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

Percentage Accurate: 99.3% → 99.0%

Time: 10.8s

Alternatives: 7

Speedup: N/A×

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: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ -2.5 \cdot \left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right) + \frac{\frac{1}{\pi}}{\sqrt{2} \cdot t} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (+
  (* -2.5 (* (/ v (* PI (sqrt 2.0))) (/ v t)))
  (/ (/ 1.0 PI) (* (sqrt 2.0) t))))

C

double code(double v, double t) {
	return (-2.5 * ((v / (((double) M_PI) * sqrt(2.0))) * (v / t))) + ((1.0 / ((double) M_PI)) / (sqrt(2.0) * t));
}

Java

public static double code(double v, double t) {
	return (-2.5 * ((v / (Math.PI * Math.sqrt(2.0))) * (v / t))) + ((1.0 / Math.PI) / (Math.sqrt(2.0) * t));
}

Python

def code(v, t):
	return (-2.5 * ((v / (math.pi * math.sqrt(2.0))) * (v / t))) + ((1.0 / math.pi) / (math.sqrt(2.0) * t))

Julia

function code(v, t)
	return Float64(Float64(-2.5 * Float64(Float64(v / Float64(pi * sqrt(2.0))) * Float64(v / t))) + Float64(Float64(1.0 / pi) / Float64(sqrt(2.0) * t)))
end

MATLAB

function tmp = code(v, t)
	tmp = (-2.5 * ((v / (pi * sqrt(2.0))) * (v / t))) + ((1.0 / pi) / (sqrt(2.0) * t));
end

Wolfram

code[v_, t_] := N[(N[(-2.5 * N[(N[(v / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / Pi), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in v around 0 98.9%

   \[\leadsto \color{blue}{-2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
4. Step-by-step derivation

   1. inv-pow 98.9%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \color{blue}{{\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right)}^{-1}} \]

   2. associate-*r* 98.9%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + {\color{blue}{\left(\left(t \cdot \pi\right) \cdot \sqrt{2}\right)}}^{-1} \]

   3. *-commutative 98.9%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + {\left(\color{blue}{\left(\pi \cdot t\right)} \cdot \sqrt{2}\right)}^{-1} \]

   4. associate-*r* 98.8%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + {\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2}\right)\right)}}^{-1} \]

   5. unpow-prod-down 99.1%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \color{blue}{{\pi}^{-1} \cdot {\left(t \cdot \sqrt{2}\right)}^{-1}} \]

5. Applied egg-rr 99.1%

   \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \color{blue}{{\pi}^{-1} \cdot {\left(t \cdot \sqrt{2}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 99.1%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \color{blue}{\frac{1}{\pi}} \cdot {\left(t \cdot \sqrt{2}\right)}^{-1} \]

   2. unpow-1 99.1%

      \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{\pi} \cdot \color{blue}{\frac{1}{t \cdot \sqrt{2}}} \]

7. Simplified 99.1%

   \[\leadsto -2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \color{blue}{\frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}}} \]
8. Step-by-step derivation

   1. pow2 99.1%

      \[\leadsto -2.5 \cdot \frac{\color{blue}{v \cdot v}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

   2. *-commutative 99.1%

      \[\leadsto -2.5 \cdot \frac{v \cdot v}{\color{blue}{\left(\pi \cdot \sqrt{2}\right) \cdot t}} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

   3. times-frac 99.1%

      \[\leadsto -2.5 \cdot \color{blue}{\left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right)} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

9. Applied egg-rr 99.1%

   \[\leadsto -2.5 \cdot \color{blue}{\left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right)} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]
10. Step-by-step derivation

    1. un-div-inv 99.3%

       \[\leadsto -2.5 \cdot \left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right) + \color{blue}{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}} \]

11. Applied egg-rr 99.3%

    \[\leadsto -2.5 \cdot \left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right) + \color{blue}{\frac{\frac{1}{\pi}}{t \cdot \sqrt{2}}} \]

12. Final simplification 99.3%

    \[\leadsto -2.5 \cdot \left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right) + \frac{\frac{1}{\pi}}{\sqrt{2} \cdot t} \]
13. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \pi \cdot \sqrt{2}\\ -2.5 \cdot \left(\frac{v}{t_1} \cdot \frac{v}{t}\right) + \frac{1}{t_1 \cdot t} \end{array} \end{array} \]

