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

Percentage Accurate: 99.3% → 99.4%

Time: 9.4s

Alternatives: 7

Speedup: 0.6×

Specification

Math

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

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)))))

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)));
}

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)));
}

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)))

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

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

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]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

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)))))

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)));
}

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)));
}

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)))

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

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

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]

Alternative 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (+ 1.0 (* -5.0 (* v v)))
  (*
   t
   (* (* PI (- 1.0 (pow v 2.0))) (sqrt (+ 2.0 (* 2.0 (* (* v v) -3.0))))))))

C

double code(double v, double t) {
	return (1.0 + (-5.0 * (v * v))) / (t * ((((double) M_PI) * (1.0 - pow(v, 2.0))) * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))));
}

Java

public static double code(double v, double t) {
	return (1.0 + (-5.0 * (v * v))) / (t * ((Math.PI * (1.0 - Math.pow(v, 2.0))) * Math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))));
}

Python

def code(v, t):
	return (1.0 + (-5.0 * (v * v))) / (t * ((math.pi * (1.0 - math.pow(v, 2.0))) * math.sqrt((2.0 + (2.0 * ((v * v) * -3.0))))))

Julia

function code(v, t)
	return Float64(Float64(1.0 + Float64(-5.0 * Float64(v * v))) / Float64(t * Float64(Float64(pi * Float64(1.0 - (v ^ 2.0))) * sqrt(Float64(2.0 + Float64(2.0 * Float64(Float64(v * v) * -3.0)))))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + (-5.0 * (v * v))) / (t * ((pi * (1.0 - (v ^ 2.0))) * sqrt((2.0 + (2.0 * ((v * v) * -3.0))))));
end

Wolfram

code[v_, t_] := N[(N[(1.0 + N[(-5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(Pi * N[(1.0 - N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{1 + -5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - \left(3 \cdot v\right) \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 98.5%

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

   2. pow3 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \frac{1 + -5 \cdot \left(v \cdot v\right)}{\left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - \left(3 \cdot v\right) \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)} \]
7. Step-by-step derivation

   1. pow1 98.5%

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

   2. rem-cbrt-cube 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. pow2 99.3%

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

   6. pow2 99.3%

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

8. Applied egg-rr 99.3%

   \[\leadsto \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{{\left(\left(t \cdot \pi\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot {v}^{2}\right)} \cdot \left(1 - {v}^{2}\right)\right)\right)}^{1}}} \]
9. Step-by-step derivation

   1. unpow1 99.3%

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

   2. associate-*l* 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*r* 99.5%

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

   5. sub-neg 99.5%

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

   6. distribute-lft-in 99.5%

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

   7. metadata-eval 99.5%

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

   8. *-commutative 99.5%

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

   9. distribute-rgt-neg-in 99.5%

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

   10. metadata-eval 99.5%

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

10. Simplified 99.5%

    \[\leadsto \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{t \cdot \left(\left(\pi \cdot \left(1 - {v}^{2}\right)\right) \cdot \sqrt{2 + 2 \cdot \left({v}^{2} \cdot -3\right)}\right)}} \]
11. Step-by-step derivation

    1. pow2 99.5%

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

12. Applied egg-rr 99.5%

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

Alternative 2: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi \cdot \sqrt{2}}}{t} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 (* PI (sqrt 2.0))) t))

C

double code(double v, double t) {
	return (1.0 / (((double) M_PI) * sqrt(2.0))) / t;
}

Java

public static double code(double v, double t) {
	return (1.0 / (Math.PI * Math.sqrt(2.0))) / t;
}

Python

def code(v, t):
	return (1.0 / (math.pi * math.sqrt(2.0))) / t

Julia

function code(v, t)
	return Float64(Float64(1.0 / Float64(pi * sqrt(2.0))) / t)
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / (pi * sqrt(2.0))) / t;
end

Wolfram

code[v_, t_] := N[(N[(1.0 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
6. Step-by-step derivation

   1. inv-pow 99.0%

      \[\leadsto \color{blue}{{\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right)}^{-1}} \]

   2. unpow-prod-down 99.1%

      \[\leadsto \color{blue}{{t}^{-1} \cdot {\left(\pi \cdot \sqrt{2}\right)}^{-1}} \]

   3. inv-pow 99.1%

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

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{t} \cdot {\left(\pi \cdot \sqrt{2}\right)}^{-1}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. unpow-1 99.4%

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

9. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi \cdot \sqrt{2}}}{t}} \]
10. Add Preprocessing

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi}}{t \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 PI) (* t (sqrt 2.0))))

