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

Percentage Accurate: 99.3% → 99.8%

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

   2. sub-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. associate-*l* N/A

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

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

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

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

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

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

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

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

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

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

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

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

5. Simplified 98.7%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f64 98.9%

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

7. Applied egg-rr 98.9%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. sqrt-lowering-sqrt.f64 98.9%

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

9. Applied egg-rr 98.9%

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

    1. cancel-sign-sub-inv N/A

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

    2. metadata-eval N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    13. *-commutative N/A

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

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

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

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

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

    16. *-lowering-*.f64 99.1%

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

11. Applied egg-rr 99.1%

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

12. Final simplification 99.1%

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

Alternative 5: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0

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

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

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

   2. sqrt-lowering-sqrt.f64 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around 0

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

10. Simplified 98.7%

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

    1. associate-/r* N/A

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

    2. *-commutative N/A

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

    3. associate-/r* N/A

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

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

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

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

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

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

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

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

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

    8. sqrt-lowering-sqrt.f64 99.1%

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

12. Applied egg-rr 99.1%

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

Alternative 6: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in v around 0

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   6. sqrt-lowering-sqrt.f64 98.9%

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

5. Simplified 98.9%

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

Alternative 7: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Initial program 99.3%

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

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0

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

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

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

   2. sqrt-lowering-sqrt.f64 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around 0

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

10. Simplified 98.7%

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

    1. associate-*r* N/A

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

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

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

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

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

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

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

    5. sqrt-lowering-sqrt.f64 98.7%

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

12. Applied egg-rr 98.7%

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

Alternative 8: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0

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

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

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

   2. sqrt-lowering-sqrt.f64 98.7%

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

7. Simplified 98.7%

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

8. Taylor expanded in v around 0

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

10. Simplified 98.7%

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

Alternative 9: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

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

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

   2. sub-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. associate-*l* N/A

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0

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

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

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

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

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

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

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

   4. PI-lowering-PI.f64 98.4%

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

7. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
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 98.4%

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

9. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]
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 99.3%

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

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

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

   2. sub-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. metadata-eval N/A

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

   12. associate-*l* N/A

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

   13. associate-*l* N/A

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

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

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

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

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

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

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0

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

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

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

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

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

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

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

   4. PI-lowering-PI.f64 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024288 
(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)))))