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

Percentage Accurate: 99.3% → 99.3%

Time: 9.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((Math.PI * t) * Math.sqrt((2.0 * (1.0 - ((v * v) * 3.0))))) * (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 - ((v * v) * 3.0))))) * (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(Float64(v * v) * 3.0))))) * Float64(1.0 - Float64(v * v))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in v around 0 98.6%

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

   1. associate-*r* 98.6%

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

7. Simplified 98.6%

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

   1. inv-pow 98.6%

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

   2. associate-*l* 98.6%

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

   3. unpow-prod-down 98.6%

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

9. Applied egg-rr 98.6%

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

    1. pow-prod-down 98.6%

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

    2. inv-pow 98.6%

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

    3. associate-*r* 98.6%

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

    4. *-commutative 98.6%

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

    5. associate-/r* 98.5%

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

    6. associate-/r* 98.7%

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

11. Applied egg-rr 98.7%

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

Alternative 4: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in v around 0 98.6%

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

Alternative 5: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.9%

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

   1. *-un-lft-identity 97.9%

      \[\leadsto \frac{\color{blue}{1 \cdot \sqrt{0.5}}}{t \cdot \pi} \]

   2. times-frac 98.0%

      \[\leadsto \color{blue}{\frac{1}{t} \cdot \frac{\sqrt{0.5}}{\pi}} \]

7. Applied egg-rr 98.0%

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

Alternative 6: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.9%

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

   1. div-inv 98.0%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

   2. *-commutative 98.0%

      \[\leadsto \sqrt{0.5} \cdot \frac{1}{\color{blue}{\pi \cdot t}} \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{\pi \cdot t}} \]
8. 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 99.5%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.9%

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

   1. add-sqr-sqrt 96.8%

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

   2. *-commutative 96.8%

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

   3. times-frac 97.5%

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

   4. pow1/2 97.5%

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

   5. sqrt-pow1 97.5%

      \[\leadsto \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{\pi} \cdot \frac{\sqrt{\sqrt{0.5}}}{t} \]

   6. metadata-eval 97.5%

      \[\leadsto \frac{{0.5}^{\color{blue}{0.25}}}{\pi} \cdot \frac{\sqrt{\sqrt{0.5}}}{t} \]

   7. pow1/2 97.5%

      \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{\sqrt{\color{blue}{{0.5}^{0.5}}}}{t} \]

   8. sqrt-pow1 98.6%

      \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{t} \]

   9. metadata-eval 98.6%

      \[\leadsto \frac{{0.5}^{0.25}}{\pi} \cdot \frac{{0.5}^{\color{blue}{0.25}}}{t} \]

7. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{\frac{{0.5}^{0.25}}{\pi} \cdot \frac{{0.5}^{0.25}}{t}} \]
8. Step-by-step derivation

   1. associate-*l/ 98.3%

      \[\leadsto \color{blue}{\frac{{0.5}^{0.25} \cdot \frac{{0.5}^{0.25}}{t}}{\pi}} \]

   2. associate-*r/ 98.5%

      \[\leadsto \frac{\color{blue}{\frac{{0.5}^{0.25} \cdot {0.5}^{0.25}}{t}}}{\pi} \]

   3. pow-prod-up 97.9%

      \[\leadsto \frac{\frac{\color{blue}{{0.5}^{\left(0.25 + 0.25\right)}}}{t}}{\pi} \]

   4. metadata-eval 97.9%

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

   5. pow1/2 97.9%

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

9. Applied egg-rr 97.9%

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

Alternative 8: 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 99.5%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.9%

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

6. Final simplification 97.9%

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

Reproduce

herbie shell --seed 2024191 
(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)))))