Falkner and Boettcher, Equation (22+)

Percentage Accurate: 98.5% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ (/ 1.3333333333333333 PI) (- 1.0 (* v v)))
  (sqrt (fma (* v v) -6.0 2.0))))

C

double code(double v) {
	return ((1.3333333333333333 / ((double) M_PI)) / (1.0 - (v * v))) / sqrt(fma((v * v), -6.0, 2.0));
}

Julia

function code(v)
	return Float64(Float64(Float64(1.3333333333333333 / pi) / Float64(1.0 - Float64(v * v))) / sqrt(fma(Float64(v * v), -6.0, 2.0)))
end

Wolfram

code[v_] := N[(N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. *-commutative 98.4%

      \[\leadsto \frac{4}{\color{blue}{\left(\left(1 - v \cdot v\right) \cdot \left(3 \cdot \pi\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   2. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(\left(1 - \color{blue}{\left(-v\right) \cdot \left(-v\right)}\right) \cdot \left(3 \cdot \pi\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(\left(1 - \left(-v\right) \cdot \left(-v\right)\right) \cdot \left(3 \cdot \pi\right)\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}}} \]

   4. *-commutative 98.4%

      \[\leadsto \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)\right)} \cdot \sqrt{2 - 6 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}} \]

   5. remove-double-neg 98.4%

      \[\leadsto \color{blue}{-\left(-\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)\right) \cdot \sqrt{2 - 6 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}}\right)} \]

   6. associate-/r* 100.0%

      \[\leadsto -\left(-\color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}{\sqrt{2 - 6 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}}}\right) \]

   7. distribute-neg-frac 100.0%

      \[\leadsto -\color{blue}{\frac{-\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}{\sqrt{2 - 6 \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/
  (/ 1.3333333333333333 (* PI (- 1.0 (* v v))))
  (sqrt (+ 2.0 (* (* v v) -6.0)))))

C

double code(double v) {
	return (1.3333333333333333 / (((double) M_PI) * (1.0 - (v * v)))) / sqrt((2.0 + ((v * v) * -6.0)));
}

Java

public static double code(double v) {
	return (1.3333333333333333 / (Math.PI * (1.0 - (v * v)))) / Math.sqrt((2.0 + ((v * v) * -6.0)));
}

Python

def code(v):
	return (1.3333333333333333 / (math.pi * (1.0 - (v * v)))) / math.sqrt((2.0 + ((v * v) * -6.0)))

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / Float64(pi * Float64(1.0 - Float64(v * v)))) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))))
end

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / (pi * (1.0 - (v * v)))) / sqrt((2.0 + ((v * v) * -6.0)));
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / N[(Pi * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. associate-/r* 100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. sqr-neg 100.0%

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - \color{blue}{\left(-v\right) \cdot \left(-v\right)}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. associate-*l* 100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. associate-/r* 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. sub-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

   8. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 6}\right)}} \]

   10. distribute-rgt-neg-in 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-6\right)}}} \]

   11. sqr-neg 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-6\right)}} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (* 1.3333333333333333 (/ 1.0 PI)) (sqrt (+ 2.0 (* (* v v) -6.0)))))

C

double code(double v) {
	return (1.3333333333333333 * (1.0 / ((double) M_PI))) / sqrt((2.0 + ((v * v) * -6.0)));
}

Java

public static double code(double v) {
	return (1.3333333333333333 * (1.0 / Math.PI)) / Math.sqrt((2.0 + ((v * v) * -6.0)));
}

Python

def code(v):
	return (1.3333333333333333 * (1.0 / math.pi)) / math.sqrt((2.0 + ((v * v) * -6.0)))

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 * Float64(1.0 / pi)) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))))
end

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 * (1.0 / pi)) / sqrt((2.0 + ((v * v) * -6.0)));
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. associate-/r* 100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. sqr-neg 100.0%

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - \color{blue}{\left(-v\right) \cdot \left(-v\right)}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. associate-*l* 100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. associate-/r* 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. sub-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

   8. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 6}\right)}} \]

   10. distribute-rgt-neg-in 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-6\right)}}} \]

   11. sqr-neg 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-6\right)}} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 99.1%

   \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
6. Step-by-step derivation

   1. clear-num 99.1%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\pi}{1.3333333333333333}}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

   2. associate-/r/ 99.1%

      \[\leadsto \frac{\color{blue}{\frac{1}{\pi} \cdot 1.3333333333333333}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

7. Applied egg-rr 99.1%

   \[\leadsto \frac{\color{blue}{\frac{1}{\pi} \cdot 1.3333333333333333}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

8. Final simplification 99.1%

   \[\leadsto \frac{1.3333333333333333 \cdot \frac{1}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ (/ 1.3333333333333333 PI) (sqrt (+ 2.0 (* (* v v) -6.0)))))

