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

Average Accuracy: 99.3% → 99.4%

Time: 12.0s

Precision: binary64

Cost: 14592

Math

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

↓

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

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 (* t (* PI (sqrt (+ 2.0 (* 2.0 (* (* v v) -3.0)))))))
  (/ (- 1.0 (* (* v v) 5.0)) (- 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 / (t * (((double) M_PI) * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (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 / (t * (Math.PI * Math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (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 / (t * (math.pi * math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (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(t * Float64(pi * sqrt(Float64(2.0 + Float64(2.0 * Float64(Float64(v * v) * -3.0))))))) * Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / 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 / (t * (pi * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))) * ((1.0 - ((v * v) * 5.0)) / (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[(t * N[(Pi * N[Sqrt[N[(2.0 + N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $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 99.4%

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

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

3. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.3%
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
Accuracy	99.3%
Cost	14336

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

Alternative 3
Accuracy	98.2%
Cost	13184

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

Alternative 4
Accuracy	98.2%
Cost	13184

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

Alternative 5
Accuracy	98.4%
Cost	13184

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

Alternative 6
Accuracy	98.7%
Cost	13184

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

Alternative 7
Accuracy	97.8%
Cost	13056

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

Alternative 8
Accuracy	97.8%
Cost	13056

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

Reproduce

herbie shell --seed 2023137 
(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)))))