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

Percentage Accurate: 99.3% → 99.8%

Time: 10.7s

Alternatives: 6

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right), \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\left(5 \cdot v\right) \cdot v\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(v \cdot \left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(v \cdot \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(v \cdot 5\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(v \cdot \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(t \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

3. Simplified 99.5%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right), \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\left(5 \cdot v\right) \cdot v\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(v \cdot \left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(v \cdot \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(v \cdot 5\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(v \cdot \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(t \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 98.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.4%

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

6. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 98.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\sqrt{2} \cdot \mathsf{PI}\left(\right)}\right), \color{blue}{t}\right) \]

   6. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)}^{-1}\right), t\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}^{-1}\right), t\right) \]

   8. pow-prod-down N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\mathsf{PI}\left(\right)}^{-1} \cdot {\left(\sqrt{2}\right)}^{-1}\right), t\right) \]

   9. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot {\left(\sqrt{2}\right)}^{-1}\right), t\right) \]

   10. inv-pow N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \frac{1}{\sqrt{2}}\right), t\right) \]

   11. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{2}}\right), t\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right), \left(\sqrt{2}\right)\right), t\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right), \left(\sqrt{2}\right)\right), t\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \left(\sqrt{2}\right)\right), t\right) \]

   15. sqrt-lowering-sqrt.f64 98.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), t\right) \]

10. Applied egg-rr 98.9%

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

Alternative 4: 98.6% 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%

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

3. Taylor expanded in v around 0

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

   1. associate-/r* N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{t}\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 98.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

5. Simplified 98.6%

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

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

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(1.0 / Float64(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[(1.0 / N[(Pi * N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 98.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.4%

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

6. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]
7. Step-by-step derivation

8. Simplified 98.4%

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

Alternative 6: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] / N[(Pi * t), $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. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right), \color{blue}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right), \left(\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\left(5 \cdot v\right) \cdot v\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(v \cdot \left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(v \cdot \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5 \cdot v\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(v \cdot 5\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(v \cdot \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(t \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)\right)}\right)\right) \]

3. Simplified 99.5%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(t \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{t} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   4. PI-lowering-PI.f64 98.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right) \]

7. Simplified 98.1%

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

8. Final simplification 98.1%

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

Reproduce

herbie shell --seed 2024139 
(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)))))