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

Percentage Accurate: 99.3% → 99.6%

Time: 3.0s

Alternatives: 4

Speedup: 1.6×

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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2} \cdot \pi\\ \frac{\mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot -15.0625 - 6.625}{t\_1} \cdot \left(v \cdot v\right) - \frac{2.5}{t\_1}, v \cdot v, {t\_1}^{-1}\right)}{t} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) PI)))
   (/
    (fma
     (- (* (/ (- (* (* v v) -15.0625) 6.625) t_1) (* v v)) (/ 2.5 t_1))
     (* v v)
     (pow t_1 -1.0))
    t)))

C

double code(double v, double t) {
	double t_1 = sqrt(2.0) * ((double) M_PI);
	return fma(((((((v * v) * -15.0625) - 6.625) / t_1) * (v * v)) - (2.5 / t_1)), (v * v), pow(t_1, -1.0)) / t;
}

Julia

function code(v, t)
	t_1 = Float64(sqrt(2.0) * pi)
	return Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(v * v) * -15.0625) - 6.625) / t_1) * Float64(v * v)) - Float64(2.5 / t_1)), Float64(v * v), (t_1 ^ -1.0)) / t)
end

Wolfram

code[v_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[(v * v), $MachinePrecision] * -15.0625), $MachinePrecision] - 6.625), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(v * v), $MachinePrecision]), $MachinePrecision] - N[(2.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[Power[t$95$1, -1.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

   \[\leadsto \color{blue}{{v}^{2} \cdot \left({v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{135}{16} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} - \frac{47}{2} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) - \frac{53}{8} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) - \frac{5}{2} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) + \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left({v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{135}{16} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} - \frac{47}{2} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) - \frac{53}{8} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) - \frac{5}{2} \cdot \frac{1}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)}\right) \cdot {v}^{2} + \frac{\color{blue}{1}}{t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)} \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left({\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right)}^{-1} \cdot -15.0625\right) \cdot \left(v \cdot v\right) - \frac{6.625}{\left(\sqrt{2} \cdot \pi\right) \cdot t}\right) \cdot \left(v \cdot v\right) - \frac{2.5}{\left(\sqrt{2} \cdot \pi\right) \cdot t}, v \cdot v, {\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right)}^{-1}\right)} \]

5. Taylor expanded in t around 0

   \[\leadsto \frac{{v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-241}{16} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right) \cdot \sqrt{2}} - \frac{53}{8} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}\right) - \frac{5}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}\right) + \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}{\color{blue}{t}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{{v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{-241}{16} \cdot \frac{{v}^{2}}{\mathsf{PI}\left(\right) \cdot \sqrt{2}} - \frac{53}{8} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}\right) - \frac{5}{2} \cdot \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}\right) + \frac{1}{\mathsf{PI}\left(\right) \cdot \sqrt{2}}}{t} \]

7. Applied rewrites 99.6%

   \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{-15.0625 \cdot \left(v \cdot v\right)}{\sqrt{2} \cdot \pi} - \frac{6.625}{\sqrt{2} \cdot \pi}\right) \cdot \left(v \cdot v\right) - \frac{2.5}{\sqrt{2} \cdot \pi}, v \cdot v, {\left(\sqrt{2} \cdot \pi\right)}^{-1}\right)}{\color{blue}{t}} \]
8. Step-by-step derivation

9. Applied rewrites 99.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(v \cdot v\right) \cdot -15.0625 - 6.625}{\sqrt{2} \cdot \pi} \cdot \left(v \cdot v\right) - \frac{2.5}{\sqrt{2} \cdot \pi}, v \cdot v, {\left(\sqrt{2} \cdot \pi\right)}^{-1}\right)}{\color{blue}{t}} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 99.3%

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

Alternative 3: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\frac{v \cdot v}{\pi}}{\sqrt{2}}, -2.5, {\left(\sqrt{2} \cdot \pi\right)}^{-1}\right)}{t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (/ (fma (/ (/ (* v v) PI) (sqrt 2.0)) -2.5 (pow (* (sqrt 2.0) PI) -1.0)) t))

C

double code(double v, double t) {
	return fma((((v * v) / ((double) M_PI)) / sqrt(2.0)), -2.5, pow((sqrt(2.0) * ((double) M_PI)), -1.0)) / t;
}

Julia

function code(v, t)
	return Float64(fma(Float64(Float64(Float64(v * v) / pi) / sqrt(2.0)), -2.5, (Float64(sqrt(2.0) * pi) ^ -1.0)) / t)
end

Wolfram

code[v_, t_] := N[(N[(N[(N[(N[(v * v), $MachinePrecision] / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -2.5 + N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi}, -2.5, {\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right)}^{-1}\right)} \]

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{v \cdot v}{\pi}}{\sqrt{2}}, -2.5, {\left(\sqrt{2} \cdot \pi\right)}^{-1}\right)}{\color{blue}{t}} \]
8. Add Preprocessing

Alternative 4: 98.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{v \cdot v}{t}}{\sqrt{2} \cdot \pi}, -2.5, {\left(\left(\sqrt{2} \cdot \pi\right) \cdot t\right)}^{-1}\right)} \]

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{v \cdot v}{\pi}}{\sqrt{2}}, -2.5, {\left(\sqrt{2} \cdot \pi\right)}^{-1}\right)}{\color{blue}{t}} \]

8. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. unpow-1 N/A

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

   6. lift-pow.f64 98.7

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

10. Applied rewrites 98.7%

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

Reproduce

herbie shell --seed 2025101 
(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)))))