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

Percentage Accurate: 99.3% → 99.3%

Time: 9.5s

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: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. pow1 99.4%

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

   2. associate-*l* 99.5%

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

   3. associate-*l* 99.5%

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

   4. pow2 99.5%

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

   5. pow2 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow1 99.5%

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

   2. *-commutative 99.5%

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

   3. cancel-sign-sub-inv 99.5%

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

   4. metadata-eval 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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(pi * Float64(t * fma(v, Float64(-v), 1.0))))
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[(Pi * N[(t * N[(v * (-v) + 1.0), $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%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)))) * ((Math.PI * t) * (1.0 - (v * v))));
}

Python

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

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(pi * t) * Float64(1.0 - Float64(v * v)))))
end

MATLAB

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

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

5. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* (* v v) 5.0))
  (* (- 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 - ((v * v) * 5.0)) / ((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 - ((v * v) * 5.0)) / ((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 - ((v * v) * 5.0)) / ((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(Float64(v * v) * 5.0)) / Float64(Float64(1.0 - Float64(v * v)) * Float64(Float64(pi * t) * sqrt(Float64(2.0 * Float64(1.0 - Float64(Float64(v * v) * 3.0)))))))
end

MATLAB

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

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

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(v, t):
	return (1.0 + (-5.0 * (v * v))) / ((math.pi * t) * ((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(-5.0 * Float64(v * v))) / Float64(Float64(pi * t) * 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 + (-5.0 * (v * v))) / ((pi * t) * ((1.0 - (v * v)) * 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[(Pi * t), $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%

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

5. Final simplification 99.4%

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

Alternative 6: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / t), $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 97.2%

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

   1. div-inv 97.1%

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

7. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. div-inv 97.2%

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

   2. *-commutative 97.2%

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

   3. associate-/r* 97.1%

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

9. Applied egg-rr 97.1%

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

    1. clear-num 98.1%

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

    2. unpow-1 98.1%

       \[\leadsto \frac{\color{blue}{{\left(\frac{\pi}{\sqrt{0.5}}\right)}^{-1}}}{t} \]

    3. div-inv 98.1%

       \[\leadsto \frac{{\color{blue}{\left(\pi \cdot \frac{1}{\sqrt{0.5}}\right)}}^{-1}}{t} \]

    4. pow1/2 98.1%

       \[\leadsto \frac{{\left(\pi \cdot \frac{1}{\color{blue}{{0.5}^{0.5}}}\right)}^{-1}}{t} \]

    5. pow-flip 98.1%

       \[\leadsto \frac{{\left(\pi \cdot \color{blue}{{0.5}^{\left(-0.5\right)}}\right)}^{-1}}{t} \]

    6. metadata-eval 98.1%

       \[\leadsto \frac{{\left(\pi \cdot {0.5}^{\color{blue}{-0.5}}\right)}^{-1}}{t} \]

    7. add-sqr-sqrt 98.1%

       \[\leadsto \frac{{\left(\pi \cdot \color{blue}{\left(\sqrt{{0.5}^{-0.5}} \cdot \sqrt{{0.5}^{-0.5}}\right)}\right)}^{-1}}{t} \]

    8. sqrt-unprod 98.1%

       \[\leadsto \frac{{\left(\pi \cdot \color{blue}{\sqrt{{0.5}^{-0.5} \cdot {0.5}^{-0.5}}}\right)}^{-1}}{t} \]

    9. pow-prod-up 98.1%

       \[\leadsto \frac{{\left(\pi \cdot \sqrt{\color{blue}{{0.5}^{\left(-0.5 + -0.5\right)}}}\right)}^{-1}}{t} \]

    10. metadata-eval 98.1%

        \[\leadsto \frac{{\left(\pi \cdot \sqrt{{0.5}^{\color{blue}{-1}}}\right)}^{-1}}{t} \]

    11. metadata-eval 98.1%

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

11. Applied egg-rr 98.1%

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

    1. unpow-1 98.1%

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

    2. *-commutative 98.1%

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

    3. associate-/r* 98.1%

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

13. Simplified 98.1%

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

Alternative 7: 98.5% 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%

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

5. Taylor expanded in v around 0 97.8%

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

Alternative 8: 98.0% 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 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 97.2%

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

6. Final simplification 97.2%

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

Reproduce

herbie shell --seed 2024100 
(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)))))