C

double code(double v) {
	return (1.3333333333333333 / ((double) M_PI)) / sqrt((2.0 + ((v * v) * -6.0)));
}

Java

public static double code(double v) {
	return (1.3333333333333333 / Math.PI) / Math.sqrt((2.0 + ((v * v) * -6.0)));
}

Python

def code(v):
	return (1.3333333333333333 / math.pi) / math.sqrt((2.0 + ((v * v) * -6.0)))

Julia

function code(v)
	return Float64(Float64(1.3333333333333333 / pi) / sqrt(Float64(2.0 + Float64(Float64(v * v) * -6.0))))
end

MATLAB

function tmp = code(v)
	tmp = (1.3333333333333333 / pi) / sqrt((2.0 + ((v * v) * -6.0)));
end

Wolfram

code[v_] := N[(N[(1.3333333333333333 / Pi), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. associate-/r* 100.0%

      \[\leadsto \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

   2. sqr-neg 100.0%

      \[\leadsto \frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - \color{blue}{\left(-v\right) \cdot \left(-v\right)}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   3. associate-*l* 100.0%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   4. associate-/r* 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - \left(-v\right) \cdot \left(-v\right)\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   6. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - \color{blue}{v \cdot v}\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

   7. sub-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6 \cdot \left(v \cdot v\right)\right)}}} \]

   8. sqr-neg 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}\right)}} \]

   9. *-commutative 100.0%

      \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(-\color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot 6}\right)}} \]

   10. distribute-rgt-neg-in 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right) \cdot \left(-6\right)}}} \]

   11. sqr-neg 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right)} \cdot \left(-6\right)}} \]

   12. metadata-eval 100.0%

       \[\leadsto \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 99.1%

   \[\leadsto \frac{\color{blue}{\frac{1.3333333333333333}{\pi}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
6. Add Preprocessing

Alternative 5: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1.3333333333333333}{\pi \cdot \sqrt{2}} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* PI (sqrt 2.0))))

C

double code(double v) {
	return 1.3333333333333333 / (((double) M_PI) * sqrt(2.0));
}

Java

public static double code(double v) {
	return 1.3333333333333333 / (Math.PI * Math.sqrt(2.0));
}

Python

def code(v):
	return 1.3333333333333333 / (math.pi * math.sqrt(2.0))

Julia

function code(v)
	return Float64(1.3333333333333333 / Float64(pi * sqrt(2.0)))
end

MATLAB

function tmp = code(v)
	tmp = 1.3333333333333333 / (pi * sqrt(2.0));
end

Wolfram

code[v_] := N[(1.3333333333333333 / N[(Pi * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. associate-*l* 98.4%

      \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}} \]

   2. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}}\right)} \]

   3. cancel-sign-sub-inv 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{2 + \left(-6\right) \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}}\right)} \]

   4. metadata-eval 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + \color{blue}{-6} \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}\right)} \]

   5. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + -6 \cdot \color{blue}{\left(v \cdot v\right)}}\right)} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + -6 \cdot \left(v \cdot v\right)}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 97.5%

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

   1. *-commutative 97.5%

      \[\leadsto \frac{4}{\color{blue}{\left(\pi \cdot \sqrt{2}\right) \cdot 3}} \]

   2. associate-*l* 99.1%

      \[\leadsto \frac{4}{\color{blue}{\pi \cdot \left(\sqrt{2} \cdot 3\right)}} \]

7. Simplified 99.1%

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

   1. *-un-lft-identity 99.1%

      \[\leadsto \color{blue}{1 \cdot \frac{4}{\pi \cdot \left(\sqrt{2} \cdot 3\right)}} \]

   2. *-commutative 99.1%

      \[\leadsto \color{blue}{\frac{4}{\pi \cdot \left(\sqrt{2} \cdot 3\right)} \cdot 1} \]

   3. associate-/r* 99.1%

      \[\leadsto \color{blue}{\frac{\frac{4}{\pi}}{\sqrt{2} \cdot 3}} \cdot 1 \]

   4. *-commutative 99.1%

      \[\leadsto \frac{\frac{4}{\pi}}{\color{blue}{3 \cdot \sqrt{2}}} \cdot 1 \]

   5. associate-/r* 99.1%

      \[\leadsto \color{blue}{\frac{\frac{\frac{4}{\pi}}{3}}{\sqrt{2}}} \cdot 1 \]

   6. associate-/r* 99.1%

      \[\leadsto \frac{\color{blue}{\frac{4}{\pi \cdot 3}}}{\sqrt{2}} \cdot 1 \]

