Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.8

Time: 9.9s

Precision: binary64

Cost: 13184

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

↓

\[\frac{\frac{\frac{1}{\sqrt{2}}}{\pi}}{t} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))

↓

(FPCore (v t) :precision binary64 (/ (/ (/ 1.0 (sqrt 2.0)) PI) t))

C

double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}

↓

double code(double v, double t) {
	return ((1.0 / sqrt(2.0)) / ((double) M_PI)) / t;
}

Java

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((Math.PI * t) * Math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
}

↓

public static double code(double v, double t) {
	return ((1.0 / Math.sqrt(2.0)) / Math.PI) / t;
}

Python

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (((math.pi * t) * math.sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)))

↓

def code(v, t):
	return ((1.0 / math.sqrt(2.0)) / math.pi) / t

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(3.0 * Float64(v * v)))))) * Float64(1.0 - Float64(v * v))))
end

↓

function code(v, t)
	return Float64(Float64(Float64(1.0 / sqrt(2.0)) / pi) / t)
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((pi * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
end

↓

function tmp = code(v, t)
	tmp = ((1.0 / sqrt(2.0)) / pi) / t;
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_, t_] := N[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

2. Taylor expanded in v around 0 1.1

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2} \cdot \left(t \cdot \pi\right)}} \]

3. Applied egg-rr 33.5

   \[\leadsto \color{blue}{{\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{-0.5} \cdot {\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{-0.5}} \]

4. Simplified 0.8

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\sqrt{2}}}{\pi}}{t}} \]
   Proof

   [Start] 33.5	\[ {\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{-0.5} \cdot {\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{-0.5} \]
   pow-sqr [=>] 1.1	\[ \color{blue}{{\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{\left(2 \cdot -0.5\right)}} \]
   metadata-eval [=>] 1.1	\[ {\left(\pi \cdot \left(\sqrt{2} \cdot t\right)\right)}^{\color{blue}{-1}} \]
   unpow-1 [=>] 1.1	\[ \color{blue}{\frac{1}{\pi \cdot \left(\sqrt{2} \cdot t\right)}} \]
   associate-*r* [=>] 1.0	\[ \frac{1}{\color{blue}{\left(\pi \cdot \sqrt{2}\right) \cdot t}} \]
   *-commutative [<=] 1.0	\[ \frac{1}{\color{blue}{\left(\sqrt{2} \cdot \pi\right)} \cdot t} \]
   associate-/r* [=>] 0.8	\[ \color{blue}{\frac{\frac{1}{\sqrt{2} \cdot \pi}}{t}} \]
   associate-/r* [=>] 0.8	\[ \frac{\color{blue}{\frac{\frac{1}{\sqrt{2}}}{\pi}}}{t} \]

5. Final simplification 0.8

   \[\leadsto \frac{\frac{\frac{1}{\sqrt{2}}}{\pi}}{t} \]

Alternatives

Alternative 1
Error	1.1
Cost	13184

\[\frac{1}{\sqrt{2} \cdot \left(\pi \cdot t\right)} \]

Alternative 2
Error	1.1
Cost	13184

\[\frac{\frac{1}{\sqrt{2}}}{\pi \cdot t} \]

Alternative 3
Error	1.4
Cost	13056

\[\frac{\sqrt{0.5}}{\pi \cdot t} \]

Alternative 4
Error	1.3
Cost	13056

\[\frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

Reproduce

herbie shell --seed 2023047 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))