Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 4

Speedup: 1.6×

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, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot 0.25\right) \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (* (sqrt (fma (* v v) -6.0 2.0)) 0.25) (- 1.0 (* v v))))

C

double code(double v) {
	return (sqrt(fma((v * v), -6.0, 2.0)) * 0.25) * (1.0 - (v * v));
}

Julia

function code(v)
	return Float64(Float64(sqrt(fma(Float64(v * v), -6.0, 2.0)) * 0.25) * Float64(1.0 - Float64(v * v)))
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision] * 0.25), $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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot 0.25\right)} \cdot \left(1 - v \cdot v\right) \]

5. Taylor expanded in v around 0

   \[\leadsto \left(\sqrt{\color{blue}{2 + -6 \cdot {v}^{2}}} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left({v}^{2}, -6, 2\right)}} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]

   4. unpow2 N/A

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

   5. lower-*.f64 100.0

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(\color{blue}{v \cdot v}, -6, 2\right)} \cdot 0.25\right) \cdot \left(1 - v \cdot v\right) \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \cdot 0.25\right) \cdot \left(1 - v \cdot v\right) \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(sqrt(2.0) * fma(Float64(-0.625 * v), v, 0.25))
end

Wolfram

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}, \sqrt{2}, \left(\left(\left(-v\right) \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}\right) \cdot 0.25\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-1}{4} \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-1}{4} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}} \]

   2. distribute-rgt-out N/A

      \[\leadsto {v}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\frac{-3}{8} + \frac{-1}{4}\right)\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   3. metadata-eval N/A

      \[\leadsto {v}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{-5}{8}}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left({v}^{2} \cdot \sqrt{2}\right) \cdot \frac{-5}{8}} + \frac{1}{4} \cdot \sqrt{2} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{-5}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}} + \frac{1}{4} \cdot \sqrt{2} \]

   7. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-5}{8}, {v}^{2}, \frac{1}{4}\right)} \]

   11. unpow2 N/A

       \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(\frac{-5}{8}, \color{blue}{v \cdot v}, \frac{1}{4}\right) \]

   12. lower-*.f64 99.5

       \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(-0.625, \color{blue}{v \cdot v}, 0.25\right) \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(-0.625, v \cdot v, 0.25\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.5%

   \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(-0.625 \cdot v, \color{blue}{v}, 0.25\right) \]
10. Add Preprocessing

Alternative 3: 99.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(v)
	return Float64(sqrt(2.0) * fma(-0.625, Float64(v * v), 0.25))
end

Wolfram

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right)}, \sqrt{2}, \left(\left(\left(-v\right) \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}\right) \cdot 0.25\right)} \]

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-1}{4} \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-1}{4} \cdot \sqrt{2}\right) + \frac{1}{4} \cdot \sqrt{2}} \]

   2. distribute-rgt-out N/A

      \[\leadsto {v}^{2} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\frac{-3}{8} + \frac{-1}{4}\right)\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   3. metadata-eval N/A

      \[\leadsto {v}^{2} \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{-5}{8}}\right) + \frac{1}{4} \cdot \sqrt{2} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left({v}^{2} \cdot \sqrt{2}\right) \cdot \frac{-5}{8}} + \frac{1}{4} \cdot \sqrt{2} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{-5}{8} \cdot \left({v}^{2} \cdot \sqrt{2}\right)} + \frac{1}{4} \cdot \sqrt{2} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-5}{8} \cdot {v}^{2}\right) \cdot \sqrt{2}} + \frac{1}{4} \cdot \sqrt{2} \]

   7. distribute-rgt-out N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right)} \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \left(\frac{-5}{8} \cdot {v}^{2} + \frac{1}{4}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sqrt{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{-5}{8}, {v}^{2}, \frac{1}{4}\right)} \]

   11. unpow2 N/A

       \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(\frac{-5}{8}, \color{blue}{v \cdot v}, \frac{1}{4}\right) \]

   12. lower-*.f64 99.5

       \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(-0.625, \color{blue}{v \cdot v}, 0.25\right) \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(-0.625, v \cdot v, 0.25\right)} \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 3.9× 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

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 99.3

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

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\sqrt{2} \cdot 0.25} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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))))