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

Percentage Accurate: 99.3% → 99.5%

Time: 9.9s

Alternatives: 7

Speedup: 0.4×

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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r* 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. times-frac 99.5%

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

   3. pow2 99.5%

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

   4. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 4: 98.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

   1. associate-/r* 98.8%

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

   2. div-inv 98.7%

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

   3. *-commutative 98.7%

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

8. Applied egg-rr 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-/r* 98.7%

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

   3. frac-2neg 98.7%

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

   4. metadata-eval 98.7%

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

   5. frac-times 98.8%

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

   6. frac-2neg 98.8%

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

   7. metadata-eval 98.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 2.4%

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

   10. sqr-neg 2.4%

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

   11. sqrt-unprod 2.4%

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

   12. add-sqr-sqrt 2.4%

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

   13. add-sqr-sqrt 1.2%

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

   14. sqrt-unprod 29.8%

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

   15. sqr-neg 29.8%

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

   16. sqrt-unprod 48.8%

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

   17. add-sqr-sqrt 98.8%

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

10. Applied egg-rr 98.8%

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

11. Taylor expanded in t around 0 98.7%

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

    1. associate-*r* 98.7%

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

    2. associate-/r* 98.6%

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

    3. associate-/l/ 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

   1. associate-/r* 98.8%

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

   2. div-inv 98.7%

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

   3. *-commutative 98.7%

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

8. Applied egg-rr 98.7%

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

   1. associate-*l/ 99.0%

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

   2. div-inv 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-/r* 99.0%

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

   5. pow1/2 99.0%

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{{2}^{0.5}}}}{\pi}}{t} \]

   6. pow-flip 98.2%

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

   7. metadata-eval 98.2%

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

10. Applied egg-rr 98.2%

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

11. Taylor expanded in t around 0 98.2%

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

12. Final simplification 98.2%

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

Alternative 6: 20.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

   1. associate-/r* 98.8%

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

   2. div-inv 98.7%

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

   3. *-commutative 98.7%

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

8. Applied egg-rr 98.7%

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

   1. *-commutative 98.7%

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

   2. associate-/r* 98.7%

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

   3. frac-2neg 98.7%

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

   4. metadata-eval 98.7%

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

   5. frac-times 98.8%

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

   6. frac-2neg 98.8%

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

   7. metadata-eval 98.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 2.4%

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

   10. sqr-neg 2.4%

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

   11. sqrt-unprod 2.4%

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

   12. add-sqr-sqrt 2.4%

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

   13. add-sqr-sqrt 1.2%

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

   14. sqrt-unprod 29.8%

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

   15. sqr-neg 29.8%

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

   16. sqrt-unprod 48.8%

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

   17. add-sqr-sqrt 98.8%

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

10. Applied egg-rr 98.8%

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

12. Applied egg-rr 20.3%

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

    1. unpow-1 20.3%

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

14. Simplified 20.3%

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

15. Final simplification 20.3%

    \[\leadsto \frac{1}{\pi \cdot t} \]

Alternative 7: 15.7% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (v t) :precision binary64 (/ PI t))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. associate-*r* 99.4%

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

   3. metadata-eval 99.4%

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

   4. associate-*l* 99.4%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 98.7%

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

   1. *-commutative 98.7%

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

6. Simplified 98.7%

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

   1. associate-/r* 98.8%

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

   2. div-inv 98.7%

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

   3. *-commutative 98.7%

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

8. Applied egg-rr 98.7%

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

   1. associate-*l/ 99.0%

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

   2. div-inv 99.0%

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

   3. *-commutative 99.0%

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

   4. associate-/r* 99.0%

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

   5. pow1/2 99.0%

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{{2}^{0.5}}}}{\pi}}{t} \]

   6. pow-flip 98.2%

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

   7. metadata-eval 98.2%

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

10. Applied egg-rr 98.2%

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

12. Applied egg-rr 15.7%

    \[\leadsto \frac{\color{blue}{\log \left(e^{\pi}\right)}}{t} \]
13. Step-by-step derivation

    1. rem-log-exp 15.7%

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

14. Simplified 15.7%

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

15. Final simplification 15.7%

    \[\leadsto \frac{\pi}{t} \]

Reproduce

herbie shell --seed 2023301 
(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)))))