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

Percentage Accurate: 99.3% → 99.6%

Time: 5.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (- 1 (* 5 (* v v)))
 (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* 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 - N[(5 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2 * N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (- 1 (* 5 (* v v)))
 (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* 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 - N[(5 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Pi * t), $MachinePrecision] * N[Sqrt[N[(2 * N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (/ (/ (- 1 (* 5 (* v v))) (* PI (- 1 (* v v)))) t)
 (sqrt (* 2 (- 1 (* 3 (* v v)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(N[(1 - N[(5 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Sqrt[N[(2 * N[(1 - N[(3 * N[(v * v), $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. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

3. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.6%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 99.6%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6%

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

5. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (/ (- 1 (* (* v v) 5)) (* (- 1 (* v v)) PI))
 (* (sqrt (* (- 1 (* 3 (* v v))) 2)) t)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1 - N[(N[(v * v), $MachinePrecision] * 5), $MachinePrecision]), $MachinePrecision] / N[(N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

3. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (/ (- 1 (* (* v v) 5)) t)
 (* PI (* (- 1 (* v v)) (sqrt (* (- 1 (* 3 (* v v))) 2))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1 - N[(N[(v * v), $MachinePrecision] * 5), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[(Pi * N[(N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $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. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (- 1 (* 5 (* v v)))
 (* (* (* (sqrt (* (- 1 (* 3 (* v v))) 2)) PI) t) (- 1 (* v v)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1 - N[(5 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Sqrt[N[(N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision] * t), $MachinePrecision] * N[(1 - 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. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.4%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4%

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

3. Applied rewrites 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/
 (- 1 (* 5 (* v v)))
 (* PI (* t (* (- 1 (* v v)) (sqrt (* (- 1 (* 3 (* v v))) 2)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1 - N[(5 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(t * N[(N[(1 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $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. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.3%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.3%

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

3. Applied rewrites 99.3%

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

Alternative 6: 98.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ (/ (- 1 (* (* v v) 5)) (* PI (sqrt (* (- 1 (* 3 (* v v))) 2)))) t))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1 - N[(N[(v * v), $MachinePrecision] * 5), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[Sqrt[N[(N[(1 - N[(3 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2), $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. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in v around 0

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

   1. lower-PI.f64 98.7%

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

6. Applied rewrites 98.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 98.9%

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

Alternative 7: 98.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ (/ 1 (* (sqrt 2) PI)) t))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1 / N[(N[Sqrt[2], $MachinePrecision] * Pi), $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. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.5%

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

4. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.9%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.9%

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

6. Applied rewrites 98.9%

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

Alternative 8: 98.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ (/ 1 PI) (* (sqrt 2) t)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1 / Pi), $MachinePrecision] / N[(N[Sqrt[2], $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. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.5%

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

4. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 98.4%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.4%

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

6. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 98.4%

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

8. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. lift-/.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 98.6%

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

   12. lift-*.f64 N/A

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

   13. *-rgt-identity 98.6%

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 98.6%

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

10. Applied rewrites 98.6%

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

Alternative 9: 98.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ (/ 1 t) (* (sqrt 2) PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1 / t), $MachinePrecision] / N[(N[Sqrt[2], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.5%

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

4. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 98.4%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.4%

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

6. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5%

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

8. Applied rewrites 98.5%

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

Alternative 10: 98.5% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (v t)
  :precision binary64
  (/ 1 (* t (* PI (sqrt 2)))))

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 / N[(t * N[(Pi * N[Sqrt[2], $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. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-sqrt.f64 98.5%

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

4. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))