C

double code(double v, double t) {
	return (1.0 / ((double) M_PI)) / (t * sqrt(2.0));
}

Java

public static double code(double v, double t) {
	return (1.0 / Math.PI) / (t * Math.sqrt(2.0));
}

Python

def code(v, t):
	return (1.0 / math.pi) / (t * math.sqrt(2.0))

Julia

function code(v, t)
	return Float64(Float64(1.0 / pi) / Float64(t * sqrt(2.0)))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / pi) / (t * sqrt(2.0));
end

Wolfram

code[v_, t_] := N[(N[(1.0 / Pi), $MachinePrecision] / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
6. Step-by-step derivation

   1. inv-pow 99.0%

      \[\leadsto \color{blue}{{\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right)}^{-1}} \]

   2. unpow-prod-down 99.1%

      \[\leadsto \color{blue}{{t}^{-1} \cdot {\left(\pi \cdot \sqrt{2}\right)}^{-1}} \]

   3. inv-pow 99.1%

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

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{t} \cdot {\left(\pi \cdot \sqrt{2}\right)}^{-1}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. unpow-1 99.4%

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

9. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi \cdot \sqrt{2}}}{t}} \]
10. Step-by-step derivation

    1. div-inv 99.1%

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

11. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{\frac{1}{\pi \cdot \sqrt{2}} \cdot \frac{1}{t}} \]
12. Step-by-step derivation

    1. *-commutative 99.1%

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

    2. associate-/r* 99.1%

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

    3. frac-times 99.1%

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

    4. *-un-lft-identity 99.1%

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

13. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}} \]
14. Add Preprocessing

Alternative 4: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 t) (* PI (sqrt 2.0))))

C

double code(double v, double t) {
	return (1.0 / t) / (((double) M_PI) * sqrt(2.0));
}

Java

public static double code(double v, double t) {
	return (1.0 / t) / (Math.PI * Math.sqrt(2.0));
}

Python

def code(v, t):
	return (1.0 / t) / (math.pi * math.sqrt(2.0))

Julia

function code(v, t)
	return Float64(Float64(1.0 / t) / Float64(pi * sqrt(2.0)))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / t) / (pi * sqrt(2.0));
end

Wolfram

code[v_, t_] := N[(N[(1.0 / t), $MachinePrecision] / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 99.0%

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

7. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}}} \]
8. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 (* t (* PI (sqrt 2.0)))))

C

double code(double v, double t) {
	return 1.0 / (t * (((double) M_PI) * sqrt(2.0)));
}

Java

public static double code(double v, double t) {
	return 1.0 / (t * (Math.PI * Math.sqrt(2.0)));
}

Python

def code(v, t):
	return 1.0 / (t * (math.pi * math.sqrt(2.0)))

Julia

function code(v, t)
	return Float64(1.0 / Float64(t * Float64(pi * sqrt(2.0))))
end

MATLAB

function tmp = code(v, t)
	tmp = 1.0 / (t * (pi * sqrt(2.0)));
end

Wolfram

code[v_, t_] := N[(1.0 / N[(t * N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
6. Add Preprocessing

Alternative 6: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sqrt{0.5}}{t}}{\pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ (sqrt 0.5) t) PI))

C

double code(double v, double t) {
	return (sqrt(0.5) / t) / ((double) M_PI);
}

Java

public static double code(double v, double t) {
	return (Math.sqrt(0.5) / t) / Math.PI;
}

Python

def code(v, t):
	return (math.sqrt(0.5) / t) / math.pi

Julia

function code(v, t)
	return Float64(Float64(sqrt(0.5) / t) / pi)
end

MATLAB

function tmp = code(v, t)
	tmp = (sqrt(0.5) / t) / pi;
end

Wolfram

code[v_, t_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\pi \cdot t}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{fma}\left(v, v, -1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 98.3%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-/r* 98.4%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]

7. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]
8. Add Preprocessing

Alternative 7: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.5}}{t \cdot \pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (sqrt 0.5) (* t PI)))

C

double code(double v, double t) {
	return sqrt(0.5) / (t * ((double) M_PI));
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) / (t * Math.PI);
}

Python

def code(v, t):
	return math.sqrt(0.5) / (t * math.pi)

Julia

function code(v, t)
	return Float64(sqrt(0.5) / Float64(t * pi))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) / (t * pi);
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] / N[(t * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(5, v \cdot v, -1\right)}{\pi \cdot t}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\mathsf{fma}\left(v, v, -1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 98.3%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024149 
(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)))))