Falkner and Boettcher, Appendix A

Average Error: 1.9 → 1.9

Time: 13.2s

Precision: binary64

Cost: 7168

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

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))) / (k ** m))
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));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 1.9

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

2. Simplified 1.9

   \[\leadsto \color{blue}{\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}} \]
   Proof
   (/.f64 a (/.f64 (+.f64 1 (+.f64 (*.f64 k 10) (*.f64 k k))) (pow.f64 k m))): 0 points increase in error, 0 points decrease in error
   (/.f64 a (/.f64 (+.f64 1 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 10 k)) (*.f64 k k))) (pow.f64 k m))): 0 points increase in error, 0 points decrease in error
   (/.f64 a (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) (pow.f64 k m))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))): 1 points increase in error, 1 points decrease in error

3. Applied egg-rr 1.9

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

4. Final simplification 1.9

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

Alternatives

Alternative 1
Error	2.6
Cost	7176

\[\begin{array}{l} t_0 := \frac{a}{\frac{k}{\frac{{k}^{m}}{k}}}\\ \mathbf{if}\;m \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.26 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.6
Cost	7048

\[\begin{array}{l} t_0 := a \cdot {k}^{\left(m - 2\right)}\\ \mathbf{if}\;m \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.26 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 1.26 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 4
Error	2.5
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -680:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -680:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1550000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 6
Error	17.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -680:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1400000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 7
Error	24.7
Cost	712

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	24.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 9
Error	24.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -9.8:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 10
Error	21.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1400000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 11
Error	39.3
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot 10}\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	24.8
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	24.8
Cost	448

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

Alternative 14
Error	46.8
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022321 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))