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

Percentage Accurate: 99.3% → 99.5%

Time: 12.8s

Alternatives: 10

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1.0 + N[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[(N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * t), $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

   1. associate-*l* N/A

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

   2. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 + v \cdot \left(v \cdot -5\right)}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(t \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + v \cdot \left(v \cdot -5\right)\right), \mathsf{PI}\left(\right)\right), \left(\color{blue}{t} \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(v \cdot \left(v \cdot -5\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(t \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(v \cdot -5\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(t \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{PI}\left(\right)\right), \left(t \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(t \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(\left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{t}\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \color{blue}{\left(\left(1 - v \cdot v\right) \cdot t\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(\left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right), \color{blue}{\left(\left(1 - v \cdot v\right) \cdot t\right)}\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 + N[(v * N[(v * -5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(N[(v * v), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(Pi * t), $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi} \cdot \left(1 + \left(v \cdot v\right) \cdot -4\right)}{t \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (* (/ 1.0 PI) (+ 1.0 (* (* v v) -4.0)))
  (* t (sqrt (+ 2.0 (* v (* v -6.0)))))))

C

double code(double v, double t) {
	return ((1.0 / ((double) M_PI)) * (1.0 + ((v * v) * -4.0))) / (t * sqrt((2.0 + (v * (v * -6.0)))));
}

Java

public static double code(double v, double t) {
	return ((1.0 / Math.PI) * (1.0 + ((v * v) * -4.0))) / (t * Math.sqrt((2.0 + (v * (v * -6.0)))));
}

Python

def code(v, t):
	return ((1.0 / math.pi) * (1.0 + ((v * v) * -4.0))) / (t * math.sqrt((2.0 + (v * (v * -6.0)))))

Julia

function code(v, t)
	return Float64(Float64(Float64(1.0 / pi) * Float64(1.0 + Float64(Float64(v * v) * -4.0))) / Float64(t * sqrt(Float64(2.0 + Float64(v * Float64(v * -6.0))))))
end

MATLAB

function tmp = code(v, t)
	tmp = ((1.0 / pi) * (1.0 + ((v * v) * -4.0))) / (t * sqrt((2.0 + (v * (v * -6.0)))));
end

Wolfram

code[v_, t_] := N[(N[(N[(1.0 / Pi), $MachinePrecision] * N[(1.0 + N[(N[(v * v), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Sqrt[N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1 + v \cdot \left(v \cdot -5\right)}{\mathsf{PI}\left(\right) \cdot t}}{1 - v \cdot v}\right), \color{blue}{\left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1 + v \cdot \left(v \cdot -5\right)}{\mathsf{PI}\left(\right) \cdot t}\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2 - \left(v \cdot v\right) \cdot 6}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 + v \cdot \left(v \cdot -5\right)\right), \left(\mathsf{PI}\left(\right) \cdot t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{\color{blue}{2} - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(v \cdot \left(v \cdot -5\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \left(v \cdot -5\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right)\right), \left(1 - v \cdot v\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right)\right), \mathsf{\_.f64}\left(1, \left(v \cdot v\right)\right)\right), \left(\sqrt{2 - \color{blue}{\left(v \cdot v\right) \cdot 6}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -5\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right), \left(\sqrt{2 - \left(v \cdot v\right) \cdot \color{blue}{6}}\right)\right) \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)} + \frac{1}{t \cdot \mathsf{PI}\left(\right)}\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{t \cdot \mathsf{PI}\left(\right)} + -4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{t \cdot \mathsf{PI}\left(\right)}\right), \left(-4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(t \cdot \mathsf{PI}\left(\right)\right)\right), \left(-4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI}\left(\right)\right)\right), \left(-4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \left(-4 \cdot \frac{{v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{-4 \cdot {v}^{2}}{t \cdot \mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)}\right)\right)\right) \]

   7. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \left(\frac{\frac{-4 \cdot {v}^{2}}{t}}{\mathsf{PI}\left(\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{-4 \cdot {v}^{2}}{t}\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-4 \cdot {v}^{2}\right), t\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(v, v\right)}, -6\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({v}^{2} \cdot -4\right), t\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{v}, v\right), -6\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({v}^{2}\right), -4\right), t\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{v}, v\right), -6\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(v \cdot v\right), -4\right), t\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -4\right), t\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

   14. PI-lowering-PI.f64 99.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(t, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -4\right), t\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \color{blue}{-6}\right)\right)\right)\right) \]

