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

Percentage Accurate: 99.3% → 99.5%

Time: 26.3s

Alternatives: 11

Speedup: 1.0×

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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. associate-/r* 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. fma-undefine 99.5%

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

   7. associate-*r* 99.5%

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

   8. cancel-sign-sub-inv 99.5%

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

   9. metadata-eval 99.5%

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

   10. *-commutative 99.5%

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

   11. +-commutative 99.5%

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

   12. fma-define 99.5%

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

5. Simplified 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / ((t * ((pi * sqrt(2.0)) * sqrt((1.0 + ((v ^ 2.0) * -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[(t * N[(N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in t around 0 99.5%

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

   1. associate-*l* 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. *-commutative 99.5%

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

5. Simplified 99.5%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* v (* v 5.0)))
  (* t (* (* PI (- 1.0 (pow v 2.0))) (sqrt (+ 2.0 (* (pow v 2.0) -6.0)))))))

C

double code(double v, double t) {
	return (1.0 - (v * (v * 5.0))) / (t * ((((double) M_PI) * (1.0 - pow(v, 2.0))) * sqrt((2.0 + (pow(v, 2.0) * -6.0)))));
}

Java

public static double code(double v, double t) {
	return (1.0 - (v * (v * 5.0))) / (t * ((Math.PI * (1.0 - Math.pow(v, 2.0))) * Math.sqrt((2.0 + (Math.pow(v, 2.0) * -6.0)))));
}

Python

def code(v, t):
	return (1.0 - (v * (v * 5.0))) / (t * ((math.pi * (1.0 - math.pow(v, 2.0))) * math.sqrt((2.0 + (math.pow(v, 2.0) * -6.0)))))

Julia

function code(v, t)
	return Float64(Float64(1.0 - Float64(v * Float64(v * 5.0))) / Float64(t * Float64(Float64(pi * Float64(1.0 - (v ^ 2.0))) * sqrt(Float64(2.0 + Float64((v ^ 2.0) * -6.0))))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (v * (v * 5.0))) / (t * ((pi * (1.0 - (v ^ 2.0))) * sqrt((2.0 + ((v ^ 2.0) * -6.0)))));
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(v * N[(v * 5.0), $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[(N[Power[v, 2.0], $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.4%

      \[\leadsto \frac{1 - v \cdot \left(5 \cdot v\right)}{\sqrt{2 + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(v \cdot v\right)\right)} \cdot -3\right) \cdot 2} \cdot \left(\left(\pi \cdot t\right) \cdot \left(1 - v \cdot v\right)\right)} \]

   2. expm1-undefine 99.4%

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

   3. log1p-undefine 99.4%

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

   4. add-exp-log 99.4%

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

   5. pow2 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in v around inf 53.1%

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

   1. distribute-rgt-in 53.1%

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

   2. *-lft-identity 53.1%

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

   3. unpow2 53.1%

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

   4. lft-mult-inverse 99.4%

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

   5. fma-undefine 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in t around 0 99.4%

    \[\leadsto \frac{1 - v \cdot \left(5 \cdot v\right)}{\color{blue}{\left(t \cdot \left(\pi \cdot \left(1 - {v}^{2}\right)\right)\right) \cdot \sqrt{2 + -6 \cdot {v}^{2}}}} \]
11. Step-by-step derivation

    1. associate-*l* 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}}{\pi \cdot \left(t \cdot \left(1 - v \cdot v\right)\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/ (fma (* v v) -5.0 1.0) (sqrt (+ 2.0 (* (* v v) -6.0))))
  (* PI (* t (- 1.0 (* v v))))))

C

double code(double v, double t) {
	return (fma((v * v), -5.0, 1.0) / sqrt((2.0 + ((v * v) * -6.0)))) / (((double) M_PI) * (t * (1.0 - (v * v))));
}

Julia

function code(v, t)
	return Float64(Float64(fma(Float64(v * v), -5.0, 1.0) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0)))) / Float64(pi * Float64(t * Float64(1.0 - Float64(v * v)))))
end

Wolfram

code[v_, t_] := N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(t * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 6: 98.8% 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 99.4%

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

3. Taylor expanded in v around 0 98.6%

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

   1. associate-*r* 98.5%

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

5. Simplified 98.5%

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

   1. inv-pow 98.5%

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

   2. associate-*l* 98.6%

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

   3. unpow-prod-down 98.6%

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

   4. inv-pow 98.6%

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

7. Applied egg-rr 98.6%

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

   1. associate-*l/ 98.9%

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

   2. *-un-lft-identity 98.9%

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

   3. unpow-1 98.9%

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

9. Applied egg-rr 98.9%

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

Alternative 7: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (/ (/ 1.0 t) (/ PI (sqrt 0.5))))

C

double code(double v, double t) {
	return (1.0 / t) / (((double) M_PI) / sqrt(0.5));
}

Java

public static double code(double v, double t) {
	return (1.0 / t) / (Math.PI / Math.sqrt(0.5));
}

Python

def code(v, t):
	return (1.0 / t) / (math.pi / math.sqrt(0.5))

Julia

function code(v, t)
	return Float64(Float64(1.0 / t) / Float64(pi / sqrt(0.5)))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 / t) / (pi / sqrt(0.5));
end

Wolfram

code[v_, t_] := N[(N[(1.0 / t), $MachinePrecision] / N[(Pi / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

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 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

      \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{{\left(\frac{t \cdot \pi}{\sqrt{0.5}}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 98.0%

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

   2. associate-/l* 98.6%

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

   3. associate-/r* 98.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}}} \]

9. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\frac{\pi}{\sqrt{0.5}}}} \]
10. Add Preprocessing

Alternative 8: 98.4% 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 99.4%

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

3. Taylor expanded in v around 0 98.6%

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

Alternative 9: 97.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \frac{1}{t \cdot \pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (sqrt 0.5) (/ 1.0 (* t PI))))

C

double code(double v, double t) {
	return sqrt(0.5) * (1.0 / (t * ((double) M_PI)));
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) * (1.0 / (t * Math.PI));
}

Python

def code(v, t):
	return math.sqrt(0.5) * (1.0 / (t * math.pi))

Julia

function code(v, t)
	return Float64(sqrt(0.5) * Float64(1.0 / Float64(t * pi)))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) * (1.0 / (t * pi));
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / N[(t * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

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 98.0%

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

   1. div-inv 98.1%

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

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]
8. Add Preprocessing

Alternative 10: 97.9% 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 99.4%

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

3. Simplified 99.5%

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

5. Taylor expanded in v around 0 98.6%

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

6. Taylor expanded in v around 0 98.0%

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

   1. associate-/r* 98.1%

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

8. Simplified 98.1%

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

Alternative 11: 97.8% 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 99.4%

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

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 98.0%

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

Reproduce

herbie shell --seed 2024143 
(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)))))