Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 97.3%

Time: 7.8s

Alternatives: 10

Speedup: 1.1×

• 23.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+146}:\\ \;\;\;\;a \cdot \frac{-{k}^{m}}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+146)
     (* a (/ (- (pow k m)) (- -1.0 (* k (+ k 10.0)))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+146) {
		tmp = a * (-pow(k, m) / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 1d+146) then
        tmp = a * (-(k ** m) / ((-1.0d0) - (k * (k + 10.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+146) {
		tmp = a * (-Math.pow(k, m) / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+146:
		tmp = a * (-math.pow(k, m) / (-1.0 - (k * (k + 10.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+146)
		tmp = Float64(a * Float64(Float64(-(k ^ m)) / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+146)
		tmp = a * (-(k ^ m) / (-1.0 - (k * (k + 10.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+146], N[(a * N[((-N[Power[k, m], $MachinePrecision]) / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 9.99999999999999934e145

   1. Initial program 97.7%

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

      1. sqr-neg 97.7%

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

      2. associate-+l+ 97.7%

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

      3. sqr-neg 97.7%

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

      4. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

      1. frac-2neg 97.7%

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

      2. div-inv 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

      4. +-commutative 97.7%

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

      5. fma-def 97.7%

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

      6. +-commutative 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. associate-*l* 97.7%

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

      2. associate-*r/ 97.7%

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

      3. *-rgt-identity 97.7%

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

      4. fma-udef 97.7%

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

      5. +-commutative 97.7%

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

      6. distribute-neg-in 97.7%

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

      7. metadata-eval 97.7%

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

      8. sub-neg 97.7%

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

   7. Simplified 97.7%

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

   if 9.99999999999999934e145 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 67.4%

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

      1. sqr-neg 67.4%

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

      2. associate-+l+ 67.4%

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

      3. sqr-neg 67.4%

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

      4. distribute-rgt-out 67.4%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+146}:\\ \;\;\;\;a \cdot \frac{-{k}^{m}}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 2.75e-11) (/ t_0 (+ 1.0 (* k (+ k 10.0)))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 2.75e-11) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 2.75d-11) then
        tmp = t_0 / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 2.75e-11) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 2.75e-11:
		tmp = t_0 / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 2.75e-11)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 2.75e-11)
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 2.75e-11], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.74999999999999987e-11

   1. Initial program 97.1%

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

      1. sqr-neg 97.1%

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

      2. associate-+l+ 97.1%

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

      3. sqr-neg 97.1%

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

      4. distribute-rgt-out 97.1%

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

   3. Simplified 97.1%

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

   if 2.74999999999999987e-11 < m

   1. Initial program 83.7%

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

      1. sqr-neg 83.7%

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

      2. associate-+l+ 83.7%

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

      3. sqr-neg 83.7%

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

      4. distribute-rgt-out 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.42 \cdot 10^{-11} \lor \neg \left(m \leq 2.7 \cdot 10^{-11}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.42e-11) (not (<= m 2.7e-11)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.42e-11) || !(m <= 2.7e-11)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-1.42d-11)) .or. (.not. (m <= 2.7d-11))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.42e-11) || !(m <= 2.7e-11)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.42e-11) or not (m <= 2.7e-11):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.42e-11) || !(m <= 2.7e-11))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -1.42e-11) || ~((m <= 2.7e-11)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.42e-11], N[Not[LessEqual[m, 2.7e-11]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.42e-11 or 2.70000000000000005e-11 < m

   1. Initial program 91.6%

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

      1. sqr-neg 91.6%

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

      2. associate-+l+ 91.6%

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

      3. sqr-neg 91.6%

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

      4. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -1.42e-11 < m < 2.70000000000000005e-11

   1. Initial program 93.7%

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

      1. sqr-neg 93.7%

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

      2. associate-+l+ 93.7%

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

      3. sqr-neg 93.7%

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

      4. distribute-rgt-out 93.7%

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

   3. Simplified 93.7%

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

      1. frac-2neg 93.7%

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

      2. div-inv 93.8%

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

      3. distribute-rgt-neg-in 93.8%

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

      4. +-commutative 93.8%

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

      5. fma-def 93.8%

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

      6. +-commutative 93.8%

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

   5. Applied egg-rr 93.8%

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

      1. associate-*l* 93.8%

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

      2. associate-*r/ 93.8%

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

      3. *-rgt-identity 93.8%

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

      4. fma-udef 93.8%

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

      5. +-commutative 93.8%

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

      6. distribute-neg-in 93.8%

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

      7. metadata-eval 93.8%

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

      8. sub-neg 93.8%

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

   7. Simplified 93.8%

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

   8. Taylor expanded in m around 0 93.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.42 \cdot 10^{-11} \lor \neg \left(m \leq 2.7 \cdot 10^{-11}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 44.9% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

