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

Percentage Accurate: 99.3% → 99.4%

Time: 10.5s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + (-5.0 * (v * v))) / ((t * (sqrt((2.0 * (1.0 - (3.0 * (v ^ 2.0))))) * pi)) * (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[(N[Sqrt[N[(2.0 * N[(1.0 - N[(3.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 67.5%

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

   2. expm1-udef 21.6%

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

   3. *-commutative 21.6%

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

   4. associate-*l* 21.6%

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

   5. pow2 21.6%

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

5. Applied egg-rr 21.6%

   \[\leadsto \frac{1 + -5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot {v}^{2}\right)}\right) \cdot \pi\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]
6. Step-by-step derivation

   1. expm1-def 67.5%

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

   2. expm1-log1p 99.4%

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

   3. associate-*l* 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + (-5.0 * (v * v))) / ((1.0 - (v * v)) * (pi * (t * sqrt((2.0 * (1.0 - (v * (v * 3.0))))))));
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[(Pi * N[(t * N[Sqrt[N[(2.0 * N[(1.0 - N[(v * N[(v * 3.0), $MachinePrecision]), $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%

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

4. Final simplification 99.4%

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

Alternative 3: 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. Final simplification 99.4%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

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))));
}

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))));
}

Python

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

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 + (-5.0 * (v * v))) / ((1.0 - (v * v)) * (t * (pi * sqrt(2.0))));
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[(t * N[(Pi * N[Sqrt[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 99.4%

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

4. Taylor expanded in v around 0 97.7%

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

   1. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 5: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in v around 0 97.7%

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

   1. associate-*r* 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in t around 0 97.7%

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

   1. associate-*r* 97.5%

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

   2. associate-/r* 97.6%

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

   3. associate-/r* 97.6%

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

9. Simplified 97.6%

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

    1. div-inv 97.6%

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

    2. *-un-lft-identity 97.6%

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

    3. times-frac 97.7%

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

11. Applied egg-rr 97.7%

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

12. Final simplification 97.7%

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

Alternative 6: 98.3% accurate, 1.1× 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%

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

4. Taylor expanded in v around 0 97.7%

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

   1. *-commutative 97.7%

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

6. Simplified 97.7%

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

7. Final simplification 97.7%

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

Alternative 7: 97.8% accurate, 1.1× 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%

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

   1. associate-*l* 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in v around 0 97.1%

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

5. Final simplification 97.1%

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

Alternative 8: 97.8% accurate, 1.1× 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%

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

   1. associate-*l* 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-*l* 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 45.3%

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

   2. pow2 45.3%

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

   3. *-commutative 45.3%

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

5. Applied egg-rr 45.3%

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

6. Taylor expanded in v around 0 97.1%

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

   1. associate-/r* 97.1%

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

8. Simplified 97.1%

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

9. Final simplification 97.1%

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

Reproduce

herbie shell --seed 2023334 
(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)))))