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

Percentage Accurate: 99.3% → 99.5%

Time: 11.0s

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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(N[(N[Power[v, 2.0], $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / t), $MachinePrecision] / N[(N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[Power[v, 2.0], $MachinePrecision] * -3.0 + 1.0), $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. Taylor expanded in t around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-/r* 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. fma-undefine 99.6%

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

   7. associate-*r* 99.6%

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

   8. cancel-sign-sub-inv 99.6%

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

   9. metadata-eval 99.6%

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

   10. *-commutative 99.6%

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

   11. +-commutative 99.6%

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

   12. fma-define 99.6%

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

5. Simplified 99.6%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(t * N[(v * (-v) + 1.0), $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 99.5%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)))) * (t * Math.PI)) * (1.0 - (v * v)));
}

Python

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

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(Float64(v * v) * 3.0)))) * Float64(t * pi)) * Float64(1.0 - Float64(v * v))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - ((v * v) * 5.0)) / ((sqrt((2.0 * (1.0 - ((v * v) * 3.0)))) * (t * pi)) * (1.0 - (v * v)));
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[(N[(v * v), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $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%

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

Alternative 4: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*r* 97.7%

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

5. Simplified 97.7%

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

6. Taylor expanded in t around 0 97.8%

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

   1. associate-*r* 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in t around 0 97.8%

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

    1. *-commutative 97.8%

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

    2. associate-/r* 98.5%

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

    3. associate-/r* 98.5%

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

11. Simplified 98.5%

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

Alternative 5: 98.4% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t), $MachinePrecision] / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0 97.8%

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

   1. inv-pow 97.8%

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

   2. associate-*r* 97.7%

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

   3. unpow-prod-down 98.0%

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

   4. inv-pow 98.0%

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

5. Applied egg-rr 98.0%

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

   1. unpow-1 98.0%

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

7. Simplified 98.0%

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

   1. *-commutative 98.0%

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

   2. associate-/r* 97.9%

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

   3. frac-times 98.2%

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

   4. *-un-lft-identity 98.2%

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

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

Alternative 6: 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 99.1%

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

3. Taylor expanded in v around 0 97.8%

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

Alternative 7: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / N[(t * Pi), $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 97.5%

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

   1. div-inv 97.6%

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

7. Applied egg-rr 97.6%

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

Alternative 8: 97.8% accurate, 1.2× speedup

Math

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

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] / N[(t * Pi), $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 97.5%

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

Reproduce

herbie shell --seed 2024089 
(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)))))