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

Percentage Accurate: 99.4% → 99.5%

Time: 3.7s

Alternatives: 8

Speedup: 1.1×

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in t around 0

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

12. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 4: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / (((math.sqrt(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(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(2.0) * pi) * t) * (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[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision] * t), $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. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 5: 98.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 6: 98.5% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ 1.0 (* (* (sqrt 2.0) 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(1.0 / Float64(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[(1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around 0

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

   1. associate-*r* 98.4

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

10. Applied rewrites 98.4%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lift-sqrt.f64 N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lift-PI.f64 98.6

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

12. Applied rewrites 98.6%

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

Alternative 7: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around 0

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

   1. associate-*r* 98.4

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

10. Applied rewrites 98.4%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lift-sqrt.f64 N/A

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

    11. lift-PI.f64 98.5

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

12. Applied rewrites 98.5%

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

Alternative 8: 98.3% accurate, 2.4× 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. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in v around 0

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

   1. associate-*r* 98.4

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

10. Applied rewrites 98.4%

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

Reproduce

herbie shell --seed 2025065 
(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)))))