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

Percentage Accurate: 99.3% → 99.4%

Time: 10.8s

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: 99.4% 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 99.4%

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

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. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (+ 1.0 (* (* v v) -5.0))
  (* (* 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 + ((v * v) * -5.0)) / ((((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 + ((v * v) * -5.0)) / ((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 + ((v * v) * -5.0)) / ((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(Float64(v * v) * -5.0)) / Float64(Float64(pi * t) * Float64(sqrt(Float64(2.0 * Float64(1.0 - Float64(v * Float64(v * 3.0))))) * Float64(1.0 - Float64(v * v)))))
end

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + ((v * v) * -5.0)) / ((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[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * t), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. pow2 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. sub-neg 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. metadata-eval 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in v around 0 99.3%

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

   1. *-commutative 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

   1. associate-/r* 99.2%

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

7. Simplified 99.2%

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

   1. add-sqr-sqrt 52.7%

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

   2. sqrt-unprod 32.6%

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

   3. frac-times 32.3%

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

   4. inv-pow 32.3%

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

   5. inv-pow 32.3%

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

   6. pow-prod-up 32.3%

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

   7. metadata-eval 32.3%

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

   8. pow2 32.3%

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

9. Applied egg-rr 32.3%

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

    1. unpow2 32.3%

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

    2. *-commutative 32.3%

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

    3. *-commutative 32.3%

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

    4. swap-sqr 32.3%

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

    5. rem-square-sqrt 32.3%

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

11. Simplified 32.3%

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

    1. *-un-lft-identity 32.3%

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

    2. times-frac 32.3%

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

    3. metadata-eval 32.3%

       \[\leadsto \sqrt{\color{blue}{0.5} \cdot \frac{{t}^{-2}}{\pi \cdot \pi}} \]

    4. rem-square-sqrt 32.0%

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

    5. sqr-pow 32.1%

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

    6. frac-times 32.4%

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

    7. metadata-eval 32.4%

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

    8. inv-pow 32.4%

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

    9. associate-/r* 32.4%

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

    10. metadata-eval 32.4%

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

    11. inv-pow 32.4%

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

    12. associate-/r* 32.4%

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

    13. swap-sqr 32.5%

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

    14. sqrt-unprod 52.5%

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

13. Applied egg-rr 98.7%

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

14. Final simplification 98.7%

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

Alternative 5: 98.4% 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 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

6. Final simplification 99.3%

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

Alternative 6: 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 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

   1. associate-/r* 99.2%

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

7. Simplified 99.2%

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

   1. add-sqr-sqrt 52.7%

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

   2. sqrt-unprod 32.6%

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

   3. frac-times 32.3%

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

   4. inv-pow 32.3%

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

   5. inv-pow 32.3%

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

   6. pow-prod-up 32.3%

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

   7. metadata-eval 32.3%

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

   8. pow2 32.3%

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

9. Applied egg-rr 32.3%

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

    1. unpow2 32.3%

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

    2. *-commutative 32.3%

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

    3. *-commutative 32.3%

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

    4. swap-sqr 32.3%

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

    5. rem-square-sqrt 32.3%

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

11. Simplified 32.3%

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

    1. sqrt-div 32.4%

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

    2. sqrt-pow1 99.2%

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

    3. sqrt-prod 99.2%

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

    4. sqrt-prod 99.2%

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

    5. add-sqr-sqrt 99.2%

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

    6. div-inv 99.2%

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

    7. metadata-eval 99.2%

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

    8. inv-pow 99.2%

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

13. Applied egg-rr 99.2%

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

    1. frac-times 99.3%

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

    2. metadata-eval 99.3%

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

    3. associate-*r* 99.1%

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

    4. *-commutative 99.1%

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

    5. *-commutative 99.1%

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

    6. associate-/r* 99.3%

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

    7. *-commutative 99.3%

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

15. Applied egg-rr 99.3%

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

16. Final simplification 99.3%

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

Alternative 7: 97.9% 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 99.4%

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

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 98.6%

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

6. Final simplification 98.6%

   \[\leadsto \frac{\sqrt{0.5}}{\pi \cdot t} \]
7. 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 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 99.3%

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

   1. associate-/r* 99.2%

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

7. Simplified 99.2%

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

   1. add-sqr-sqrt 52.7%

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

   2. sqrt-unprod 32.6%

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

   3. frac-times 32.3%

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

   4. inv-pow 32.3%

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

   5. inv-pow 32.3%

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

   6. pow-prod-up 32.3%

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

   7. metadata-eval 32.3%

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

   8. pow2 32.3%

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

9. Applied egg-rr 32.3%

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

    1. unpow2 32.3%

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

    2. *-commutative 32.3%

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

    3. *-commutative 32.3%

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

    4. swap-sqr 32.3%

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

    5. rem-square-sqrt 32.3%

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

11. Simplified 32.3%

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

    1. *-un-lft-identity 32.3%

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

    2. times-frac 32.3%

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

    3. metadata-eval 32.3%

       \[\leadsto \sqrt{\color{blue}{0.5} \cdot \frac{{t}^{-2}}{\pi \cdot \pi}} \]

    4. rem-square-sqrt 32.0%

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

    5. sqr-pow 32.1%

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

    6. frac-times 32.4%

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

    7. metadata-eval 32.4%

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

    8. inv-pow 32.4%

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

    9. associate-/r* 32.4%

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

    10. metadata-eval 32.4%

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

    11. inv-pow 32.4%

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

    12. associate-/r* 32.4%

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

    13. swap-sqr 32.5%

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

    14. sqrt-unprod 52.5%

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

13. Applied egg-rr 98.7%

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

14. Taylor expanded in t around 0 98.6%

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

    1. associate-/r* 98.7%

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

16. Simplified 98.7%

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

17. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024112 
(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)))))