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

Average Accuracy: 99.3% → 99.6%

Time: 11.8s

Precision: binary64

Cost: 14592

Math

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

↓

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

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

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

   [Start] 99.3	\[ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   cancel-sign-sub-inv [=>] 99.3	\[ \frac{\color{blue}{1 + \left(-5\right) \cdot \left(v \cdot v\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   metadata-eval [=>] 99.3	\[ \frac{1 + \color{blue}{-5} \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
   distribute-rgt-out-- [<=] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{1 \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]
   *-lft-identity [=>] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \]
   *-commutative [=>] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right)} - \left(v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \]
   *-commutative [=>] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right) - \left(v \cdot v\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right)\right)}} \]
   associate-*r* [=>] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\pi \cdot t\right) - \color{blue}{\left(\left(v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(\pi \cdot t\right)}} \]
   distribute-rgt-out-- [=>] 99.3	\[ \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} - \left(v \cdot v\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]

3. Applied egg-rr 99.3%

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

4. Simplified 99.6%

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

   [Start] 99.3	\[ \left(-1 - -5 \cdot \left(v \cdot v\right)\right) \cdot \frac{1}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.3	\[ \color{blue}{\left(-1 + \left(--5\right) \cdot \left(v \cdot v\right)\right)} \cdot \frac{1}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   metadata-eval [=>] 99.3	\[ \left(-1 + \color{blue}{5} \cdot \left(v \cdot v\right)\right) \cdot \frac{1}{\pi \cdot \left(-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)\right)} \]
   associate-/r* [=>] 99.6	\[ \left(-1 + 5 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{\frac{\frac{1}{\pi}}{-t \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   distribute-lft-neg-in [=>] 99.6	\[ \left(-1 + 5 \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{1}{\pi}}{\color{blue}{\left(-t\right) \cdot \left(\sqrt{2 + 2 \cdot \left(\left(v \cdot v\right) \cdot -3\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   associate-*l* [=>] 99.6	\[ \left(-1 + 5 \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{1}{\pi}}{\left(-t\right) \cdot \left(\sqrt{2 + 2 \cdot \color{blue}{\left(v \cdot \left(v \cdot -3\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]

5. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	14464

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

Alternative 2
Accuracy	99.3%
Cost	14464

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

Alternative 3
Accuracy	98.4%
Cost	13760

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

Alternative 4
Accuracy	98.3%
Cost	13184

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

Alternative 5
Accuracy	98.3%
Cost	13184

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

Alternative 6
Accuracy	98.2%
Cost	13184

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

Alternative 7
Accuracy	97.7%
Cost	13056

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

Alternative 8
Accuracy	97.8%
Cost	13056

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

Reproduce

herbie shell --seed 2023122 
(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)))))