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

Percentage Accurate: 99.3% → 99.4%

Time: 10.3s

Alternatives: 8

Speedup: 1.0×

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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

5. Taylor expanded in t around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. sub-neg 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(1.0 - N[(N[(v * v), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * Pi), $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. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \frac{1}{t \cdot \pi} \end{array} \]

FPCore

(FPCore (v t) :precision binary64 (* (sqrt 0.5) (/ 1.0 (* t PI))))

C

double code(double v, double t) {
	return sqrt(0.5) * (1.0 / (t * ((double) M_PI)));
}

Java

public static double code(double v, double t) {
	return Math.sqrt(0.5) * (1.0 / (t * Math.PI));
}

Python

def code(v, t):
	return math.sqrt(0.5) * (1.0 / (t * math.pi))

Julia

function code(v, t)
	return Float64(sqrt(0.5) * Float64(1.0 / Float64(t * pi)))
end

MATLAB

function tmp = code(v, t)
	tmp = sqrt(0.5) * (1.0 / (t * pi));
end

Wolfram

code[v_, t_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / N[(t * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.0%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
6. Step-by-step derivation

   1. div-inv 98.1%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

8. Final simplification 98.1%

   \[\leadsto \sqrt{0.5} \cdot \frac{1}{t \cdot \pi} \]
9. Add Preprocessing

Alternative 5: 98.4% accurate, 1.2× 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%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.6%

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

6. Taylor expanded in v around 0 98.6%

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

7. Final simplification 98.6%

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

Alternative 6: 98.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. times-frac 98.7%

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

   3. *-commutative 98.7%

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

7. Applied egg-rr 98.7%

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

8. Taylor expanded in v around 0 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*l* 98.6%

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

   3. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 7: 98.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. times-frac 98.7%

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

   3. *-commutative 98.7%

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

7. Applied egg-rr 98.7%

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

8. Taylor expanded in v around 0 98.7%

   \[\leadsto \frac{1}{t} \cdot \color{blue}{\frac{1}{\pi \cdot \sqrt{2}}} \]
9. Step-by-step derivation

   1. *-commutative 98.7%

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

   2. associate-/r* 98.7%

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

10. Simplified 98.7%

    \[\leadsto \frac{1}{t} \cdot \color{blue}{\frac{\frac{1}{\sqrt{2}}}{\pi}} \]
11. Step-by-step derivation

    1. associate-/l/ 98.7%

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

    2. associate-/r* 98.7%

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

    3. frac-times 98.7%

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

    4. *-un-lft-identity 98.7%

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

    5. *-commutative 98.7%

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

12. Applied egg-rr 98.7%

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

13. Final simplification 98.7%

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

Alternative 8: 97.9% accurate, 1.2× 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%

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

3. Simplified 99.4%

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

5. Taylor expanded in v around 0 98.0%

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

6. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024078 
(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)))))