Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 6

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

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

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(sqrt(2.0) / 4.0) * Float64(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[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_] := N[(N[Sqrt[N[(2.0 * N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 / N[(1.0 - N[(v * v), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

   3. frac-2neg 100.0%

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

   4. metadata-eval 100.0%

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

   5. associate-/l/ 100.0%

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

   6. fma-udef 100.0%

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

   7. metadata-eval 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. *-commutative 100.0%

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

   10. +-commutative 100.0%

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

   11. sub-neg 100.0%

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

   12. fma-udef 100.0%

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

   13. distribute-neg-in 100.0%

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

   14. metadata-eval 100.0%

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

   15. +-commutative 100.0%

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

   16. sub-neg 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* 0.25 (+ (sqrt 2.0) (* (* v v) (* (sqrt 2.0) -2.5)))))

C

double code(double v) {
	return 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = 0.25d0 * (sqrt(2.0d0) + ((v * v) * (sqrt(2.0d0) * (-2.5d0))))
end function

Java

public static double code(double v) {
	return 0.25 * (Math.sqrt(2.0) + ((v * v) * (Math.sqrt(2.0) * -2.5)));
}

Python

def code(v):
	return 0.25 * (math.sqrt(2.0) + ((v * v) * (math.sqrt(2.0) * -2.5)))

Julia

function code(v)
	return Float64(0.25 * Float64(sqrt(2.0) + Float64(Float64(v * v) * Float64(sqrt(2.0) * -2.5))))
end

MATLAB

function tmp = code(v)
	tmp = 0.25 * (sqrt(2.0) + ((v * v) * (sqrt(2.0) * -2.5)));
end

Wolfram

code[v_] := N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -2.5), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + 0.25 \cdot \left({v}^{2} \cdot \left(-1 \cdot \sqrt{2} + -1.5 \cdot \sqrt{2}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 99.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + {v}^{2} \cdot \left(-1 \cdot \sqrt{2} + -1.5 \cdot \sqrt{2}\right)\right)} \]

   2. unpow2 99.7%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \color{blue}{\left(v \cdot v\right)} \cdot \left(-1 \cdot \sqrt{2} + -1.5 \cdot \sqrt{2}\right)\right) \]

   3. *-commutative 99.7%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \color{blue}{\left(-1 \cdot \sqrt{2} + -1.5 \cdot \sqrt{2}\right) \cdot \left(v \cdot v\right)}\right) \]

   4. distribute-rgt-out 99.7%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \color{blue}{\left(\sqrt{2} \cdot \left(-1 + -1.5\right)\right)} \cdot \left(v \cdot v\right)\right) \]

   5. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \left(\sqrt{2} + \left(\sqrt{2} \cdot \color{blue}{-2.5}\right) \cdot \left(v \cdot v\right)\right) \]

6. Simplified 99.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{2} + \left(\sqrt{2} \cdot -2.5\right) \cdot \left(v \cdot v\right)\right)} \]

7. Final simplification 99.7%

   \[\leadsto 0.25 \cdot \left(\sqrt{2} + \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot -2.5\right)\right) \]

Alternative 4: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(0.0625 + \left(v \cdot v\right) \cdot -0.3125\right)} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (sqrt (* 2.0 (+ 0.0625 (* (* v v) -0.3125)))))

C

double code(double v) {
	return sqrt((2.0 * (0.0625 + ((v * v) * -0.3125))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = sqrt((2.0d0 * (0.0625d0 + ((v * v) * (-0.3125d0)))))
end function

Java

public static double code(double v) {
	return Math.sqrt((2.0 * (0.0625 + ((v * v) * -0.3125))));
}

Python

def code(v):
	return math.sqrt((2.0 * (0.0625 + ((v * v) * -0.3125))))

Julia

function code(v)
	return sqrt(Float64(2.0 * Float64(0.0625 + Float64(Float64(v * v) * -0.3125))))
end

MATLAB

function tmp = code(v)
	tmp = sqrt((2.0 * (0.0625 + ((v * v) * -0.3125))));
end

Wolfram

code[v_] := N[Sqrt[N[(2.0 * N[(0.0625 + N[(N[(v * v), $MachinePrecision] * -0.3125), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{-0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) + 0.25 \cdot \sqrt{2}} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + -0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right)} \]

