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

Percentage Accurate: 99.3% → 98.9%

Time: 6.5s

Alternatives: 6

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

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

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(Float64(1.0 / Float64(pi / sqrt(0.5))) / t)
end

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 / N[(Pi / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $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. Step-by-step derivation

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.5%

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

   3. associate-/l/ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.5%

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

   9. metadata-eval 99.5%

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

   10. sub-neg 99.5%

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

   11. distribute-rgt-in 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. associate-/r* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-udef 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 98.7%

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

   1. associate-/l/ 98.8%

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

8. Simplified 98.8%

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

   1. clear-num 99.8%

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

   2. inv-pow 99.8%

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

10. Applied egg-rr 99.8%

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

    1. unpow-1 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(Pi * t), $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. Step-by-step derivation

   1. associate-*l* 99.4%

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

   2. sub-neg 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 99.4%

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

5. Taylor expanded in v around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 98.6% accurate, 1.1× 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%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate-*r* 99.4%

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

   4. associate-/r* 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in v around 0 99.5%

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

5. Taylor expanded in v around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 97.9% 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 99.4%

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

   1. associate-*l* 99.4%

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

   2. sub-neg 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.0% accurate, 1.1× 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%

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

   1. associate-*l* 99.4%

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

   2. sub-neg 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 98.7%

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

   1. add-sqr-sqrt 51.7%

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

   2. pow2 51.7%

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

   3. *-commutative 51.7%

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

6. Applied egg-rr 51.7%

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

   1. unpow2 51.7%

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

   2. add-sqr-sqrt 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-/r* 98.8%

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

8. Applied egg-rr 98.8%

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

9. Final simplification 98.8%

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

Alternative 6: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. associate-/r* 99.5%

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

   3. associate-/l/ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.5%

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

   9. metadata-eval 99.5%

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

   10. sub-neg 99.5%

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

   11. distribute-rgt-in 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. associate-/r* 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-udef 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in v around 0 98.7%

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

   1. associate-/l/ 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2023256 
(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)))))