Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 99.7%

Time: 16.3s

Alternatives: 5

Speedup: 1.4×

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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4} - \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.28125, v \cdot v, -0.375\right), v \cdot v, 0.25 \cdot \sqrt{2}\right) \cdot \left(v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (-
  (/ (sqrt (* (fma -3.0 (* v v) 1.0) 2.0)) 4.0)
  (*
   (fma
    (* (sqrt 2.0) (fma -0.28125 (* v v) -0.375))
    (* v v)
    (* 0.25 (sqrt 2.0)))
   (* v v))))

C

double code(double v) {
	return (sqrt((fma(-3.0, (v * v), 1.0) * 2.0)) / 4.0) - (fma((sqrt(2.0) * fma(-0.28125, (v * v), -0.375)), (v * v), (0.25 * sqrt(2.0))) * (v * v));
}

Julia

function code(v)
	return Float64(Float64(sqrt(Float64(fma(-3.0, Float64(v * v), 1.0) * 2.0)) / 4.0) - Float64(fma(Float64(sqrt(2.0) * fma(-0.28125, Float64(v * v), -0.375)), Float64(v * v), Float64(0.25 * sqrt(2.0))) * Float64(v * v)))
end

Wolfram

code[v_] := N[(N[(N[Sqrt[N[(N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 4.0), $MachinePrecision] - N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.28125 * N[(v * v), $MachinePrecision] + -0.375), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(0.25 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4} - \color{blue}{{v}^{2} \cdot \left(\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-9}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4} - \color{blue}{\left(\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-9}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot {v}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4} - \color{blue}{\left(\frac{1}{4} \cdot \sqrt{2} + {v}^{2} \cdot \left(\frac{-3}{8} \cdot \sqrt{2} + \frac{-9}{32} \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)\right) \cdot {v}^{2}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4} - \color{blue}{\mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(-0.28125, v \cdot v, -0.375\right), v \cdot v, 0.25 \cdot \sqrt{2}\right) \cdot \left(v \cdot v\right)} \]
8. Add Preprocessing

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\mathsf{fma}\left(-3, v \cdot v, 1\right) \cdot 2}}{4}\\ t\_0 - t\_0 \cdot \left(v \cdot v\right) \end{array} \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* (fma -3.0 (* v v) 1.0) 2.0)) 4.0)))
   (- t_0 (* t_0 (* v v)))))

C

double code(double v) {
	double t_0 = sqrt((fma(-3.0, (v * v), 1.0) * 2.0)) / 4.0;
	return t_0 - (t_0 * (v * v));
}

Julia

function code(v)
	t_0 = Float64(sqrt(Float64(fma(-3.0, Float64(v * v), 1.0) * 2.0)) / 4.0)
	return Float64(t_0 - Float64(t_0 * Float64(v * v)))
end

Wolfram

code[v_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(-3.0 * N[(v * v), $MachinePrecision] + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / 4.0), $MachinePrecision]}, N[(t$95$0 - N[(t$95$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%

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

Alternative 3: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 4: 99.4% 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%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + \frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. 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. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 5: 98.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt 0.125))

C

double code(double v) {
	return sqrt(0.125);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt(0.125d0)
end function

Java

public static double code(double v) {
	return Math.sqrt(0.125);
}

Python

def code(v):
	return math.sqrt(0.125)

Julia

function code(v)
	return sqrt(0.125)
end

MATLAB

function tmp = code(v)
	tmp = sqrt(0.125);
end

Wolfram

code[v_] := N[Sqrt[0.125], $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%

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

5. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{2} + \frac{1}{4} \cdot \left({v}^{2} \cdot \left(\frac{-3}{2} \cdot \sqrt{2} + -1 \cdot \sqrt{2}\right)\right)} \]
6. 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. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-fma.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \left(v \cdot v\right), \color{blue}{-0.625}, \sqrt{0.125}\right) \]

10. Taylor expanded in v around 0

    \[\leadsto \sqrt{\frac{1}{8}} \]
11. Step-by-step derivation

12. Applied rewrites 99.0%

    \[\leadsto \sqrt{0.125} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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))))