ a_m (+ 1.0 (* k (+ 10.0 k))))
      (* a_m (+ 1.0 (* k (- (* 99.0 k) 10.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -3.7e+76) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else if (m <= 1.75) {
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	} else {
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-3.7d+76)) then
        tmp = a_m / ((10.0d0 * k) + (k * k))
    else if (m <= 1.75d0) then
        tmp = a_m / (1.0d0 + (k * (10.0d0 + k)))
    else
        tmp = a_m * (1.0d0 + (k * ((99.0d0 * k) - 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -3.7e+76) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else if (m <= 1.75) {
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	} else {
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -3.7e+76:
		tmp = a_m / ((10.0 * k) + (k * k))
	elif m <= 1.75:
		tmp = a_m / (1.0 + (k * (10.0 + k)))
	else:
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -3.7e+76)
		tmp = Float64(a_m / Float64(Float64(10.0 * k) + Float64(k * k)));
	elseif (m <= 1.75)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(10.0 + k))));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -3.7e+76)
		tmp = a_m / ((10.0 * k) + (k * k));
	elseif (m <= 1.75)
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	else
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -3.7e+76], N[(a$95$m / N[(N[(10.0 * k), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.75], N[(a$95$m / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -3.6999999999999999e76

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 46.8%

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

   6. Taylor expanded in k around inf 55.4%

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

   if -3.6999999999999999e76 < m < 1.75

   1. Initial program 91.9%

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

      1. associate-/l* 91.8%

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

      2. remove-double-neg 91.8%

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

      3. distribute-frac-neg2 91.8%

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

      4. distribute-neg-frac2 91.8%

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

      5. remove-double-neg 91.8%

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

      6. sqr-neg 91.8%

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

      7. associate-+l+ 91.8%

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

      8. sqr-neg 91.8%

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

      9. distribute-rgt-out 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in m around 0 84.2%

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

   if 1.75 < m

   1. Initial program 66.7%

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

      1. associate-/l* 66.7%

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

      2. remove-double-neg 66.7%

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

      3. distribute-frac-neg2 66.7%

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

      4. distribute-neg-frac2 66.7%

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

      5. remove-double-neg 66.7%

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

      6. sqr-neg 66.7%

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

      7. associate-+l+ 66.7%

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

      8. sqr-neg 66.7%

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

      9. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in m around 0 2.8%

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

   6. Taylor expanded in k around 0 37.3%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\frac{a\_m}{10 \cdot k + k \cdot k}\\ \mathbf{elif}\;m \leq 2.1:\\ \;\;\;\;\frac{a\_m}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot \left(99 \cdot k - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -4e+76)
    (/ a_m (+ (* 10.0 k) (* k k)))
    (if (<= m 2.1)
      (/ a_m (+ (+ 1.0 (* k 10.0)) (* k k)))
      (* a_m (+ 1.0 (* k (- (* 99.0 k) 10.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4e+76) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else if (m <= 2.1) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4d+76)) then
        tmp = a_m / ((10.0d0 * k) + (k * k))
    else if (m <= 2.1d0) then
        tmp = a_m / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = a_m * (1.0d0 + (k * ((99.0d0 * k) - 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4e+76) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else if (m <= 2.1) {
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -4e+76:
		tmp = a_m / ((10.0 * k) + (k * k))
	elif m <= 2.1:
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -4e+76)
		tmp = Float64(a_m / Float64(Float64(10.0 * k) + Float64(k * k)));
	elseif (m <= 2.1)
		tmp = Float64(a_m / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -4e+76)
		tmp = a_m / ((10.0 * k) + (k * k));
	elseif (m <= 2.1)
		tmp = a_m / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = a_m * (1.0 + (k * ((99.0 * k) - 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -4e+76], N[(a$95$m / N[(N[(10.0 * k), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.1], N[(a$95$m / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.0000000000000002e76

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 46.8%

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

   6. Taylor expanded in k around inf 55.4%

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

   if -4.0000000000000002e76 < m < 2.10000000000000009

   1. Initial program 91.9%

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

      1. *-commutative 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in m around 0 84.2%

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

   if 2.10000000000000009 < m

   1. Initial program 66.7%

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

      1. associate-/l* 66.7%

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

      2. remove-double-neg 66.7%

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

      3. distribute-frac-neg2 66.7%

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

      4. distribute-neg-frac2 66.7%

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

      5. remove-double-neg 66.7%

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

      6. sqr-neg 66.7%

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

      7. associate-+l+ 66.7%

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

      8. sqr-neg 66.7%

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

      9. distribute-rgt-out 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in m around 0 2.8%