   2. unpow2 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + -0.25 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) \]

   3. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)} \]

   4. distribute-rgt1-in 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{\left(\left(1.5 + 1\right) \cdot \sqrt{2}\right)} \]

   5. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{2.5} \cdot \sqrt{2}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{\frac{5}{2}} \cdot \sqrt{2}\right) \]

   7. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right) \cdot \sqrt{2}} \]

   8. distribute-rgt-out 99.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{2.5}\right) \]

6. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 98.2%

      \[\leadsto \color{blue}{\sqrt{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \cdot \sqrt{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)}} \]

   2. sqrt-unprod 99.7%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right) \cdot \left(\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}} \]

   3. swap-sqr 99.7%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(\left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right) \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}} \]

   4. add-sqr-sqrt 99.7%

      \[\leadsto \sqrt{\color{blue}{2} \cdot \left(\left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right) \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)} \]

   5. pow2 99.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{{\left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)}^{2}}} \]

   6. +-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot {\color{blue}{\left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5 + 0.25\right)}}^{2}} \]

   7. *-commutative 99.7%

      \[\leadsto \sqrt{2 \cdot {\left(\color{blue}{2.5 \cdot \left(-0.25 \cdot \left(v \cdot v\right)\right)} + 0.25\right)}^{2}} \]

   8. associate-*r* 99.7%

      \[\leadsto \sqrt{2 \cdot {\left(\color{blue}{\left(2.5 \cdot -0.25\right) \cdot \left(v \cdot v\right)} + 0.25\right)}^{2}} \]

   9. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in v around 0 99.7%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(0.0625 + -0.3125 \cdot {v}^{2}\right)}} \]
10. Step-by-step derivation

    1. unpow2 99.7%

       \[\leadsto \sqrt{2 \cdot \left(0.0625 + -0.3125 \cdot \color{blue}{\left(v \cdot v\right)}\right)} \]

11. Simplified 99.7%

    \[\leadsto \sqrt{2 \cdot \color{blue}{\left(0.0625 + -0.3125 \cdot \left(v \cdot v\right)\right)}} \]

12. Final simplification 99.7%

    \[\leadsto \sqrt{2 \cdot \left(0.0625 + \left(v \cdot v\right) \cdot -0.3125\right)} \]

Alternative 5: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{-0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) + 0.25 \cdot \sqrt{2}} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + -0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right)} \]

   2. unpow2 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + -0.25 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) \]

   3. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)} \]

   4. distribute-rgt1-in 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{\left(\left(1.5 + 1\right) \cdot \sqrt{2}\right)} \]

   5. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{2.5} \cdot \sqrt{2}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{\frac{5}{2}} \cdot \sqrt{2}\right) \]

   7. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right) \cdot \sqrt{2}} \]

   8. distribute-rgt-out 99.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{2.5}\right) \]

6. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

7. Taylor expanded in v around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 6: 98.9% accurate, 2.1× 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. Step-by-step derivation

   1. associate-*l/ 100.0%

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

   2. associate-/r/ 100.0%

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

   3. associate-*r/ 100.0%

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

   4. sub-neg 100.0%

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

   5. +-commutative 100.0%

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

   6. *-commutative 100.0%

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

   7. distribute-rgt-neg-in 100.0%

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

   8. fma-def 100.0%

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

   9. metadata-eval 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. neg-sub0 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{\left(0 - v \cdot v\right)} + 1}} \]

   13. associate-+l- 100.0%

       \[\leadsto \sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}{\frac{4}{\color{blue}{0 - \left(v \cdot v - 1\right)}}} \]

   14. sub0-neg 100.0%

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

   15. neg-mul-1 100.0%

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

   16. associate-/r* 100.0%

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

   17. metadata-eval 100.0%

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

   18. fma-neg 100.0%

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

   19. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.7%

   \[\leadsto \color{blue}{-0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) + 0.25 \cdot \sqrt{2}} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} + -0.25 \cdot \left({v}^{2} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right)} \]

   2. unpow2 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + -0.25 \cdot \left(\color{blue}{\left(v \cdot v\right)} \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)\right) \]

   3. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{2} + 1.5 \cdot \sqrt{2}\right)} \]

