Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 4

Speedup: 1.7×

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, v \cdot v, 2\right)\\ \left(1 - v \cdot v\right) \cdot \left({\left(t\_0 \cdot t\_0\right)}^{0.25} \cdot 0.25\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (fma -6.0 (* v v) 2.0)))
   (* (- 1.0 (* v v)) (* (pow (* t_0 t_0) 0.25) 0.25))))

C

double code(double v) {
	double t_0 = fma(-6.0, (v * v), 2.0);
	return (1.0 - (v * v)) * (pow((t_0 * t_0), 0.25) * 0.25);
}

Julia

function code(v)
	t_0 = fma(-6.0, Float64(v * v), 2.0)
	return Float64(Float64(1.0 - Float64(v * v)) * Float64((Float64(t_0 * t_0) ^ 0.25) * 0.25))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(-6.0 * N[(v * v), $MachinePrecision] + 2.0), $MachinePrecision]}, N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $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%

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot 0.25\right)} \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

      \[\leadsto \left(\color{blue}{{\left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right)}^{\frac{1}{2}}} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]

   3. metadata-eval N/A

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

   4. pow-prod-up N/A

      \[\leadsto \left(\color{blue}{\left({\left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right)}^{\frac{1}{4}} \cdot {\left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right)}^{\frac{1}{4}}\right)} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]

   5. pow-prod-down N/A

      \[\leadsto \left(\color{blue}{{\left(\left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right)\right)}^{\frac{1}{4}}} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \left(\color{blue}{{\left(\left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2\right)\right)}^{\frac{1}{4}}} \cdot \frac{1}{4}\right) \cdot \left(1 - v \cdot v\right) \]

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \cdot 0.25\right)} \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

      \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(v \cdot v\right)\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   8. +-commutative N/A

      \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(v \cdot v\right)\right) + 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   9. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \left(\mathsf{neg}\left(v \cdot v\right)\right) + \frac{1}{4} \cdot 1\right)} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\color{blue}{\mathsf{neg}\left(4\right)}} \cdot \left(v \cdot v\right) + \frac{1}{4} \cdot 1\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   15. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\mathsf{neg}\left(4\right)} \cdot \left(v \cdot v\right) + \color{blue}{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   16. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{neg}\left(4\right)}, v \cdot v, \frac{1}{4}\right)} \cdot \sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2} \]

   17. metadata-eval N/A

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

   18. metadata-eval 100.0

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-fma.f64 N/A

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

   22. distribute-lft-in N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 98.9% 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 98.8

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

5. Applied rewrites 98.8%

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

Reproduce

herbie shell --seed 2024304 
(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))))