   7. *-commutative 99.1%

      \[\leadsto \frac{\frac{4}{\color{blue}{3 \cdot \pi}}}{\sqrt{2}} \cdot 1 \]

   8. associate-/r* 99.1%

      \[\leadsto \frac{\color{blue}{\frac{\frac{4}{3}}{\pi}}}{\sqrt{2}} \cdot 1 \]

   9. metadata-eval 99.1%

      \[\leadsto \frac{\frac{\color{blue}{1.3333333333333333}}{\pi}}{\sqrt{2}} \cdot 1 \]

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2}} \cdot 1} \]
10. Step-by-step derivation

    1. *-rgt-identity 99.1%

       \[\leadsto \color{blue}{\frac{\frac{1.3333333333333333}{\pi}}{\sqrt{2}}} \]

    2. associate-/r* 99.1%

       \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \sqrt{2}}} \]

11. Simplified 99.1%

    \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \sqrt{2}}} \]
12. Add Preprocessing

Alternative 6: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{0.8888888888888888}}{\pi} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (/ (sqrt 0.8888888888888888) PI))

C

double code(double v) {
	return sqrt(0.8888888888888888) / ((double) M_PI);
}

Java

public static double code(double v) {
	return Math.sqrt(0.8888888888888888) / Math.PI;
}

Python

def code(v):
	return math.sqrt(0.8888888888888888) / math.pi

Julia

function code(v)
	return Float64(sqrt(0.8888888888888888) / pi)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.8888888888888888) / pi;
end

Wolfram

code[v_] := N[(N[Sqrt[0.8888888888888888], $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
2. Step-by-step derivation

   1. associate-*l* 98.4%

      \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}\right)}} \]

   2. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(\left(-v\right) \cdot \left(-v\right)\right)}}\right)} \]

   3. cancel-sign-sub-inv 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{2 + \left(-6\right) \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}}\right)} \]

   4. metadata-eval 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + \color{blue}{-6} \cdot \left(\left(-v\right) \cdot \left(-v\right)\right)}\right)} \]

   5. sqr-neg 98.4%

      \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + -6 \cdot \color{blue}{\left(v \cdot v\right)}}\right)} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 + -6 \cdot \left(v \cdot v\right)}\right)}} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 97.5%

   \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 97.5%

      \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}} \]

7. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}} \]
8. Step-by-step derivation

   1. expm1-log1p-u 97.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}\right)\right)} \]

   2. expm1-undefine 97.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1.3333333333333333 \cdot \sqrt{0.5}}{\pi}\right)} - 1} \]

   3. add-sqr-sqrt 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{1.3333333333333333 \cdot \sqrt{0.5}} \cdot \sqrt{1.3333333333333333 \cdot \sqrt{0.5}}}}{\pi}\right)} - 1 \]

   4. sqrt-unprod 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(1.3333333333333333 \cdot \sqrt{0.5}\right) \cdot \left(1.3333333333333333 \cdot \sqrt{0.5}\right)}}}{\pi}\right)} - 1 \]

   5. swap-sqr 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\left(1.3333333333333333 \cdot 1.3333333333333333\right) \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}}}{\pi}\right)} - 1 \]

   6. metadata-eval 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{1.7777777777777777} \cdot \left(\sqrt{0.5} \cdot \sqrt{0.5}\right)}}{\pi}\right)} - 1 \]

   7. rem-square-sqrt 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{1.7777777777777777 \cdot \color{blue}{0.5}}}{\pi}\right)} - 1 \]

   8. metadata-eval 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{0.8888888888888888}}}{\pi}\right)} - 1 \]

9. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 97.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)} + \left(-1\right)} \]

    2. metadata-eval 97.5%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)} + \color{blue}{-1} \]

    3. +-commutative 97.5%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{\sqrt{0.8888888888888888}}{\pi}\right)}} \]

    4. log1p-undefine 97.5%

       \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \frac{\sqrt{0.8888888888888888}}{\pi}\right)}} \]

    5. rem-exp-log 97.5%

       \[\leadsto -1 + \color{blue}{\left(1 + \frac{\sqrt{0.8888888888888888}}{\pi}\right)} \]

    6. associate-+r+ 97.5%

       \[\leadsto \color{blue}{\left(-1 + 1\right) + \frac{\sqrt{0.8888888888888888}}{\pi}} \]

    7. metadata-eval 97.5%

       \[\leadsto \color{blue}{0} + \frac{\sqrt{0.8888888888888888}}{\pi} \]

    8. +-lft-identity 97.5%

       \[\leadsto \color{blue}{\frac{\sqrt{0.8888888888888888}}{\pi}} \]

11. Simplified 97.5%

    \[\leadsto \color{blue}{\frac{\sqrt{0.8888888888888888}}{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))