Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = ((sqrt(2.0d0) / 4.0d0) * sqrt((1.0d0 - (3.0d0 * (v * v))))) * (1.0d0 - (v * v))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot \left({v}^{2} \cdot -0.625 + 0.25\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) (+ (* (pow v 2.0) -0.625) 0.25)))

C

double code(double v) {
	return sqrt(2.0) * ((pow(v, 2.0) * -0.625) + 0.25);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * (((v ** 2.0d0) * (-0.625d0)) + 0.25d0)
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * ((Math.pow(v, 2.0) * -0.625) + 0.25);
}

Python

def code(v):
	return math.sqrt(2.0) * ((math.pow(v, 2.0) * -0.625) + 0.25)

Julia

function code(v)
	return Float64(sqrt(2.0) * Float64(Float64((v ^ 2.0) * -0.625) + 0.25))
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * (((v ^ 2.0) * -0.625) + 0.25);
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Power[v, 2.0], $MachinePrecision] * -0.625), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 98.9%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

   1. associate-*r* 98.9%

      \[\leadsto \left(\color{blue}{\left(-0.375 \cdot {v}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \sqrt{2}\right) \cdot \left(1 - v \cdot v\right) \]

   2. distribute-rgt-out 98.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(-0.375 \cdot {v}^{2} + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   3. *-commutative 98.9%

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{{v}^{2} \cdot -0.375} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({v}^{2} \cdot -0.375 + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

6. Taylor expanded in v around 0 99.0%

   \[\leadsto \color{blue}{-0.625 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}} \]
7. Step-by-step derivation

   1. associate-*r* 99.0%

      \[\leadsto \color{blue}{\left(-0.625 \cdot {v}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \sqrt{2} \]

   2. distribute-rgt-out 99.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-0.625 \cdot {v}^{2} + 0.25\right)} \]

   3. *-commutative 99.0%

      \[\leadsto \sqrt{2} \cdot \left(\color{blue}{{v}^{2} \cdot -0.625} + 0.25\right) \]

8. Simplified 99.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left({v}^{2} \cdot -0.625 + 0.25\right)} \]

9. Final simplification 99.0%

   \[\leadsto \sqrt{2} \cdot \left({v}^{2} \cdot -0.625 + 0.25\right) \]
10. Add Preprocessing

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot 0.25\right) \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (- 1.0 (* v v)) (* (sqrt 2.0) 0.25)))

C

double code(double v) {
	return (1.0 - (v * v)) * (sqrt(2.0) * 0.25);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (sqrt(2.0d0) * 0.25d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (Math.sqrt(2.0) * 0.25);
}

Python

def code(v):
	return (1.0 - (v * v)) * (math.sqrt(2.0) * 0.25)

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * Float64(sqrt(2.0) * 0.25))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (sqrt(2.0) * 0.25);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 98.9%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
4. Step-by-step derivation

   1. associate-*r* 98.9%

      \[\leadsto \left(\color{blue}{\left(-0.375 \cdot {v}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \sqrt{2}\right) \cdot \left(1 - v \cdot v\right) \]

   2. distribute-rgt-out 98.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(-0.375 \cdot {v}^{2} + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   3. *-commutative 98.9%

      \[\leadsto \left(\sqrt{2} \cdot \left(\color{blue}{{v}^{2} \cdot -0.375} + 0.25\right)\right) \cdot \left(1 - v \cdot v\right) \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left({v}^{2} \cdot -0.375 + 0.25\right)\right)} \cdot \left(1 - v \cdot v\right) \]

6. Taylor expanded in v around 0 98.2%

   \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]

7. Final simplification 98.2%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot 0.25\right) \]
8. Add Preprocessing

Alternative 4: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot 0.25 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) 0.25))

C

double code(double v) {
	return sqrt(2.0) * 0.25;
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(2.0d0) * 0.25d0
end function

Java

public static double code(double v) {
	return Math.sqrt(2.0) * 0.25;
}

Python

def code(v):
	return math.sqrt(2.0) * 0.25

Julia

function code(v)
	return Float64(sqrt(2.0) * 0.25)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(2.0) * 0.25;
end

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in v around 0 98.2%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2}} \]

4. Final simplification 98.2%

   \[\leadsto \sqrt{2} \cdot 0.25 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))