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

Percentage Accurate: 99.3% → 99.8%

Time: 3.8s

Alternatives: 4

Speedup: N/A×

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: 99.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{0.5} \cdot {2}^{0.5}}}{\pi}}{t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (/
   (/
    (/
     (fma (* v v) -5.0 1.0)
     (* (pow (fma (* v v) -3.0 1.0) 0.5) (pow 2.0 0.5)))
    PI)
   t)
  (/ (- 1.0 (pow v 4.0)) (+ 1.0 (* v v)))))

C

double code(double v, double t) {
	return (((fma((v * v), -5.0, 1.0) / (pow(fma((v * v), -3.0, 1.0), 0.5) * pow(2.0, 0.5))) / ((double) M_PI)) / t) / ((1.0 - pow(v, 4.0)) / (1.0 + (v * v)));
}

Julia

function code(v, t)
	return Float64(Float64(Float64(Float64(fma(Float64(v * v), -5.0, 1.0) / Float64((fma(Float64(v * v), -3.0, 1.0) ^ 0.5) * (2.0 ^ 0.5))) / pi) / t) / Float64(Float64(1.0 - (v ^ 4.0)) / Float64(1.0 + Float64(v * v))))
end

Wolfram

code[v_, t_] := N[(N[(N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / N[(N[Power[N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / t), $MachinePrecision] / N[(N[(1.0 - N[Power[v, 4.0], $MachinePrecision]), $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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\left({2}^{0.5} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{0.5}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\left({2}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{-5 \cdot \left(v \cdot v\right) + 1}}{\left({2}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \color{blue}{\left(v \cdot v\right)} + 1}{\left({2}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\color{blue}{\left({2}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\color{blue}{\left({2}^{\frac{1}{2}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right)} \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\left(\color{blue}{{2}^{\frac{1}{2}}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\left({2}^{\frac{1}{2}} \cdot \color{blue}{{\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{\frac{1}{2}}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\left({2}^{\frac{1}{2}} \cdot {\color{blue}{\left(-3 \cdot \left(v \cdot v\right) + 1\right)}}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{\left({2}^{\frac{1}{2}} \cdot {\left(-3 \cdot \color{blue}{\left(v \cdot v\right)} + 1\right)}^{\frac{1}{2}}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   10. associate-/r* N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{{2}^{\frac{1}{2}} \cdot {\left(-3 \cdot \left(v \cdot v\right) + 1\right)}^{\frac{1}{2}}}}{\pi \cdot t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{-5 \cdot \left(v \cdot v\right) + 1}{{2}^{\frac{1}{2}} \cdot {\left(-3 \cdot \left(v \cdot v\right) + 1\right)}^{\frac{1}{2}}}}{\pi \cdot t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

6. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{0.5} \cdot {2}^{0.5}}}{\pi \cdot t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(v \cdot v\right) \cdot -5 + 1}}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{\color{blue}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{\color{blue}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{\frac{1}{2}}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\mathsf{fma}\left(\color{blue}{v \cdot v}, -3, 1\right)\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   8. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\color{blue}{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}}{\pi \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot t}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\color{blue}{\mathsf{PI}\left(\right) \cdot t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   12. associate-/r* N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\mathsf{PI}\left(\right)}}{t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\left(v \cdot v\right) \cdot -5 + 1}{{\left(\left(v \cdot v\right) \cdot -3 + 1\right)}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\mathsf{PI}\left(\right)}}{t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{{\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{0.5} \cdot {2}^{0.5}}}{\pi}}{t}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left({\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{0.5} \cdot \left(\left({2}^{0.5} \cdot \pi\right) \cdot t\right)\right) \cdot \left(1 - v \cdot v\right)} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (*
   (* (pow (fma -3.0 (* v v) 1.0) 0.5) (* (* (pow 2.0 0.5) PI) t))
   (- 1.0 (* v v)))))

C

double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / ((pow(fma(-3.0, (v * v), 1.0), 0.5) * ((pow(2.0, 0.5) * ((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((fma(-3.0, Float64(v * v), 1.0) ^ 0.5) * Float64(Float64((2.0 ^ 0.5) * 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[Power[N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[(N[Power[2.0, 0.5], $MachinePrecision] * Pi), $MachinePrecision] * t), $MachinePrecision]), $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. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/2 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. pow2 N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

5. Applied rewrites 99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left({\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{0.5} \cdot \left(\left({2}^{0.5} \cdot \pi\right) \cdot t\right)\right)} \cdot \left(1 - v \cdot v\right)} \]
6. Add Preprocessing

Alternative 3: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t}}{{2}^{0.5} \cdot \pi} \cdot {\left({\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{-1}\right)}^{0.5}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (*
   (/ (/ (fma (* v v) -5.0 1.0) t) (* (pow 2.0 0.5) PI))
   (pow (pow (fma (* v v) -3.0 1.0) -1.0) 0.5))
  (/ (- 1.0 (pow v 4.0)) (+ 1.0 (* v v)))))

C

double code(double v, double t) {
	return (((fma((v * v), -5.0, 1.0) / t) / (pow(2.0, 0.5) * ((double) M_PI))) * pow(pow(fma((v * v), -3.0, 1.0), -1.0), 0.5)) / ((1.0 - pow(v, 4.0)) / (1.0 + (v * v)));
}

Julia

function code(v, t)
	return Float64(Float64(Float64(Float64(fma(Float64(v * v), -5.0, 1.0) / t) / Float64((2.0 ^ 0.5) * pi)) * ((fma(Float64(v * v), -3.0, 1.0) ^ -1.0) ^ 0.5)) / Float64(Float64(1.0 - (v ^ 4.0)) / Float64(1.0 + Float64(v * v))))
end

Wolfram

code[v_, t_] := N[(N[(N[(N[(N[(N[(v * v), $MachinePrecision] * -5.0 + 1.0), $MachinePrecision] / t), $MachinePrecision] / N[(N[Power[2.0, 0.5], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[Power[v, 4.0], $MachinePrecision]), $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

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-5, v \cdot v, 1\right)}{\left({2}^{0.5} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right)\right)}^{0.5}\right) \cdot \left(\pi \cdot t\right)}}{\frac{1 - {v}^{4}}{1 + v \cdot v}}} \]

5. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1 + -5 \cdot {v}^{2}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \cdot \color{blue}{\sqrt{\frac{1}{1 + -3 \cdot {v}^{2}}}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{t}}{{2}^{0.5} \cdot \pi} \cdot {\left({\left(\mathsf{fma}\left(v \cdot v, -3, 1\right)\right)}^{-1}\right)}^{0.5}}}{\frac{1 - {v}^{4}}{1 + v \cdot v}} \]
8. Add Preprocessing

Alternative 4: 49.2% accurate, N/A× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (* (- (pow (* v v) -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 ((pow((v * v), -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 ((Math.pow((v * v), -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 ((math.pow((v * v), -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(Float64((Float64(v * v) ^ -1.0) - 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 = ((((v * v) ^ -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[(N[(N[Power[N[(v * v), $MachinePrecision], -1.0], $MachinePrecision] - 5.0), $MachinePrecision] * N[(v * v), $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]

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 inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. inv-pow N/A

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

   5. lower-pow.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 55.9

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

5. Applied rewrites 55.9%

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

Reproduce

herbie shell --seed 2025066 
(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)))))