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

Percentage Accurate: 99.3% → 98.8%

Time: 10.4s

Alternatives: 9

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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}}{t} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. add-sqr-sqrt 96.9%

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

   2. *-un-lft-identity 96.9%

      \[\leadsto \frac{\sqrt{\sqrt{0.5}} \cdot \sqrt{\sqrt{0.5}}}{\color{blue}{1 \cdot \left(t \cdot \pi\right)}} \]

   3. times-frac 97.2%

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

   4. pow1/2 97.2%

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

   5. sqrt-pow1 98.2%

      \[\leadsto \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{1} \cdot \frac{\sqrt{\sqrt{0.5}}}{t \cdot \pi} \]

   6. metadata-eval 98.2%

      \[\leadsto \frac{{0.5}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{0.5}}}{t \cdot \pi} \]

   7. pow1/2 98.2%

      \[\leadsto \frac{{0.5}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{0.5}^{0.5}}}}{t \cdot \pi} \]

   8. sqrt-pow1 98.3%

      \[\leadsto \frac{{0.5}^{0.25}}{1} \cdot \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{t \cdot \pi} \]

   9. metadata-eval 98.3%

      \[\leadsto \frac{{0.5}^{0.25}}{1} \cdot \frac{{0.5}^{\color{blue}{0.25}}}{t \cdot \pi} \]

7. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{{0.5}^{0.25}}{1} \cdot \frac{{0.5}^{0.25}}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. /-rgt-identity 98.3%

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

   2. associate-*r/ 98.4%

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

   3. times-frac 98.6%

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

   4. associate-*l/ 99.2%

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

   5. associate-*r/ 99.2%

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

   6. pow-sqr 98.2%

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

   7. metadata-eval 98.2%

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

9. Simplified 98.2%

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

    1. pow1/2 98.2%

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

    2. clear-num 99.2%

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

    3. inv-pow 99.2%

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

11. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{{\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}}}{t} \]
12. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 4: 98.5% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / pi) / Float64(t * sqrt(2.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. div-inv 98.0%

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

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. associate-*r/ 98.0%

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

   2. frac-times 98.0%

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

   3. clear-num 98.0%

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

   4. associate-*l/ 98.1%

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

   5. *-un-lft-identity 98.1%

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

   6. div-inv 97.9%

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

   7. pow1/2 97.9%

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

   8. pow-flip 98.9%

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

   9. metadata-eval 98.9%

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

   10. add-sqr-sqrt 97.9%

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

   11. sqrt-unprod 98.9%

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

   12. pow-prod-up 98.9%

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

   13. metadata-eval 98.9%

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

   14. metadata-eval 98.9%

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

9. Applied egg-rr 98.9%

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

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

7. Applied egg-rr 98.0%

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

   1. unpow-1 98.0%

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

9. Simplified 98.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot \pi}{\sqrt{0.5}}}} \]
10. Step-by-step derivation

    1. associate-/l* 98.6%

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

    2. *-commutative 98.6%

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

11. Applied egg-rr 98.6%

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

Alternative 6: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. div-inv 98.0%

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

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. associate-/r* 97.9%

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

   2. *-commutative 97.9%

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

   3. associate-/r* 98.0%

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

   4. pow1/2 98.0%

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

   5. metadata-eval 98.0%

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

   6. pow-flip 98.3%

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

   7. frac-times 98.4%

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

   8. metadata-eval 98.4%

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

   9. *-commutative 98.4%

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

   10. add-sqr-sqrt 97.8%

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

   11. sqrt-unprod 98.4%

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

   12. pow-prod-up 98.4%

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

   13. metadata-eval 98.4%

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

   14. metadata-eval 98.4%

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

9. Applied egg-rr 98.4%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

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

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(Float64(sqrt(0.5) / pi) / t)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. add-sqr-sqrt 96.9%

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

   2. *-commutative 96.9%

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

   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. Taylor expanded in t around 0 98.0%

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

   1. associate-/l/ 98.2%

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

10. Simplified 98.2%

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

Alternative 8: 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

   1. add-sqr-sqrt 96.9%

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

   2. *-commutative 96.9%

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

   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/ 98.2%

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

   2. associate-*r/ 98.8%

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

   3. pow-sqr 98.2%

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

   4. metadata-eval 98.2%

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

   5. pow1/2 98.2%

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

9. Applied egg-rr 98.2%

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

Alternative 9: 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in v around 0 98.0%

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

6. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024106 
(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)))))