Falkner and Boettcher, Appendix A

Average Error: 2.3 → 0.1

Time: 11.1s

Precision: binary64

Cost: 13700

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{k}\\ \end{array} \]

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (if (<= k 6.8e+131)
   (/ (* a (pow k m)) (fma k (+ k 10.0) 1.0))
   (* (/ a k) (/ (pow (- (pow (/ -1.0 k) -1.0)) m) 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 tmp;
	if (k <= 6.8e+131) {
		tmp = (a * pow(k, m)) / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (a / k) * (pow(-pow((-1.0 / k), -1.0), m) / 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)
	tmp = 0.0
	if (k <= 6.8e+131)
		tmp = Float64(Float64(a * (k ^ m)) / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64(a / k) * Float64((Float64(-(Float64(-1.0 / k) ^ -1.0)) ^ m) / k));
	end
	return 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_] := If[LessEqual[k, 6.8e+131], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[(-N[Power[N[(-1.0 / k), $MachinePrecision], -1.0], $MachinePrecision]), m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.79999999999999972e131

   1. Initial program 0.1

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      Proof
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (+.f64 k 10) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (Rewrite<= +-commutative_binary64 (+.f64 10 k)) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 k (+.f64 10 k)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 10 k) (*.f64 k k))) 1)): 1 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= +-commutative_binary64 (+.f64 1 (+.f64 (*.f64 10 k) (*.f64 k k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))): 0 points increase in error, 0 points decrease in error

   if 6.79999999999999972e131 < k

   1. Initial program 9.9

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

   2. Simplified 9.9

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      Proof
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (+.f64 k 10) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (Rewrite<= +-commutative_binary64 (+.f64 10 k)) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 k (+.f64 10 k)) 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 10 k) (*.f64 k k))) 1)): 1 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= +-commutative_binary64 (+.f64 1 (+.f64 (*.f64 10 k) (*.f64 k k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in k around -inf 64.0

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{\left(-1 \cdot {\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{k}} \]
      Proof
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (*.f64 -1 (pow.f64 (/.f64 -1 k) -1)) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (*.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 -1))) (pow.f64 (/.f64 -1 k) -1)) m) k)): 205 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (*.f64 (exp.f64 (log.f64 -1)) (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 -1 k)) -1)))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (*.f64 (exp.f64 (log.f64 -1)) (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (log.f64 (/.f64 -1 k)))))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (Rewrite<= exp-sum_binary64 (exp.f64 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 (/.f64 -1 k)))))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 (/.f64 -1 k)))) m))) k)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a (exp.f64 (*.f64 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 (/.f64 -1 k)))) m))) (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (exp.f64 (*.f64 (+.f64 (log.f64 -1) (*.f64 -1 (log.f64 (/.f64 -1 k)))) m))) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{+94}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13508

\[\begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{a} \cdot \frac{k}{{k}^{m}}\right)}^{-1}\\ \end{array} \]

Alternative 3
Error	2.3
Cost	7296

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

Alternative 4
Error	3.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 5
Error	3.2
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -6 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-305}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 7
Error	18.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{-303}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \mathbf{elif}\;k \leq 0.14:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 8
Error	18.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 8.8 \cdot 10^{-305}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 9
Error	11.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -225000:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 0.98:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]

Alternative 10
Error	18.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-304}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	13.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{+34}:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k} \cdot 0.1\right)\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]

Alternative 12
Error	11.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -225000:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]

Alternative 13
Error	24.2
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 14
Error	23.2
Cost	584

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

Alternative 15
Error	15.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot k\right) \cdot 99\right)\\ \end{array} \]

Alternative 16
Error	40.7
Cost	452

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

Alternative 17
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022338 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))