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

Percentage Accurate: 99.3% → 99.4%

Time: 11.6s

Alternatives: 6

Speedup: 1.2×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-neg-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. PI-lowering-PI.f64 99.6

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

4. Applied egg-rr 99.6%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. +-commutative N/A

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

   7. associate-*r* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 99.6

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

6. Applied egg-rr 99.6%

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

   1. *-rgt-identity N/A

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

   2. frac-2neg N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. neg-mul-1 N/A

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

   10. remove-double-neg N/A

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

   11. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.6%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-neg-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. PI-lowering-PI.f64 99.6

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

4. Applied egg-rr 99.6%

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. +-commutative N/A

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

   7. associate-*r* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 99.6

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

6. Applied egg-rr 99.6%

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

   1. *-rgt-identity N/A

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

   2. frac-2neg N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. neg-mul-1 N/A

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

   10. remove-double-neg N/A

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

   11. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in t around 0

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

    1. associate-*r* N/A

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

    2. associate-*l* N/A

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

    3. *-lowering-*.f64 N/A

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

    4. *-lowering-*.f64 N/A

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

    5. PI-lowering-PI.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. sub-neg N/A

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

    8. unpow2 N/A

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

    9. metadata-eval N/A

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

    10. accelerator-lowering-fma.f64 N/A

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

    11. sqrt-lowering-sqrt.f64 N/A

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

    12. +-commutative N/A

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

    13. *-commutative N/A

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

    14. accelerator-lowering-fma.f64 N/A

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 99.5

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-neg-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. PI-lowering-PI.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 97.8

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

7. Simplified 97.8%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. PI-lowering-PI.f64 97.8

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

9. Applied egg-rr 97.8%

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

    1. add-sqr-sqrt N/A

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

    2. sqrt-unprod N/A

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

    3. sqrt-undiv N/A

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

    4. sqrt-lowering-sqrt.f64 N/A

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

    5. /-lowering-/.f64 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. PI-lowering-PI.f64 N/A

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

    8. PI-lowering-PI.f64 98.7

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

11. Applied egg-rr 98.7%

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{0.5}{\pi \cdot \pi}}}}{t} \]
12. Add Preprocessing

Alternative 4: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 98.3

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

5. Simplified 98.3%

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 98.3

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

7. Applied egg-rr 98.3%

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

Alternative 5: 98.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 98.3

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

5. Simplified 98.3%

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

Alternative 6: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. distribute-lft-neg-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. PI-lowering-PI.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 97.8

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

7. Simplified 97.8%

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

8. Final simplification 97.8%

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

Reproduce

herbie shell --seed 2024198 
(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)))))