9. Simplified 99.2%

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

10. Taylor expanded in t around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{-4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \frac{1}{\mathsf{PI}\left(\right)}}{t}\right)}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(-1 \cdot 4\right) \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \frac{1}{\mathsf{PI}\left(\right)}}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)}\right) + \frac{1}{\mathsf{PI}\left(\right)}}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)}\right) + \frac{-1 \cdot -1}{\mathsf{PI}\left(\right)}}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \frac{-1}{\mathsf{PI}\left(\right)}}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \frac{\mathsf{neg}\left(1\right)}{\mathsf{PI}\left(\right)}}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    6. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)}\right) + -1 \cdot \left(\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    7. distribute-lft-in N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\frac{1}{\mathsf{PI}\left(\right)}\right)\right)\right)}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    8. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} - \frac{1}{\mathsf{PI}\left(\right)}\right)}{t}\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(4 \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right)} - \frac{1}{\mathsf{PI}\left(\right)}\right)\right), t\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -6\right)\right)}\right)\right) \]

12. Simplified 99.3%

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

    1. associate-/l/ N/A

       \[\leadsto \frac{\frac{-4 \cdot \left(v \cdot v\right)}{\mathsf{PI}\left(\right)} + \frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t}} \]

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

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-4 \cdot \left(v \cdot v\right)}{\mathsf{PI}\left(\right)} + \frac{1}{\mathsf{PI}\left(\right)}\right), \color{blue}{\left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t\right)}\right) \]

    3. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(-4 \cdot \left(v \cdot v\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)} + \frac{1}{\mathsf{PI}\left(\right)}\right), \left(\sqrt{\color{blue}{2 + \left(v \cdot v\right) \cdot -6}} \cdot t\right)\right) \]

    4. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(-4 \cdot \left(v \cdot v\right)\right) \cdot \frac{1}{\mathsf{PI}\left(\right)} + 1 \cdot \frac{1}{\mathsf{PI}\left(\right)}\right), \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t\right)\right) \]

    5. distribute-rgt-out N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \left(-4 \cdot \left(v \cdot v\right) + 1\right)\right), \left(\color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot t\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right), \left(-4 \cdot \left(v \cdot v\right) + 1\right)\right), \left(\color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \cdot t\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right), \left(-4 \cdot \left(v \cdot v\right) + 1\right)\right), \left(\sqrt{\color{blue}{2 + \left(v \cdot v\right) \cdot -6}} \cdot t\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \left(-4 \cdot \left(v \cdot v\right) + 1\right)\right), \left(\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot -6}} \cdot t\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\left(-4 \cdot \left(v \cdot v\right)\right), 1\right)\right), \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \left(v \cdot v\right)\right), 1\right)\right), \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \left(\sqrt{2 + \left(v \cdot v\right) \cdot -6} \cdot t\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \left(t \cdot \color{blue}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2 + \left(v \cdot v\right) \cdot -6}\right)}\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\left(2 + \left(v \cdot v\right) \cdot -6\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(v \cdot v\right) \cdot -6\right)\right)\right)\right)\right) \]

    16. associate-*l* N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(v \cdot \left(v \cdot -6\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \left(v \cdot -6\right)\right)\right)\right)\right)\right) \]

    18. *-lowering-*.f64 99.3%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(v, v\right)\right), 1\right)\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, -6\right)\right)\right)\right)\right)\right) \]

14. Applied egg-rr 99.3%

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

15. Final simplification 99.3%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1.0 + N[(-5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.6%

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

7. Simplified 98.6%

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

   1. associate-*l* N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   12. sqrt-lowering-sqrt.f64 98.8%

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

9. Applied egg-rr 98.8%

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

    1. associate-/l/ N/A

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

    2. associate-/r* N/A

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

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

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

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

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

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

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

    6. *-commutative N/A

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

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

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

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

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

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

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

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

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

    11. sqrt-lowering-sqrt.f64 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 5: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 / N[(Pi / N[Sqrt[0.5], $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f64 98.1%

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

9. Applied egg-rr 98.1%

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

    1. clear-num N/A

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

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

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 99.0%

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

11. Applied egg-rr 99.0%

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

Alternative 6: 98.2% accurate, 1.2× speedup

Math

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

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 / Float64(pi * 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 / N[(Pi * t), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around 0

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

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

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

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

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.6%

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

10. Simplified 98.6%

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

    1. associate-*r* N/A

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

    2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

    7. sqrt-lowering-sqrt.f64 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 7: 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 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around 0

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

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

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

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

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

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

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

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

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

   5. sqrt-lowering-sqrt.f64 98.6%

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

10. Simplified 98.6%

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

Alternative 8: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] / Pi), $MachinePrecision] * N[(1.0 / t), $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. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

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

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

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

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

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

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

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

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

   8. /-lowering-/.f64 98.2%

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

9. Applied egg-rr 98.2%

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

Alternative 9: 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 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

   5. PI-lowering-PI.f64 98.1%

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

9. Applied egg-rr 98.1%

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

Alternative 10: 97.8% 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 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1 + v \cdot \left(v \cdot -5\right)}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 - \left(v \cdot v\right) \cdot 6} \cdot \left(1 - v \cdot v\right)\right)}} \]
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 2024163 
(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)))))