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

   6. Taylor expanded in k around 0 37.3%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 47.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{+77}:\\ \;\;\;\;\frac{a\_m}{10 \cdot k + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(10 + k\right)}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -6e+77)
    (/ a_m (+ (* 10.0 k) (* k k)))
    (/ a_m (+ 1.0 (* k (+ 10.0 k)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -6e+77) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else {
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-6d+77)) then
        tmp = a_m / ((10.0d0 * k) + (k * k))
    else
        tmp = a_m / (1.0d0 + (k * (10.0d0 + k)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -6e+77) {
		tmp = a_m / ((10.0 * k) + (k * k));
	} else {
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -6e+77:
		tmp = a_m / ((10.0 * k) + (k * k))
	else:
		tmp = a_m / (1.0 + (k * (10.0 + k)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -6e+77)
		tmp = Float64(a_m / Float64(Float64(10.0 * k) + Float64(k * k)));
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(10.0 + k))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -6e+77)
		tmp = a_m / ((10.0 * k) + (k * k));
	else
		tmp = a_m / (1.0 + (k * (10.0 + k)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -6e+77], N[(a$95$m / N[(N[(10.0 * k), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -5.9999999999999996e77

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 46.8%

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

   6. Taylor expanded in k around inf 55.4%

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

   if -5.9999999999999996e77 < m

   1. Initial program 83.2%

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

      1. associate-/l* 83.1%

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

      2. remove-double-neg 83.1%

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

      3. distribute-frac-neg2 83.1%

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

      4. distribute-neg-frac2 83.1%

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

      5. remove-double-neg 83.1%

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

      6. sqr-neg 83.1%

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

      7. associate-+l+ 83.1%

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

      8. sqr-neg 83.1%

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

      9. distribute-rgt-out 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in m around 0 56.1%

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

Alternative 13: 45.5% accurate, 12.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \frac{a\_m}{1 + k \cdot \left(10 + k\right)} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (/ a_m (+ 1.0 (* k (+ 10.0 k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (k * (10.0 + k))));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * (a_m / (1.0d0 + (k * (10.0d0 + k))))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (k * (10.0 + k))));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (a_m / (1.0 + (k * (10.0 + k))))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m / Float64(1.0 + Float64(k * Float64(10.0 + k)))))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (a_m / (1.0 + (k * (10.0 + k))));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(a$95$m / N[(1.0 + N[(k * N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. associate-/l* 87.4%

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

   2. remove-double-neg 87.4%

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

   3. distribute-frac-neg2 87.4%

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

   4. distribute-neg-frac2 87.4%

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

   5. remove-double-neg 87.4%

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

   6. sqr-neg 87.4%

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

   7. associate-+l+ 87.4%

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

   8. sqr-neg 87.4%

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

   9. distribute-rgt-out 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in m around 0 53.7%

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

Alternative 14: 28.8% accurate, 16.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \frac{a\_m}{1 + 10 \cdot k} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (/ a_m (+ 1.0 (* 10.0 k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (10.0 * k)));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * (a_m / (1.0d0 + (10.0d0 * k)))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (10.0 * k)));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (a_m / (1.0 + (10.0 * k)))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m / Float64(1.0 + Float64(10.0 * k))))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (a_m / (1.0 + (10.0 * k)));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(a$95$m / N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. associate-/l* 87.4%

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

   2. remove-double-neg 87.4%

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

   3. distribute-frac-neg2 87.4%

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

   4. distribute-neg-frac2 87.4%

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

   5. remove-double-neg 87.4%

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

   6. sqr-neg 87.4%

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

   7. associate-+l+ 87.4%

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

   8. sqr-neg 87.4%

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

   9. distribute-rgt-out 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in m around 0 53.7%

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

6. Taylor expanded in k around 0 32.3%

   \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
7. Add Preprocessing

Alternative 15: 44.7% accurate, 16.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \frac{a\_m}{1 + k \cdot k} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (/ a_m (+ 1.0 (* k k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (k * k)));
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * (a_m / (1.0d0 + (k * k)))
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * (a_m / (1.0 + (k * k)));
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * (a_m / (1.0 + (k * k)))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m / Float64(1.0 + Float64(k * k))))
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * (a_m / (1.0 + (k * k)));
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. *-commutative 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in k around 0 86.3%

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

6. Taylor expanded in m around 0 52.6%

   \[\leadsto \frac{\color{blue}{a}}{1 + k \cdot k} \]
7. Add Preprocessing

Alternative 16: 20.6% accurate, 114.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot a\_m \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s a_m))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * a_m
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * a_m

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * a_m)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * a_m;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * a$95$m), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. associate-/l* 87.4%

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

   2. remove-double-neg 87.4%

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

   3. distribute-frac-neg2 87.4%

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

   4. distribute-neg-frac2 87.4%

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

   5. remove-double-neg 87.4%

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

   6. sqr-neg 87.4%

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

   7. associate-+l+ 87.4%

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

   8. sqr-neg 87.4%

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

   9. distribute-rgt-out 87.4%

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

3. Simplified 87.4%

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

5. Taylor expanded in m around 0 53.7%

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

6. Taylor expanded in k around 0 21.9%

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))