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

Percentage Accurate: 99.3% → 99.4%

Time: 7.4s

Alternatives: 9

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, t_] := N[(1.0 / N[(N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(t * N[(Pi * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v * N[(v * -5.0), $MachinePrecision] + 1.0), $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 99.3%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. metadata-eval 99.4%

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

   4. unpow2 99.4%

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

   5. *-commutative 99.4%

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

   6. unpow2 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in t around 0 99.4%

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

   1. associate-*l* 99.4%

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

   2. unpow2 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. sqr-neg 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. sqr-neg 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return ((1.0 + ((v * v) * -5.0)) / (Math.PI * ((1.0 - (v * v)) * t))) * Math.sqrt((1.0 / (2.0 + (v * (v * -6.0)))));
}

Python

def code(v, t):
	return ((1.0 + ((v * v) * -5.0)) / (math.pi * ((1.0 - (v * v)) * t))) * math.sqrt((1.0 / (2.0 + (v * (v * -6.0)))))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = ((1.0 + ((v * v) * -5.0)) / (pi * ((1.0 - (v * v)) * t))) * sqrt((1.0 / (2.0 + (v * (v * -6.0)))));
end

Wolfram

code[v_, t_] := N[(N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(v * N[(v * -6.0), $MachinePrecision]), $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 99.3%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. metadata-eval 99.4%

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

   4. unpow2 99.4%

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

   5. *-commutative 99.4%

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

   6. unpow2 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in t around 0 99.4%

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

   1. associate-*l* 99.4%

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

   2. unpow2 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in t around 0 98.9%

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

    1. *-commutative 98.9%

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

    2. unpow2 98.9%

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

    3. *-commutative 98.9%

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

    4. unpow2 98.9%

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

    5. associate-*r* 98.9%

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

    6. *-commutative 98.9%

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

    7. unpow2 98.9%

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

    8. associate-*l* 98.9%

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

13. Simplified 98.9%

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

14. Final simplification 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return ((1.0 + ((v * v) * -5.0)) * Math.sqrt((1.0 / (2.0 + ((v * v) * -6.0))))) / (t * ((1.0 - (v * v)) * Math.PI));
}

Python

def code(v, t):
	return ((1.0 + ((v * v) * -5.0)) * math.sqrt((1.0 / (2.0 + ((v * v) * -6.0))))) / (t * ((1.0 - (v * v)) * math.pi))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = ((1.0 + ((v * v) * -5.0)) * sqrt((1.0 / (2.0 + ((v * v) * -6.0))))) / (t * ((1.0 - (v * v)) * pi));
end

Wolfram

code[v_, t_] := N[(N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * Pi), $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 99.3%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. metadata-eval 99.4%

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

   4. unpow2 99.4%

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

   5. *-commutative 99.4%

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

   6. unpow2 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in t around 0 98.9%

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

   1. associate-*l/ 99.0%

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

   2. *-commutative 99.0%

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

   3. unpow2 99.0%

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

   4. unpow2 99.0%

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

   5. unpow2 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

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(Float64(v * 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[(N[(v * v), $MachinePrecision] * -5.0), $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 99.4%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 6: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

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 * (1.0 - ((v * v) * 3.0))))));
}

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 * (1.0 - ((v * v) * 3.0))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * ((t * pi) * sqrt((2.0 * (1.0 - ((v * v) * 3.0))))));
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[(N[(t * Pi), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(1.0 - N[(N[(v * v), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $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. Final simplification 99.4%

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

Alternative 7: 98.4% 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 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.3%

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

   1. *-commutative 98.3%

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

6. Simplified 98.3%

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

7. Final simplification 98.3%

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

Alternative 8: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in v around 0 98.3%

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

   1. associate-/r* 98.3%

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

   2. *-commutative 98.3%

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

6. Simplified 98.3%

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

   1. div-inv 98.4%

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

   2. *-commutative 98.4%

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

8. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.7%

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

   2. *-un-lft-identity 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 9: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.3%

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

   2. associate-/r* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. sqr-neg 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-neg-in 99.4%

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

   8. fma-def 99.4%

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

   9. sqr-neg 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.9%

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

5. Final simplification 97.9%

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

Reproduce

herbie shell --seed 2023272 
(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)))))