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

Average Accuracy: 99.3% → 99.6%

Time: 13.8s

Precision: binary64

Cost: 14336

Math

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

↓

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

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 (- 1.0 (* v v))) (sqrt (+ 2.0 (* (* v v) -6.0))))))

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 * (1.0 - (v * v))) * sqrt((2.0 + ((v * v) * -6.0))));
}

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 * (1.0 - (v * v))) * Math.sqrt((2.0 + ((v * v) * -6.0))));
}

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 * (1.0 - (v * v))) * math.sqrt((2.0 + ((v * v) * -6.0))))

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(Float64(v * v) * -5.0)) / pi) / Float64(Float64(t * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))))
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 * (1.0 - (v * v))) * sqrt((2.0 + ((v * v) * -6.0))));
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[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[(N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $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. Simplified 99.3%

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

   [Start] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   associate-*l* [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   sub-neg [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]
   distribute-lft-in [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{\color{blue}{2 \cdot 1 + 2 \cdot \left(-3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{\color{blue}{2} + 2 \cdot \left(-3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   *-commutative [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(-\color{blue}{\left(v \cdot v\right) \cdot 3}\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   distribute-rgt-neg-in [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \color{blue}{\left(\left(v \cdot v\right) \cdot \left(-3\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot \color{blue}{-3}\right)} \cdot \left(1 - v \cdot v\right)\right)} \]

3. Applied egg-rr 99.6%

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

   [Start] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
   *-un-lft-identity [=>] 99.3	\[ \color{blue}{1 \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   *-commutative [=>] 99.3	\[ \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)} \cdot 1} \]

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	14464

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

Alternative 2
Accuracy	99.4%
Cost	14400

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

Alternative 3
Accuracy	98.6%
Cost	13568

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

Alternative 4
Accuracy	97.9%
Cost	13184

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

Alternative 5
Accuracy	98.3%
Cost	13184

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

Alternative 6
Accuracy	98.3%
Cost	13184

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

Alternative 7
Accuracy	98.4%
Cost	13184

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

Alternative 8
Accuracy	98.4%
Cost	13184

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

Alternative 9
Accuracy	97.9%
Cost	13056

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

Reproduce

herbie shell --seed 2023146 
(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)))))