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

Average Error: 0.5 → 0.3

Time: 12.8s

Precision: binary64

Cost: 14464

Math

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

↓

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

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 (* 5.0 (* v v))) PI)
  (* (sqrt (* 2.0 (- 1.0 (* v (* v 3.0))))) (* t (- 1.0 (* v v))))))

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

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 - (5.0 * (v * v))) / Math.PI) / (Math.sqrt((2.0 * (1.0 - (v * (v * 3.0))))) * (t * (1.0 - (v * v))));
}

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 - (5.0 * (v * v))) / math.pi) / (math.sqrt((2.0 * (1.0 - (v * (v * 3.0))))) * (t * (1.0 - (v * v))))

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 - Float64(5.0 * Float64(v * v))) / pi) / Float64(sqrt(Float64(2.0 * Float64(1.0 - Float64(v * Float64(v * 3.0))))) * Float64(t * Float64(1.0 - Float64(v * v)))))
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 - (5.0 * (v * v))) / pi) / (sqrt((2.0 * (1.0 - (v * (v * 3.0))))) * (t * (1.0 - (v * v))));
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[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[(N[Sqrt[N[(2.0 * N[(1.0 - N[(v * N[(v * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

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

2. Simplified 0.5

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

   [Start] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   rational.json-simplify-2 [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(1 - v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   rational.json-simplify-43 [<=] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot t\right)\right)}} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \color{blue}{\left(\pi \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)}} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\pi \cdot \left(\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   rational.json-simplify-43 [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\left(t \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 \cdot \left(1 - \color{blue}{v \cdot \left(v \cdot 3\right)}\right)}\right)} \]

3. Applied egg-rr 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi} \cdot \frac{\frac{1}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}}{t \cdot \left(1 - v \cdot v\right)} \]
   rational.json-simplify-44 [=>] 0.4	\[ \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi} \cdot \color{blue}{\frac{\frac{1}{t \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}} \]
   rational.json-simplify-46 [=>] 0.4	\[ \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi} \cdot \frac{\color{blue}{\frac{\frac{1}{t}}{1 - v \cdot v}}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}} \]
   rational.json-simplify-47 [=>] 0.4	\[ \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi} \cdot \color{blue}{\frac{\frac{1}{t}}{\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}} \]
   rational.json-simplify-46 [<=] 0.4	\[ \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi} \cdot \color{blue}{\frac{1}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}\right)}} \]
   rational.json-simplify-49 [<=] 0.3	\[ \color{blue}{\frac{1 \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}\right)}} \]
   rational.json-simplify-6 [=>] 0.3	\[ \frac{\color{blue}{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}\right)} \]
   rational.json-simplify-2 [=>] 0.3	\[ \frac{\frac{1 - \color{blue}{5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}\right)} \]
   rational.json-simplify-43 [<=] 0.3	\[ \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\color{blue}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)}} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	14464

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

Alternative 2
Error	0.7
Cost	13568

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

Alternative 3
Error	1.3
Cost	13184

\[\sqrt{2} \cdot \frac{\frac{0.5}{t}}{\pi} \]

Alternative 4
Error	1.0
Cost	13184

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

Alternative 5
Error	0.9
Cost	13184

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

Reproduce

herbie shell --seed 2023075 
(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)))))