FPCore

(FPCore (v t)
 :precision binary64
 (let* ((t_1 (* PI (sqrt 2.0))))
   (+ (* -2.5 (* (/ v t_1) (/ v t))) (/ 1.0 (* t_1 t)))))

C

double code(double v, double t) {
	double t_1 = ((double) M_PI) * sqrt(2.0);
	return (-2.5 * ((v / t_1) * (v / t))) + (1.0 / (t_1 * t));
}

Java

public static double code(double v, double t) {
	double t_1 = Math.PI * Math.sqrt(2.0);
	return (-2.5 * ((v / t_1) * (v / t))) + (1.0 / (t_1 * t));
}

Python

def code(v, t):
	t_1 = math.pi * math.sqrt(2.0)
	return (-2.5 * ((v / t_1) * (v / t))) + (1.0 / (t_1 * t))

Julia

function code(v, t)
	t_1 = Float64(pi * sqrt(2.0))
	return Float64(Float64(-2.5 * Float64(Float64(v / t_1) * Float64(v / t))) + Float64(1.0 / Float64(t_1 * t)))
end

MATLAB

function tmp = code(v, t)
	t_1 = pi * sqrt(2.0);
	tmp = (-2.5 * ((v / t_1) * (v / t))) + (1.0 / (t_1 * t));
end

Wolfram

code[v_, t_] := Block[{t$95$1 = N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-2.5 * N[(N[(v / t$95$1), $MachinePrecision] * N[(v / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$1 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in v around 0 98.9%

   \[\leadsto \color{blue}{-2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]
4. Step-by-step derivation

   1. pow2 99.1%

      \[\leadsto -2.5 \cdot \frac{\color{blue}{v \cdot v}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

   2. *-commutative 99.1%

      \[\leadsto -2.5 \cdot \frac{v \cdot v}{\color{blue}{\left(\pi \cdot \sqrt{2}\right) \cdot t}} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

   3. times-frac 99.1%

      \[\leadsto -2.5 \cdot \color{blue}{\left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right)} + \frac{1}{\pi} \cdot \frac{1}{t \cdot \sqrt{2}} \]

5. Applied egg-rr 98.9%

   \[\leadsto -2.5 \cdot \color{blue}{\left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right)} + \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \]

6. Final simplification 98.9%

   \[\leadsto -2.5 \cdot \left(\frac{v}{\pi \cdot \sqrt{2}} \cdot \frac{v}{t}\right) + \frac{1}{\left(\pi \cdot \sqrt{2}\right) \cdot t} \]
7. Add Preprocessing

Alternative 3: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around 0 97.5%

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

   1. *-un-lft-identity 97.5%

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

   2. times-frac 98.0%

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

7. Applied egg-rr 98.0%

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

8. Final simplification 98.0%

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

Alternative 4: 97.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around 0 97.5%

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

   1. associate-/r* 98.0%

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

7. Simplified 98.0%

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

   1. clear-num 97.7%

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

   2. associate-/r/ 98.0%

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

9. Applied egg-rr 98.0%

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

10. Final simplification 98.0%

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

Alternative 5: 98.8% accurate, 1.2× 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 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in t around 0 98.8%

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

6. Taylor expanded in v around 0 97.5%

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

   1. associate-/r* 98.0%

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

   2. *-rgt-identity 98.0%

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

   3. associate-*r/ 97.8%

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

   4. associate-*l/ 97.9%

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

   5. associate-*r/ 97.9%

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

   6. *-rgt-identity 97.9%

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

8. Simplified 97.9%

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

   1. metadata-eval 97.9%

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

   2. metadata-eval 97.9%

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

   3. rem-square-sqrt 98.9%

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

   4. frac-times 98.9%

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

   5. sqrt-unprod 98.9%

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

   6. add-sqr-sqrt 98.9%

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

   7. associate-/r* 98.9%

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

   8. *-commutative 98.9%

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

   9. inv-pow 98.9%

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

10. Applied egg-rr 98.9%

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

    1. unpow-1 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 6: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 7: 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 98.9%

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

3. Simplified 98.8%

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

5. Taylor expanded in v around 0 97.5%

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

   1. associate-/r* 98.0%

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

7. Simplified 98.0%

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

8. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024016 
(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)))))