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

Percentage Accurate: 99.3% → 99.4%

Time: 11.2s

Alternatives: 9

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, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(1.0 + N[(1.0 + N[(-1.0 - N[(N[Power[v, 2.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t * N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[Power[v, 2.0], $MachinePrecision] * 3.0), $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 t around 0 99.6%

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

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)\right)}}{\left(\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right) \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-undefine 99.6%

      \[\leadsto \frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)} - 1\right)}}{\left(\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. log1p-undefine 99.6%

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

   4. add-exp-log 99.6%

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

   5. *-commutative 99.6%

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

   6. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (*
   (* (* t (* PI (sqrt 2.0))) (sqrt (- 1.0 (* (pow v 2.0) 3.0))))
   (- 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))) * sqrt((1.0 - (pow(v, 2.0) * 3.0)))) * (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))) * Math.sqrt((1.0 - (Math.pow(v, 2.0) * 3.0)))) * (1.0 - (v * v)));
}

Python

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

Julia

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

Wolfram

code[v_, t_] := N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t * N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[Power[v, 2.0], $MachinePrecision] * 3.0), $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 t around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)\right)}}{\left(\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right) \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-undefine 99.6%

      \[\leadsto \frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(5 \cdot \left(v \cdot v\right)\right)} - 1\right)}}{\left(\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - 3 \cdot {v}^{2}}\right) \cdot \left(1 - v \cdot v\right)} \]

   3. log1p-undefine 99.6%

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

   4. add-exp-log 99.6%

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

   5. *-commutative 99.6%

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

   6. pow2 99.6%

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

4. Applied egg-rr 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Final simplification 99.4%

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 6: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in v around 0 98.5%

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

6. Final simplification 98.5%

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

Alternative 7: 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%

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

5. Taylor expanded in v around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 8: 97.8% 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%

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

5. Taylor expanded in v around 0 97.8%

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

6. Final simplification 97.8%

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

Alternative 9: 97.9% accurate, 1.2× speedup

Math

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

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(Float64(sqrt(0.5) / t) / pi)
end

MATLAB

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

Wolfram

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

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

5. Taylor expanded in v around 0 97.8%

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

   1. associate-/r* 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Reproduce

herbie shell --seed 2024034 
(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)))))