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

Average Error: 0.7% → 0.68%

Time: 10.5s

Precision: binary64

Cost: 14464

Math

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

↓

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

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

Derivation

1. Initial program 0.7

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

2. Applied egg-rr 74.15

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{\pi \cdot t}}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}}}{1 - v \cdot v}\right)} - 1} \]

3. Simplified 0.68

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

   [Start] 74.15	\[ e^{\mathsf{log1p}\left(\frac{\frac{\frac{1 + \left(v \cdot v\right) \cdot -5}{\pi \cdot t}}{\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}}}{1 - v \cdot v}\right)} - 1 \]

4. Final simplification 0.68

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

Alternatives

Alternative 1
Error	0.7%
Cost	14336

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

Alternative 2
Error	1.59%
Cost	13312

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

Alternative 3
Error	1.81%
Cost	13184

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

Alternative 4
Error	1.69%
Cost	13184

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

Alternative 5
Error	1.75%
Cost	13184

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

Alternative 6
Error	2.25%
Cost	13056

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

Alternative 7
Error	2.16%
Cost	13056

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

Reproduce

herbie shell --seed 2023088 
(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)))))