   4. distribute-rgt1-in 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{\left(\left(1.5 + 1\right) \cdot \sqrt{2}\right)} \]

   5. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{2.5} \cdot \sqrt{2}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{\frac{5}{2}} \cdot \sqrt{2}\right) \]

   7. associate-*r* 99.7%

      \[\leadsto 0.25 \cdot \sqrt{2} + \color{blue}{\left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right) \cdot \sqrt{2}} \]

   8. distribute-rgt-out 99.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \frac{5}{2}\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot \color{blue}{2.5}\right) \]

6. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(0.25 + \left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 99.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot 0.25 + \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   2. flip-+ 98.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right) \cdot \left(\sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)}} \]

   3. *-commutative 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \color{blue}{\left(2.5 \cdot \left(-0.25 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left(\sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   4. associate-*r* 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \color{blue}{\left(\left(2.5 \cdot -0.25\right) \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(\sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   5. metadata-eval 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(\color{blue}{-0.625} \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   6. *-commutative 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(-0.625 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(2.5 \cdot \left(-0.25 \cdot \left(v \cdot v\right)\right)\right)}\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   7. associate-*r* 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(-0.625 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\left(2.5 \cdot -0.25\right) \cdot \left(v \cdot v\right)\right)}\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   8. metadata-eval 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(-0.625 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{-0.625} \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \left(\left(-0.25 \cdot \left(v \cdot v\right)\right) \cdot 2.5\right)} \]

   9. *-commutative 98.2%

      \[\leadsto \frac{\left(\sqrt{2} \cdot 0.25\right) \cdot \left(\sqrt{2} \cdot 0.25\right) - \left(\sqrt{2} \cdot \left(-0.625 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(-0.625 \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{2} \cdot 0.25 - \sqrt{2} \cdot \color{blue}{\left(2.5 \cdot \left(-0.25 \cdot \left(v \cdot v\right)\right)\right)}} \]

8. Applied egg-rr 98.2%

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

10. Simplified 98.2%

    \[\leadsto \color{blue}{\frac{0.125 + -2 \cdot \left({v}^{4} \cdot 0.390625\right)}{\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot 0.625\right)}} \]

11. Taylor expanded in v around 0 97.7%

    \[\leadsto \color{blue}{\frac{0.5}{\sqrt{2}}} \]
12. Step-by-step derivation

    1. add-log-exp 99.2%

       \[\leadsto \color{blue}{\log \left(e^{\frac{0.5}{\sqrt{2}}}\right)} \]

    2. *-un-lft-identity 99.2%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{0.5}{\sqrt{2}}}\right)} \]

    3. log-prod 99.2%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{0.5}{\sqrt{2}}}\right)} \]

    4. metadata-eval 99.2%

       \[\leadsto \color{blue}{0} + \log \left(e^{\frac{0.5}{\sqrt{2}}}\right) \]

    5. add-log-exp 97.7%

       \[\leadsto 0 + \color{blue}{\frac{0.5}{\sqrt{2}}} \]

    6. add-sqr-sqrt 97.7%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{0.5}{\sqrt{2}}} \cdot \sqrt{\frac{0.5}{\sqrt{2}}}} \]

    7. sqrt-unprod 97.7%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{0.5}{\sqrt{2}} \cdot \frac{0.5}{\sqrt{2}}}} \]

    8. frac-times 97.7%

       \[\leadsto 0 + \sqrt{\color{blue}{\frac{0.5 \cdot 0.5}{\sqrt{2} \cdot \sqrt{2}}}} \]

    9. metadata-eval 97.7%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{0.25}}{\sqrt{2} \cdot \sqrt{2}}} \]

    10. add-sqr-sqrt 99.2%

        \[\leadsto 0 + \sqrt{\frac{0.25}{\color{blue}{2}}} \]

    11. metadata-eval 99.2%

        \[\leadsto 0 + \sqrt{\color{blue}{0.125}} \]

13. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{0 + \sqrt{0.125}} \]
14. Step-by-step derivation

    1. +-lft-identity 99.2%

       \[\leadsto \color{blue}{\sqrt{0.125}} \]

15. Simplified 99.2%

    \[\leadsto \color{blue}{\sqrt{0.125}} \]

16. Final simplification 99.2%

    \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023214 
(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))))