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

Average Error: 0.5 → 0.3

Time: 11.7s

Precision: binary64

Cost: 14592

Math

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

↓

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

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 PI) t)
  (/
   (+ 1.0 (* (* v v) -5.0))
   (* (sqrt (+ 2.0 (* 2.0 (* (* v v) -3.0)))) (- 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 / ((double) M_PI)) / t) * ((1.0 + ((v * v) * -5.0)) / (sqrt((2.0 + (2.0 * ((v * v) * -3.0)))) * (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 / Math.PI) / t) * ((1.0 + ((v * v) * -5.0)) / (Math.sqrt((2.0 + (2.0 * ((v * v) * -3.0)))) * (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 / math.pi) / t) * ((1.0 + ((v * v) * -5.0)) / (math.sqrt((2.0 + (2.0 * ((v * v) * -3.0)))) * (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 / pi) / t) * Float64(Float64(1.0 + Float64(Float64(v * v) * -5.0)) / Float64(sqrt(Float64(2.0 + Float64(2.0 * Float64(Float64(v * v) * -3.0)))) * 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 / pi) / t) * ((1.0 + ((v * v) * -5.0)) / (sqrt((2.0 + (2.0 * ((v * v) * -3.0)))) * (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 / Pi), $MachinePrecision] / t), $MachinePrecision] * N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(2.0 + N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 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. Applied egg-rr 0.5

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

3. Taylor expanded in t around 0 0.5

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

4. Simplified 0.3

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

   [Start] 0.5	\[ \frac{1}{t \cdot \pi} \cdot \frac{1 + \left(v \cdot v\right) \cdot -5}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-/l/ [<=] 0.3	\[ \color{blue}{\frac{\frac{1}{\pi}}{t}} \cdot \frac{1 + \left(v \cdot v\right) \cdot -5}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.4
Cost	13184

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

Alternative 2
Error	1.1
Cost	13184

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

Alternative 3
Error	0.9
Cost	13184

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

Alternative 4
Error	1.4
Cost	13056

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

Alternative 5
Error	1.4
Cost	13056

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

Reproduce

herbie shell --seed 2023059 
(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)))))