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

Average Error: 0.5 → 0.5

Time: 9.3s

Precision: binary64

Cost: 14336

Math

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

↓

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

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))
  (* (* (sqrt (+ 2.0 (* (* v v) -6.0))) (* t PI)) (- 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)) / ((sqrt((2.0 + ((v * v) * -6.0))) * (t * ((double) M_PI))) * (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.sqrt((2.0 + ((v * v) * -6.0))) * (t * Math.PI)) * (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.sqrt((2.0 + ((v * v) * -6.0))) * (t * math.pi)) * (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(sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))) * Float64(t * pi)) * 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)) / ((sqrt((2.0 + ((v * v) * -6.0))) * (t * pi)) * (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[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $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 48.8

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

3. Simplified 0.5

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

   [Start] 48.8	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]
   expm1-def [=>] 17.8	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   expm1-log1p [=>] 0.4	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(t \cdot \left(\pi \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   associate-*r* [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(t \cdot \pi\right) \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [<=] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\left(\pi \cdot t\right)} \cdot \sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(\pi \cdot t\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \color{blue}{\left(t \cdot \pi\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   *-commutative [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + \color{blue}{\left(\left(v \cdot v\right) \cdot -3\right) \cdot 2}} \cdot \left(t \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   unpow2 [<=] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + \left(\color{blue}{{v}^{2}} \cdot -3\right) \cdot 2} \cdot \left(t \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + \color{blue}{{v}^{2} \cdot \left(-3 \cdot 2\right)}} \cdot \left(t \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   unpow2 [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-3 \cdot 2\right)} \cdot \left(t \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}} \cdot \left(t \cdot \pi\right)\right) \cdot \left(1 - v \cdot v\right)} \]

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	1.1
Cost	13184

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

Alternative 2
Error	0.8
Cost	13184

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

Alternative 3
Error	1.4
Cost	13056

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

Alternative 4
Error	1.3
Cost	13056

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

Reproduce

herbie shell --seed 2023237 
(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)))))