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

Average Accuracy: 99.3% → 99.3%

Time: 11.3s

Precision: binary64

Cost: 14336

Math

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

↓

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

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

Derivation

1. Initial program 99.3%

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

2. Applied egg-rr 24.7%

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

   [Start] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   expm1-log1p-u [=>] 72.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   expm1-udef [=>] 24.7	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]

3. Simplified 99.3%

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

   [Start] 24.7	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]
   expm1-def [=>] 72.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   expm1-log1p [=>] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 99.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right) \cdot t\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot t\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [<=] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \color{blue}{\left(t \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right) \cdot 2}}\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   unpow2 [<=] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 + \left(\color{blue}{{v}^{2}} \cdot -3\right) \cdot 2}\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 + \color{blue}{{v}^{2} \cdot \left(-3 \cdot 2\right)}}\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   unpow2 [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-3 \cdot 2\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	98.0%
Cost	13184

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

Alternative 2
Accuracy	98.3%
Cost	13184

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

Alternative 3
Accuracy	98.9%
Cost	13184

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

Alternative 4
Accuracy	97.9%
Cost	13056

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

Alternative 5
Accuracy	98.0%
Cost	13056

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

Reproduce

herbie shell --seed 2023139 
(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)))))