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

Percentage Accurate: 99.3% → 99.5%

Time: 12.6s

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return ((1.0 + (v * (v * -5.0))) / Math.PI) / ((t * (1.0 - (v * v))) * Math.sqrt((2.0 + ((v * v) * -6.0))));
}

Python

def code(v, t):
	return ((1.0 + (v * (v * -5.0))) / math.pi) / ((t * (1.0 - (v * v))) * math.sqrt((2.0 + ((v * v) * -6.0))))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = ((1.0 + (v * (v * -5.0))) / pi) / ((t * (1.0 - (v * v))) * sqrt((2.0 + ((v * v) * -6.0))));
end

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/r* N/A

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

   5. frac-times N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{PI}\left(\right)} \cdot 1\right), \color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double v, double t) {
	return (1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (t * (Math.PI * Math.sqrt((2.0 + ((v * v) * -6.0))))));
}

Python

def code(v, t):
	return (1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (t * (math.pi * math.sqrt((2.0 + ((v * v) * -6.0))))))

Julia

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

MATLAB

function tmp = code(v, t)
	tmp = (1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (t * (pi * sqrt((2.0 + ((v * v) * -6.0))))));
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[(t * N[(Pi * N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $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. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot t\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot t\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(1 + \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot 1 + 2 \cdot \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 + 2 \cdot \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \left(\mathsf{neg}\left(3 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(2 \cdot \left(\left(\mathsf{neg}\left(3\right)\right) \cdot \left(v \cdot v\right)\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(2 \cdot \left(\mathsf{neg}\left(3\right)\right)\right) \cdot \left(v \cdot v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(2 \cdot \left(\mathsf{neg}\left(3\right)\right)\right), \left(v \cdot v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(2 \cdot -3\right), \left(v \cdot v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-6, \left(v \cdot v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-6, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{PI}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

   16. PI-lowering-PI.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(5, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(-6, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), \mathsf{PI.f64}\left(\right)\right), t\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 98.7% accurate, 1.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / Pi), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.5%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{\mathsf{PI}\left(\right)}}{\sqrt{2}}\right), \color{blue}{t}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right), \left(\sqrt{2}\right)\right), t\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right), \left(\sqrt{2}\right)\right), t\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \left(\sqrt{2}\right)\right), t\right) \]

   8. sqrt-lowering-sqrt.f64 98.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), t\right) \]

7. Applied egg-rr 98.9%

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

Alternative 4: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_, t_] := N[(N[(N[(1.0 / Pi), $MachinePrecision] / t), $MachinePrecision] / N[Sqrt[2.0], $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 v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.5%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right) \cdot t}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{\mathsf{PI}\left(\right)}}{t}\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{\mathsf{PI}\left(\right)}\right), t\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI}\left(\right)\right), t\right), \left(\sqrt{2}\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), t\right), \left(\sqrt{2}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 98.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{PI.f64}\left(\right)\right), t\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

7. Applied egg-rr 98.7%

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

Alternative 5: 98.2% accurate, 1.2× speedup

Math

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

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(1.0 / Float64(sqrt(2.0) * Float64(pi * t)))
end

MATLAB

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

Wolfram

code[v_, t_] := N[(1.0 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(Pi * t), $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 v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.5%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot t\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), t\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right) \]

   4. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right), \left(\sqrt{2}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 98.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 6: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(t \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(t \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{2}}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{PI}\left(\right) \cdot t\right) \cdot \sqrt{\color{blue}{2}}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(t \cdot \sqrt{2}\right)}\right)\right) \]

   6. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{t} \cdot \sqrt{2}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 98.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 98.5%

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

Alternative 7: 97.8% 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%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/r* N/A

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

   5. frac-times N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{PI}\left(\right)} \cdot 1\right), \color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(t \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{t} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{t}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{t}\right)\right) \]

   5. PI-lowering-PI.f64 98.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right)\right) \]

7. Simplified 98.1%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sqrt{\frac{1}{2}}}{t}\right), \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right), \mathsf{PI}\left(\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{PI}\left(\right)\right) \]

   6. PI-lowering-PI.f64 98.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{PI.f64}\left(\right)\right) \]

9. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]
10. Add Preprocessing

Alternative 8: 97.7% 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%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. associate-/r* N/A

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

   5. frac-times N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{PI}\left(\right)} \cdot 1\right), \color{blue}{\left(\left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(t \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\color{blue}{t} \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{t}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{t}\right)\right) \]

   5. PI-lowering-PI.f64 98.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), t\right)\right) \]

7. Simplified 98.1%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\pi \cdot t}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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)))))