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

Percentage Accurate: 99.2% → 99.3%

Time: 10.4s

Alternatives: 7

Speedup: 0.6×

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: 99.2% 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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{-2.5 \cdot \left(\frac{v}{\sqrt{2}} \cdot \frac{v}{\pi}\right) + \frac{1}{\sqrt{2} \cdot \pi}}{t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (+ (* -2.5 (* (/ v (sqrt 2.0)) (/ v PI))) (/ 1.0 (* (sqrt 2.0) PI))) t))

C

double code(double v, double t) {
	return ((-2.5 * ((v / sqrt(2.0)) * (v / ((double) M_PI)))) + (1.0 / (sqrt(2.0) * ((double) M_PI)))) / t;
}

Java

public static double code(double v, double t) {
	return ((-2.5 * ((v / Math.sqrt(2.0)) * (v / Math.PI))) + (1.0 / (Math.sqrt(2.0) * Math.PI))) / t;
}

Python

def code(v, t):
	return ((-2.5 * ((v / math.sqrt(2.0)) * (v / math.pi))) + (1.0 / (math.sqrt(2.0) * math.pi))) / t

Julia

function code(v, t)
	return Float64(Float64(Float64(-2.5 * Float64(Float64(v / sqrt(2.0)) * Float64(v / pi))) + Float64(1.0 / Float64(sqrt(2.0) * pi))) / t)
end

MATLAB

function tmp = code(v, t)
	tmp = ((-2.5 * ((v / sqrt(2.0)) * (v / pi))) + (1.0 / (sqrt(2.0) * pi))) / t;
end

Wolfram

code[v_, t_] := N[(N[(N[(-2.5 * N[(N[(v / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(v / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around 0 98.5%

   \[\leadsto \color{blue}{-2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
6. Step-by-step derivation

   1. fma-def 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2.5, \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}, \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)} \]

   2. associate-/r* 98.5%

      \[\leadsto \mathsf{fma}\left(-2.5, \color{blue}{\frac{\frac{{v}^{2}}{t}}{\pi \cdot \sqrt{2}}}, \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right) \]

   3. associate-/r* 98.7%

      \[\leadsto \mathsf{fma}\left(-2.5, \frac{\frac{{v}^{2}}{t}}{\pi \cdot \sqrt{2}}, \color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}}}\right) \]

7. Simplified 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2.5, \frac{\frac{{v}^{2}}{t}}{\pi \cdot \sqrt{2}}, \frac{\frac{1}{t}}{\pi \cdot \sqrt{2}}\right)} \]

8. Taylor expanded in t around 0 99.2%

   \[\leadsto \color{blue}{\frac{-2.5 \cdot \frac{{v}^{2}}{\pi \cdot \sqrt{2}} + \frac{1}{\pi \cdot \sqrt{2}}}{t}} \]
9. Step-by-step derivation

   1. unpow2 99.2%

      \[\leadsto \frac{-2.5 \cdot \frac{\color{blue}{v \cdot v}}{\pi \cdot \sqrt{2}} + \frac{1}{\pi \cdot \sqrt{2}}}{t} \]

   2. *-commutative 99.2%

      \[\leadsto \frac{-2.5 \cdot \frac{v \cdot v}{\color{blue}{\sqrt{2} \cdot \pi}} + \frac{1}{\pi \cdot \sqrt{2}}}{t} \]

   3. times-frac 99.2%

      \[\leadsto \frac{-2.5 \cdot \color{blue}{\left(\frac{v}{\sqrt{2}} \cdot \frac{v}{\pi}\right)} + \frac{1}{\pi \cdot \sqrt{2}}}{t} \]

10. Applied egg-rr 99.2%

    \[\leadsto \frac{-2.5 \cdot \color{blue}{\left(\frac{v}{\sqrt{2}} \cdot \frac{v}{\pi}\right)} + \frac{1}{\pi \cdot \sqrt{2}}}{t} \]

11. Final simplification 99.2%

    \[\leadsto \frac{-2.5 \cdot \left(\frac{v}{\sqrt{2}} \cdot \frac{v}{\pi}\right) + \frac{1}{\sqrt{2} \cdot \pi}}{t} \]
12. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 98.3% accurate, 1.2× 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around 0 97.9%

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

6. Final simplification 97.9%

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

Alternative 4: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around 0 97.9%

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

   1. associate-/r* 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

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

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in t around 0 99.1%

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

   1. associate-*l* 99.1%

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

   2. *-commutative 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in v around 0 97.9%

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

   1. associate-/r* 98.2%

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

   2. *-rgt-identity 98.2%

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

   3. associate-*r/ 98.3%

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

   4. associate-*l/ 98.6%

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

   5. *-lft-identity 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around 0 97.4%

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

6. Final simplification 97.4%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in v around 0 97.4%

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

   1. expm1-log1p-u 74.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{t \cdot \pi}\right)\right)} \]

   2. expm1-udef 25.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{0.5}}{t \cdot \pi}\right)} - 1} \]

7. Applied egg-rr 25.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{0.5}}{t \cdot \pi}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 74.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{0.5}}{t \cdot \pi}\right)\right)} \]

   2. expm1-log1p 97.4%

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

   3. associate-/r* 97.7%

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Reproduce

herbie shell --seed 2024027 
(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)))))