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

Percentage Accurate: 99.3% → 99.8%

Time: 5.3s

Alternatives: 8

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.4%

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

Alternative 5: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.9%

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

Alternative 6: 98.9% 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 99.3%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in v around 0

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

6. Applied rewrites 98.9%

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

Alternative 7: 98.5% 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 99.3%

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

2. Taylor expanded in v around 0

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

4. Applied rewrites 98.5%

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

Alternative 8: 20.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Applied rewrites 98.5%

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

6. Applied rewrites 20.3%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 20.3%

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

Reproduce

herbie shell --seed 2025161 
(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)))))