Falkner and Boettcher, Appendix A

Average Accuracy: 90.5% → 99.1%

Time: 11.9s

Precision: binary64

Cost: 7172

• 23.0% 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

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 0.00024:\\ \;\;\;\;t_0 \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{t_0}{k}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 0.00024) (* t_0 (+ (* k -10.0) 1.0)) (/ 1.0 (/ k (/ t_0 k))))))

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) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 0.00024) {
		tmp = t_0 * ((k * -10.0) + 1.0);
	} else {
		tmp = 1.0 / (k / (t_0 / 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
    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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= 0.00024d0) then
        tmp = t_0 * ((k * (-10.0d0)) + 1.0d0)
    else
        tmp = 1.0d0 / (k / (t_0 / k))
    end if
    code = tmp
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) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (k <= 0.00024) {
		tmp = t_0 * ((k * -10.0) + 1.0);
	} else {
		tmp = 1.0 / (k / (t_0 / k));
	}
	return tmp;
}

Python

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

↓

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 0.00024:
		tmp = t_0 * ((k * -10.0) + 1.0)
	else:
		tmp = 1.0 / (k / (t_0 / k))
	return tmp

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)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 0.00024)
		tmp = Float64(t_0 * Float64(Float64(k * -10.0) + 1.0));
	else
		tmp = Float64(1.0 / Float64(k / Float64(t_0 / k)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 0.00024)
		tmp = t_0 * ((k * -10.0) + 1.0);
	else
		tmp = 1.0 / (k / (t_0 / k));
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 0.00024], N[(t$95$0 * N[(N[(k * -10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k / N[(t$95$0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.40000000000000006e-4

   1. Initial program 96.4%

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

   2. Simplified 96.4%

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

      [Start] 96.4	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 96.4	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 96.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 96.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 96.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 96.4	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 96.4	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around 0 36.7%

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

   4. Simplified 100.0%

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

      [Start] 36.7	\[ -10 \cdot \left(k \cdot \left(e^{\log k \cdot m} \cdot a\right)\right) + e^{\log k \cdot m} \cdot a \]
      associate-*r* [=>] 36.7	\[ \color{blue}{\left(-10 \cdot k\right) \cdot \left(e^{\log k \cdot m} \cdot a\right)} + e^{\log k \cdot m} \cdot a \]
      distribute-lft1-in [=>] 52.1	\[ \color{blue}{\left(-10 \cdot k + 1\right) \cdot \left(e^{\log k \cdot m} \cdot a\right)} \]
      *-commutative [=>] 52.1	\[ \left(\color{blue}{k \cdot -10} + 1\right) \cdot \left(e^{\log k \cdot m} \cdot a\right) \]
      exp-to-pow [=>] 100.0	\[ \left(k \cdot -10 + 1\right) \cdot \left(\color{blue}{{k}^{m}} \cdot a\right) \]
      *-commutative [=>] 100.0	\[ \left(k \cdot -10 + 1\right) \cdot \color{blue}{\left(a \cdot {k}^{m}\right)} \]

   if 2.40000000000000006e-4 < k

   1. Initial program 88.6%

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

   2. Simplified 88.6%

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

      [Start] 88.6	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 88.6	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 88.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 88.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 88.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 88.6	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 88.6	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Applied egg-rr 87.8%

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

      [Start] 88.6	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)} \]
      associate-*r/ [=>] 88.6	\[ \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      clear-num [=>] 87.8	\[ \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{a \cdot {k}^{m}}}} \]
      *-commutative [=>] 87.8	\[ \frac{1}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{\color{blue}{{k}^{m} \cdot a}}} \]

   4. Taylor expanded in k around inf 87.3%

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

   5. Simplified 97.5%

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

      [Start] 87.3	\[ \frac{1}{\frac{{k}^{2}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}} \]
      unpow2 [=>] 87.3	\[ \frac{1}{\frac{\color{blue}{k \cdot k}}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}} \]
      *-commutative [=>] 87.3	\[ \frac{1}{\frac{k \cdot k}{\color{blue}{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}}} \]
      associate-/l* [=>] 97.5	\[ \frac{1}{\color{blue}{\frac{k}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}}}} \]
      associate-*r* [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{k}\right)\right) \cdot m}}}{k}}} \]
      exp-prod [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot \color{blue}{{\left(e^{-1 \cdot \log \left(\frac{1}{k}\right)}\right)}^{m}}}{k}}} \]
      mul-1-neg [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot {\left(e^{\color{blue}{-\log \left(\frac{1}{k}\right)}}\right)}^{m}}{k}}} \]
      log-rec [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot {\left(e^{-\color{blue}{\left(-\log k\right)}}\right)}^{m}}{k}}} \]
      remove-double-neg [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot {\left(e^{\color{blue}{\log k}}\right)}^{m}}{k}}} \]
      rem-exp-log [=>] 97.5	\[ \frac{1}{\frac{k}{\frac{a \cdot {\color{blue}{k}}^{m}}{k}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00024:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{a \cdot {k}^{m}}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.8%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.56 \cdot 10^{-75}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 2
Accuracy	82.8%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-5} \lor \neg \left(m \leq 0.0058\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 3
Accuracy	41.2%
Cost	980

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq 4.8 \cdot 10^{-72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-112}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.7 \cdot 10^{-182}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 175000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 4
Accuracy	41.4%
Cost	980

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;m \leq 3.55 \cdot 10^{-68}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.5 \cdot 10^{-111}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 3.7 \cdot 10^{-181}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 225000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 5
Accuracy	41.4%
Cost	980

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ t_1 := a \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{if}\;m \leq 1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 8.4 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq 5.6 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.45 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;m \leq 340000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 6
Accuracy	41.5%
Cost	980

\[\begin{array}{l} t_0 := a \cdot \left(k \cdot -10 + 1\right)\\ \mathbf{if}\;m \leq 4.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.2 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.48 \cdot 10^{-288}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;m \leq 3.45 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 175000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 7
Accuracy	41.4%
Cost	980

\[\begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;m \leq 2.75 \cdot 10^{-64}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.7 \cdot 10^{-285}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 225000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8
Accuracy	41.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 2700:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 175000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9
Accuracy	19.7%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot 10}\\ \mathbf{if}\;m \leq 2.7 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-138}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 225000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10
Accuracy	41.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 2500000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 7 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11
Accuracy	41.8%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 2700:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 225000000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 12
Accuracy	25.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 1.3 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13
Accuracy	20.1%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023159 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))