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

Percentage Accurate: 99.4% → 99.5%

Time: 1.8s

Alternatives: 5

Speedup: 0.8×

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / (((pi * t) * sqrt((2.0 * (1.0 - (3.0 * (v * v)))))) * (1.0 - (v * v)));
end

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\pi \cdot \sqrt{2}}\\ \frac{\mathsf{fma}\left({v}^{2}, 3 \cdot \frac{1}{\pi \cdot {\left(\sqrt{2}\right)}^{3}} - 4 \cdot t\_1, t\_1\right)}{t} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (* PI (sqrt 2.0)))))
   (/
    (fma
     (pow v 2.0)
     (- (* 3.0 (/ 1.0 (* PI (pow (sqrt 2.0) 3.0)))) (* 4.0 t_1))
     t_1)
    t)))

C

double code(double v, double t) {
	double t_1 = 1.0 / (((double) M_PI) * sqrt(2.0));
	return fma(pow(v, 2.0), ((3.0 * (1.0 / (((double) M_PI) * pow(sqrt(2.0), 3.0)))) - (4.0 * t_1)), t_1) / t;
}

Julia

function code(v, t)
	t_1 = Float64(1.0 / Float64(pi * sqrt(2.0)))
	return Float64(fma((v ^ 2.0), Float64(Float64(3.0 * Float64(1.0 / Float64(pi * (sqrt(2.0) ^ 3.0)))) - Float64(4.0 * t_1)), t_1) / t)
end

Wolfram

code[v_, t_] := Block[{t$95$1 = N[(1.0 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[v, 2.0], $MachinePrecision] * N[(N[(3.0 * N[(1.0 / N[(Pi * N[Power[N[Sqrt[2.0], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$1), $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. Taylor expanded in v around 0

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

   1. lower-fma.f64 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 99.2%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / ((t * (Math.PI * Math.sqrt((2.0 + (-6.0 * Math.pow(v, 2.0)))))) * (1.0 - (v * v)));
}

Python

def code(v, t):
	return (1.0 - (5.0 * (v * v))) / ((t * (math.pi * math.sqrt((2.0 + (-6.0 * math.pow(v, 2.0)))))) * (1.0 - (v * v)))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - (5.0 * (v * v))) / ((t * (pi * sqrt((2.0 + (-6.0 * (v ^ 2.0)))))) * (1.0 - (v * v)));
end

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(Pi * N[Sqrt[N[(2.0 + N[(-6.0 * N[Power[v, 2.0], $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. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in v around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lift-pow.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.2% 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]

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

Alternative 4: 98.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $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. Taylor expanded in v around 0

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

   1. lower-fma.f64 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 99.2%

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

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. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-/.f64 98.6

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

10. Applied rewrites 98.6%

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

Alternative 5: 98.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 98.3

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

4. Applied rewrites 98.3%

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

Reproduce

herbie shell --seed 2025135 
(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)))))