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

Percentage Accurate: 99.3% → 99.4%

Time: 8.2s

Alternatives: 6

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: 99.4% 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 99.3%

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

3. Simplified 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* (* v v) 5.0))
  (* (* (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0)))) (* PI t)) (- 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)))) * (((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 * (1.0 - ((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 * (1.0 - ((v * v) * 3.0)))) * (math.pi * t)) * (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(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 * (1.0 - ((v * v) * 3.0)))) * (pi * t)) * (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[(Pi * t), $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. Final simplification 99.3%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in v around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.8%

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

   1. add-sqr-sqrt 44.9%

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

   2. sqrt-unprod 27.9%

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

   3. frac-times 27.7%

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

   4. rem-square-sqrt 27.9%

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

   5. pow2 27.9%

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

7. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{\sqrt{\frac{0.5}{{\left(t \cdot \pi\right)}^{2}}}} \]
8. Step-by-step derivation

   1. sqrt-div 28.0%

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

   2. sqrt-pow1 97.8%

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

   3. metadata-eval 97.8%

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

   4. pow1 97.8%

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

   5. pow1/2 97.8%

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

   6. metadata-eval 97.8%

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

   7. pow-prod-up 98.3%

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

   8. frac-times 98.4%

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

   9. clear-num 98.4%

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

   10. un-div-inv 98.4%

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

9. Applied egg-rr 98.4%

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

    1. associate-/r/ 98.1%

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

    2. associate-*l/ 98.1%

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

    3. clear-num 98.0%

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

    4. associate-/r/ 98.0%

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

    5. associate-*r* 98.1%

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

    6. pow-prod-up 97.8%

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

    7. metadata-eval 97.8%

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

    8. pow1/2 97.8%

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

    9. clear-num 97.9%

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

    10. associate-*l/ 97.9%

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

    11. *-un-lft-identity 97.9%

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

    12. frac-2neg 97.9%

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

    13. metadata-eval 97.9%

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

    14. distribute-frac-neg2 97.9%

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

    15. associate-/r/ 98.4%

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

    16. *-commutative 98.4%

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

    \[\leadsto \frac{-1}{t \cdot \frac{\pi}{-\sqrt{0.5}}} \]
13. Add Preprocessing

Alternative 5: 97.7% accurate, 1.2× speedup

Math

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

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(pi / Float64(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[(Pi / N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.8%

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

   1. add-sqr-sqrt 44.9%

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

   2. sqrt-unprod 27.9%

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

   3. frac-times 27.7%

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

   4. rem-square-sqrt 27.9%

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

   5. pow2 27.9%

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

7. Applied egg-rr 27.9%

   \[\leadsto \color{blue}{\sqrt{\frac{0.5}{{\left(t \cdot \pi\right)}^{2}}}} \]
8. Step-by-step derivation

   1. sqrt-div 28.0%

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

   2. sqrt-pow1 97.8%

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

   3. metadata-eval 97.8%

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

   4. pow1 97.8%

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

   5. pow1/2 97.8%

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

   6. metadata-eval 97.8%

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

   7. pow-prod-up 98.3%

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

   8. frac-times 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.1%

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

   11. clear-num 98.0%

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

   12. associate-*r/ 98.1%

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

   13. pow-prod-up 97.9%

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

   14. metadata-eval 97.9%

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

   15. pow1/2 97.9%

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

9. Applied egg-rr 97.9%

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

Alternative 6: 97.7% 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 99.3%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 97.8%

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

6. Final simplification 97.8%

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

Reproduce

herbie shell --seed 2024111 
(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)))))