Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * 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]), $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. Final simplification 100.0%

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

Alternative 3: 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 99.9%

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

   2. un-div-inv 99.9%

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

   3. frac-2neg 99.9%

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

   4. metadata-eval 99.9%

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

   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 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(v * N[(v * -0.625), $MachinePrecision]), $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. 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 99.4%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + -2.5 \cdot {v}^{2}\right)} \]
5. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{v}^{2} \cdot -2.5}\right) \]

   2. unpow2 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot -2.5\right) \]

6. Simplified 99.4%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(1 + \left(v \cdot v\right) \cdot -2.5\right)} \]
7. Step-by-step derivation

   1. pow1 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{\left(\left(v \cdot v\right) \cdot -2.5\right)}^{1}}\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + {\color{blue}{\left(v \cdot \left(v \cdot -2.5\right)\right)}}^{1}\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{{\left(v \cdot \left(v \cdot -2.5\right)\right)}^{1}}\right) \]
9. Step-by-step derivation

   1. pow1 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \left(1 + \color{blue}{v \cdot \left(v \cdot -2.5\right)}\right) \]

   2. +-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\left(v \cdot \left(v \cdot -2.5\right) + 1\right)} \]

   3. fma-udef 99.4%

      \[\leadsto \frac{\sqrt{2}}{4} \cdot \color{blue}{\mathsf{fma}\left(v, v \cdot -2.5, 1\right)} \]

   4. associate-/r/ 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{4}{\mathsf{fma}\left(v, v \cdot -2.5, 1\right)}}} \]

   5. fma-udef 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{4}{\color{blue}{v \cdot \left(v \cdot -2.5\right) + 1}}} \]

   6. associate-*r* 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{4}{\color{blue}{\left(v \cdot v\right) \cdot -2.5} + 1}} \]

   7. *-commutative 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{4}{\color{blue}{-2.5 \cdot \left(v \cdot v\right)} + 1}} \]

   8. fma-def 99.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{4}{\color{blue}{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}}} \]

10. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{4}{\mathsf{fma}\left(-2.5, v \cdot v, 1\right)}}} \]
11. Step-by-step derivation

    1. associate-/l* 99.4%

       \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{4}} \]

    2. metadata-eval 99.4%

       \[\leadsto \frac{\sqrt{2} \cdot \mathsf{fma}\left(-2.5, v \cdot v, 1\right)}{\color{blue}{\frac{1}{0.25}}} \]

    3. associate-/l* 99.4%

       \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot \mathsf{fma}\left(-2.5, v \cdot v, 1\right)\right) \cdot 0.25}{1}} \]

    4. associate-*r* 99.4%

       \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \left(\mathsf{fma}\left(-2.5, v \cdot v, 1\right) \cdot 0.25\right)}}{1} \]

    5. /-rgt-identity 99.4%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\mathsf{fma}\left(-2.5, v \cdot v, 1\right) \cdot 0.25\right)} \]

    6. *-commutative 99.4%

       \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0.25 \cdot \mathsf{fma}\left(-2.5, v \cdot v, 1\right)\right)} \]

    7. fma-udef 99.4%

       \[\leadsto \sqrt{2} \cdot \left(0.25 \cdot \color{blue}{\left(-2.5 \cdot \left(v \cdot v\right) + 1\right)}\right) \]

    8. distribute-lft-in 99.4%

       \[\leadsto \sqrt{2} \cdot \color{blue}{\left(0.25 \cdot \left(-2.5 \cdot \left(v \cdot v\right)\right) + 0.25 \cdot 1\right)} \]

    9. associate-*r* 99.4%

       \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(0.25 \cdot -2.5\right) \cdot \left(v \cdot v\right)} + 0.25 \cdot 1\right) \]

    10. metadata-eval 99.4%

        \[\leadsto \sqrt{2} \cdot \left(\color{blue}{-0.625} \cdot \left(v \cdot v\right) + 0.25 \cdot 1\right) \]

    11. associate-*r* 99.4%

        \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(-0.625 \cdot v\right) \cdot v} + 0.25 \cdot 1\right) \]

    12. metadata-eval 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - v \cdot v\right) \cdot \sqrt{0.125} \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (- 1.0 (* v v)) (sqrt 0.125)))

C

double code(double v) {
	return (1.0 - (v * v)) * sqrt(0.125);
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * sqrt(0.125d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * Math.sqrt(0.125);
}

Python

def code(v):
	return (1.0 - (v * v)) * math.sqrt(0.125)

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * sqrt(0.125))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * sqrt(0.125);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.125], $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 99.9%

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

   2. un-div-inv 99.9%

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

   3. frac-2neg 99.9%

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

   4. metadata-eval 99.9%

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

   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. Taylor expanded in v around 0 98.3%

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

   1. associate-/r/ 98.3%

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

   2. *-commutative 98.3%

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

   3. add-sqr-sqrt 96.8%

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

   4. sqrt-unprod 98.3%

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

   5. frac-times 98.3%

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

   6. add-sqr-sqrt 98.3%

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

   7. metadata-eval 98.3%

      \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\frac{2}{\color{blue}{16}}} \]

   8. metadata-eval 98.3%

      \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{0.125}} \]

10. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\left(1 - v \cdot v\right) \cdot \sqrt{0.125}} \]

11. Final simplification 98.3%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]

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. Step-by-step derivation

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

   3. frac-2neg 99.9%

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

   4. metadata-eval 99.9%

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

   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. Taylor expanded in v around 0 98.3%

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

   1. associate-/r/ 98.3%

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

   2. *-commutative 98.3%

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

   3. add-sqr-sqrt 96.8%

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

   4. sqrt-unprod 98.3%

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

   5. frac-times 98.3%

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

   6. add-sqr-sqrt 98.3%

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

   7. metadata-eval 98.3%

      \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\frac{2}{\color{blue}{16}}} \]

   8. metadata-eval 98.3%

      \[\leadsto \left(1 - v \cdot v\right) \cdot \sqrt{\color{blue}{0.125}} \]

10. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\left(1 - v \cdot v\right) \cdot \sqrt{0.125}} \]

11. Taylor expanded in v around 0 98.3%

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

12. Final simplification 98.3%

    \[\leadsto \sqrt{0.125} \]

Reproduce

herbie shell --seed 2023189 
(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))))