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

Percentage Accurate: 99.3% → 99.4%

Time: 12.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

4. Taylor expanded in t around 0 99.3%

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

   1. associate-/r* 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. unpow2 99.4%

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

   5. associate-*r* 99.4%

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

   6. fma-def 99.5%

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

   7. *-commutative 99.5%

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

   8. unpow2 99.5%

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

   9. cancel-sign-sub-inv 99.5%

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

   10. metadata-eval 99.5%

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

   11. unpow2 99.5%

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

   12. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

   1. pow1 99.4%

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

5. Applied egg-rr 99.3%

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

   1. unpow1 99.3%

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

   2. associate-*r* 99.3%

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

   3. +-commutative 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. unpow2 99.3%

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

   6. associate-*r* 99.3%

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

   7. metadata-eval 99.3%

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

   8. *-commutative 99.3%

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

   9. unpow2 99.3%

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

   10. metadata-eval 99.3%

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

   11. fma-udef 99.3%

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

   12. associate-*r* 99.3%

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

   13. associate-*l* 99.4%

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

   14. associate-*r* 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (+ 1.0 (* -5.0 (* v v)))
  (* 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 + (-5.0 * (v * v))) / (((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 + (-5.0 * (v * v))) / (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 + (-5.0 * (v * v))) / (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(-5.0 * Float64(v * v))) / 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 + (-5.0 * (v * v))) / (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[(-5.0 * N[(v * v), $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 99.3%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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 97.4%

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

   1. div-inv 97.4%

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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 97.4%

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

   1. clear-num 97.4%

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

   2. inv-pow 97.4%

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

6. Applied egg-rr 97.4%

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

   1. unpow-1 97.4%

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

   2. associate-/l* 97.4%

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

8. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{t}{\frac{\sqrt{0.5}}{\pi}}}} \]
9. Step-by-step derivation

   1. associate-/r/ 97.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.8%

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

5. Final simplification 97.8%

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

Alternative 7: 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 99.3%

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

3. Simplified 99.4%

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

   1. pow1 99.4%

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

5. Applied egg-rr 99.3%

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

   1. unpow1 99.3%

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

   2. associate-*r* 99.3%

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

   3. +-commutative 99.3%

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

   4. distribute-lft-in 99.3%

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

   5. unpow2 99.3%

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

   6. associate-*r* 99.3%

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

   7. metadata-eval 99.3%

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

   8. *-commutative 99.3%

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

   9. unpow2 99.3%

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

   10. metadata-eval 99.3%

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

   11. fma-udef 99.3%

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

   12. associate-*r* 99.3%

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

   13. associate-*l* 99.4%

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

   14. associate-*r* 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in v around 0 97.4%

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

   1. associate-/l/ 97.3%

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

10. Simplified 97.3%

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

    1. clear-num 98.3%

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

    2. inv-pow 98.3%

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

12. Applied egg-rr 98.3%

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

    1. unpow-1 98.3%

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

14. Simplified 98.3%

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

15. Final simplification 98.3%

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

Alternative 8: 98.0% 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 99.3%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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 97.4%

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

5. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2023274 
(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)))))