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 * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. sqr-neg 92.3%

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

   2. associate-+l+ 92.3%

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

   3. sqr-neg 92.3%

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

   4. distribute-rgt-out 92.3%

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

3. Simplified 92.3%

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

   1. frac-2neg 92.3%

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

   2. div-inv 92.3%

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

   3. distribute-rgt-neg-in 92.3%

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

   4. +-commutative 92.3%

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

   5. fma-def 92.3%

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

   6. +-commutative 92.3%

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

5. Applied egg-rr 92.3%

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

   1. associate-*l* 92.3%

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

   2. associate-*r/ 92.3%

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

   3. *-rgt-identity 92.3%

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

   4. fma-udef 92.3%

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

   5. +-commutative 92.3%

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

   6. distribute-neg-in 92.3%

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

   7. metadata-eval 92.3%

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

   8. sub-neg 92.3%

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

7. Simplified 92.3%

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

8. Taylor expanded in m around 0 43.3%

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

9. Final simplification 43.3%

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

Alternative 5: 44.9% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. sqr-neg 92.3%

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

   2. associate-+l+ 92.3%

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

   3. sqr-neg 92.3%

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

   4. distribute-rgt-out 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in m around 0 43.3%

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

   1. clear-num 43.7%

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

   2. inv-pow 43.7%

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

   3. +-commutative 43.7%

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

   4. +-commutative 43.7%

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

   5. fma-def 43.7%

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

6. Applied egg-rr 43.7%

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

   1. unpow-1 43.7%

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

8. Simplified 43.7%

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

9. Taylor expanded in a around 0 43.7%

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

10. Final simplification 43.7%

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

Alternative 6: 27.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (* a (+ 1.0 (* k -10.0))) (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.075d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.075)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. frac-2neg 96.5%

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

      2. div-inv 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

      4. +-commutative 96.5%

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

      5. fma-def 96.5%

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

      6. +-commutative 96.5%

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

   5. Applied egg-rr 96.5%

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

      1. associate-*l* 96.5%

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

      2. associate-*r/ 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. fma-udef 96.5%

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

      5. +-commutative 96.5%

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

      6. distribute-neg-in 96.5%

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

      7. metadata-eval 96.5%

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

      8. sub-neg 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in m around 0 32.2%

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

   9. Taylor expanded in k around 0 27.1%

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

       1. *-commutative 27.1%

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

   11. Simplified 27.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 84.0%

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

      1. sqr-neg 84.0%

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

      2. associate-+l+ 84.0%

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

      3. sqr-neg 84.0%

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

      4. distribute-rgt-out 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in m around 0 65.3%

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

   5. Taylor expanded in k around 0 24.6%

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

      1. *-commutative 24.6%

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

   7. Simplified 24.6%

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

   8. Taylor expanded in k around inf 24.5%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 7: 45.0% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

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 / (1.0d0 + (k * (k + 10.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. sqr-neg 92.3%

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

   2. associate-+l+ 92.3%

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

   3. sqr-neg 92.3%

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

   4. distribute-rgt-out 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in m around 0 43.3%

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

5. Final simplification 43.3%

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

Alternative 8: 25.7% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 96.5%

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

      1. sqr-neg 96.5%

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

      2. associate-+l+ 96.5%

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

      3. sqr-neg 96.5%

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

      4. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 32.2%

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

   5. Taylor expanded in k around 0 24.6%

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

   if 0.10000000000000001 < k

   1. Initial program 84.0%

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

      1. sqr-neg 84.0%

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

      2. associate-+l+ 84.0%

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

      3. sqr-neg 84.0%

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

      4. distribute-rgt-out 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in m around 0 65.3%

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

   5. Taylor expanded in k around 0 24.6%

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

      1. *-commutative 24.6%

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

   7. Simplified 24.6%

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

   8. Taylor expanded in k around inf 24.5%

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

4. Final simplification 24.6%

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

Alternative 9: 28.1% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1 + k \cdot 10} \end{array} \]

FPCore

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

C

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

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 / (1.0d0 + (k * 10.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. sqr-neg 92.3%

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

   2. associate-+l+ 92.3%

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

   3. sqr-neg 92.3%

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

   4. distribute-rgt-out 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in m around 0 43.3%

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

5. Taylor expanded in k around 0 25.8%

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

   1. *-commutative 25.8%

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

7. Simplified 25.8%

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

8. Final simplification 25.8%

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

Alternative 10: 19.4% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 92.3%

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

   1. sqr-neg 92.3%

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

   2. associate-+l+ 92.3%

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

   3. sqr-neg 92.3%

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

   4. distribute-rgt-out 92.3%

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

3. Simplified 92.3%

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

4. Taylor expanded in m around 0 43.3%

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

5. Taylor expanded in k around 0 17.8%

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

6. Final simplification 17.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023300 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))