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

Percentage Accurate: 99.3% → 99.5%

Time: 9.2s

Alternatives: 8

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. frac-times N/A

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg2 N/A

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

   4. distribute-frac-neg N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

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

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

   16. lower-fma.f64 N/A

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

   17. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.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. Applied rewrites 99.5%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. remove-double-neg N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. sub-neg N/A

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

   14. unpow2 N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-PI.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 5: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.7

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 99.1%

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

Alternative 6: 98.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 7: 98.4% accurate, 2.4× speedup

Math

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

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. 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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 98.7

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.6%

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

Alternative 8: 98.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

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. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Reproduce

herbie shell --seed 2024235 
(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)))))