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

Percentage Accurate: 99.3% → 99.5%

Time: 10.5s

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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. times-frac 99.5%

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

   3. fma-undefine 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-define 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. associate-/l/ 99.5%

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

   3. associate-*l/ 99.4%

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

   4. *-un-lft-identity 99.4%

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

   5. fma-undefine 99.4%

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

   6. pow2 99.4%

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

   7. *-commutative 99.4%

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

   8. fma-define 99.4%

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

   9. pow2 99.4%

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

   10. pow2 99.4%

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

8. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

   2. associate-/l/ 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. metadata-eval 99.5%

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

   6. fma-neg 99.5%

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

   7. unpow2 99.5%

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

   8. *-commutative 99.5%

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

   9. unpow2 99.5%

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

   10. fma-neg 99.5%

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

   11. metadata-eval 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. times-frac 99.5%

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

   3. fma-undefine 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-define 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Final simplification 99.1%

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

Alternative 5: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0 98.2%

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

4. Final simplification 98.2%

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

Alternative 6: 98.4% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t), $MachinePrecision] / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0 98.2%

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

4. Taylor expanded in v around 0 98.2%

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

5. Taylor expanded in v around 0 98.2%

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

   1. associate-/r* 98.6%

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

7. Simplified 98.6%

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

8. Final simplification 98.6%

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

Alternative 7: 98.7% accurate, 1.2× 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 99.1%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. times-frac 99.5%

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

   3. fma-undefine 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-define 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 97.7%

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

   1. associate-/r* 98.0%

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

9. Simplified 98.0%

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

    1. associate-/l/ 97.7%

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

    2. add-sqr-sqrt 96.7%

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

    3. *-commutative 96.7%

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

    4. times-frac 97.6%

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

    5. pow1/2 97.6%

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

    6. sqrt-pow1 98.6%

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

    7. metadata-eval 98.6%

       \[\leadsto \frac{{0.5}^{\color{blue}{0.25}}}{t} \cdot \frac{\sqrt{\sqrt{0.5}}}{\pi} \]

    8. pow1/2 98.6%

       \[\leadsto \frac{{0.5}^{0.25}}{t} \cdot \frac{\sqrt{\color{blue}{{0.5}^{0.5}}}}{\pi} \]

    9. sqrt-pow1 98.6%

       \[\leadsto \frac{{0.5}^{0.25}}{t} \cdot \frac{\color{blue}{{0.5}^{\left(\frac{0.5}{2}\right)}}}{\pi} \]

    10. metadata-eval 98.6%

        \[\leadsto \frac{{0.5}^{0.25}}{t} \cdot \frac{{0.5}^{\color{blue}{0.25}}}{\pi} \]

11. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\frac{{0.5}^{0.25}}{t} \cdot \frac{{0.5}^{0.25}}{\pi}} \]
12. Step-by-step derivation

    1. associate-*l/ 98.9%

       \[\leadsto \color{blue}{\frac{{0.5}^{0.25} \cdot \frac{{0.5}^{0.25}}{\pi}}{t}} \]

    2. associate-*r/ 98.9%

       \[\leadsto \frac{\color{blue}{\frac{{0.5}^{0.25} \cdot {0.5}^{0.25}}{\pi}}}{t} \]

    3. pow-sqr 97.9%

       \[\leadsto \frac{\frac{\color{blue}{{0.5}^{\left(2 \cdot 0.25\right)}}}{\pi}}{t} \]

    4. metadata-eval 97.9%

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

13. Simplified 97.9%

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

    1. pow1/2 97.9%

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

    2. clear-num 98.9%

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

    3. inv-pow 98.9%

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

15. Applied egg-rr 98.9%

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

    1. unpow-1 98.9%

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

17. Simplified 98.9%

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

18. Final simplification 98.9%

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

Alternative 8: 97.8% accurate, 1.2× 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 99.1%

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

3. Simplified 99.2%

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

5. Taylor expanded in v around 0 97.7%

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

6. Final simplification 97.7%

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

Alternative 9: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. times-frac 99.5%

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

   3. fma-undefine 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-define 99.5%

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

   6. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in v around 0 97.7%

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

   1. associate-/r* 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024067 
(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)))))