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

Average Error: 0.4 → 0.3

Time: 16.5s

Precision: binary64

Cost: 14464

Math

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

↓

\[\frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{\pi \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot \frac{1}{t}}{1 - v \cdot v} \]

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 (* (* v v) -5.0)) (* PI (sqrt (+ 2.0 (* (* v v) -6.0)))))
   (/ 1.0 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 + ((v * v) * -5.0)) / (((double) M_PI) * sqrt((2.0 + ((v * v) * -6.0))))) * (1.0 / 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 + ((v * v) * -5.0)) / (Math.PI * Math.sqrt((2.0 + ((v * v) * -6.0))))) * (1.0 / 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 + ((v * v) * -5.0)) / (math.pi * math.sqrt((2.0 + ((v * v) * -6.0))))) * (1.0 / 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(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / Float64(pi * sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))))) * Float64(1.0 / 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 + ((v * v) * -5.0)) / (pi * sqrt((2.0 + ((v * v) * -6.0))))) * (1.0 / 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[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $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. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{1 - v \cdot v}} \]
   Proof

   [Start] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   rational.json-simplify-28 [=>] 0.4	\[ \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v}} \]
   rational.json-simplify-28 [=>] 0.4	\[ \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v} \]
   rational.json-simplify-42 [=>] 0.4	\[ \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{\color{blue}{1 \cdot 2 - 2 \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}}}{1 - v \cdot v} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{\color{blue}{2} - 2 \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}}{1 - v \cdot v} \]
   rational.json-simplify-50 [=>] 0.4	\[ \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 - 2 \cdot \color{blue}{\left(\left(v \cdot v\right) \cdot 3\right)}}}}{1 - v \cdot v} \]
   rational.json-simplify-24 [=>] 0.4	\[ \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 - \color{blue}{\left(v \cdot v\right) \cdot \left(2 \cdot 3\right)}}}}{1 - v \cdot v} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 - \left(v \cdot v\right) \cdot \color{blue}{6}}}}{1 - v \cdot v} \]

3. Applied egg-rr 0.3

   \[\leadsto \frac{\color{blue}{\frac{1 + \left(v \cdot v\right) \cdot -5}{\pi \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot \frac{1}{t}}}{1 - v \cdot v} \]

4. Final simplification 0.3

   \[\leadsto \frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{\pi \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot \frac{1}{t}}{1 - v \cdot v} \]

Alternatives

Alternative 1
Error	0.5
Cost	14336

\[\frac{1 + -5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(\sqrt{2 + \frac{v \cdot v}{-0.16666666666666666}} \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)\right)} \]

Alternative 2
Error	0.4
Cost	14336

\[\frac{-1 - \left(v \cdot v\right) \cdot -5}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot \left(\left(v \cdot v + -1\right) \cdot \left(\pi \cdot t\right)\right)} \]

Alternative 3
Error	0.4
Cost	14336

\[\frac{\frac{1 + -5 \cdot \left(v \cdot v\right)}{\left(t \cdot \pi\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{1 - v \cdot v} \]

Alternative 4
Error	0.4
Cost	14336

\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{1 - v \cdot v} \]

Alternative 5
Error	0.4
Cost	14336

\[\frac{\frac{\frac{\frac{1 + v \cdot \left(v \cdot -5\right)}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{t}}{\pi}}{1 - v \cdot v} \]

Alternative 6
Error	0.7
Cost	13312

\[\frac{\frac{4}{\frac{\pi}{\sqrt{0.5}}} \cdot 0.25}{t} \]

Alternative 7
Error	1.3
Cost	13184

\[\frac{\frac{1}{t}}{\pi} \cdot \sqrt{0.5} \]

Alternative 8
Error	1.0
Cost	13184

\[\frac{1}{t \cdot \frac{\pi}{\sqrt{0.5}}} \]

Alternative 9
Error	1.0
Cost	13184

\[\frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}} \]

Alternative 10
Error	1.3
Cost	13056

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

Alternative 11
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2023073 
(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)))))