Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 5.5s

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v)
	tmp = ((sqrt(2.0) / 4.0) * sqrt((1.0 - ((1.0 + (3.0 * (v ^ 2.0))) + -1.0)))) * (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[(N[(1.0 + N[(3.0 * N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot v\right) \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

   2. expm1-udef 100.0%

      \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(3 \cdot v\right) \cdot v\right)} - 1\right)}}\right) \cdot \left(1 - v \cdot v\right) \]

   3. log1p-udef 100.0%

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

   4. add-exp-log 100.0%

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

   5. +-commutative 100.0%

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

   6. associate-*l* 100.0%

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

   7. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

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

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (0.25d0 * sqrt((2.0d0 + ((v ** 2.0d0) * (-6.0d0)))))
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (0.25 * Math.sqrt((2.0 + (Math.pow(v, 2.0) * -6.0))));
}

Python

def code(v):
	return (1.0 - (v * v)) * (0.25 * math.sqrt((2.0 + (math.pow(v, 2.0) * -6.0))))

Julia

function code(v)
	return Float64(Float64(1.0 - Float64(v * v)) * Float64(0.25 * sqrt(Float64(2.0 + Float64((v ^ 2.0) * -6.0)))))
end

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (0.25 * sqrt((2.0 + ((v ^ 2.0) * -6.0))));
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[Sqrt[N[(2.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -6.0), $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. Step-by-step derivation

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

   1. associate-*l/ 100.0%

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

   2. clear-num 100.0%

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

   3. sqrt-unprod 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-lft-in 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*l* 100.0%

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

   8. distribute-lft-neg-in 100.0%

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

   9. metadata-eval 100.0%

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

   10. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-/r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-*r* 100.0%

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

   4. *-commutative 100.0%

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

   5. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \left(1 - v \cdot v\right) \cdot \left(0.25 \cdot \sqrt{2 + {v}^{2} \cdot -6}\right) \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[v_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[v, 2.0], $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-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

   \[\leadsto \color{blue}{\left(-0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \sqrt{2}\right)} \cdot \left(1 - v \cdot v\right) \]
5. Step-by-step derivation

   1. +-commutative 99.6%

      \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{2} + -0.375 \cdot \left({v}^{2} \cdot \sqrt{2}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   2. associate-*r* 99.6%

      \[\leadsto \left(0.25 \cdot \sqrt{2} + \color{blue}{\left(-0.375 \cdot {v}^{2}\right) \cdot \sqrt{2}}\right) \cdot \left(1 - v \cdot v\right) \]

   3. distribute-rgt-out 99.6%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(0.25 + -0.375 \cdot {v}^{2}\right)\right)} \cdot \left(1 - v \cdot v\right) \]

   4. *-commutative 99.6%

      \[\leadsto \left(\sqrt{2} \cdot \left(0.25 + \color{blue}{{v}^{2} \cdot -0.375}\right)\right) \cdot \left(1 - v \cdot v\right) \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(0.25 + {v}^{2} \cdot -0.375\right)\right)} \cdot \left(1 - v \cdot v\right) \]

7. Taylor expanded in v around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. distribute-rgt-out 99.6%

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

   4. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (sqrt(2.0d0) * 0.25d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (Math.sqrt(2.0) * 0.25);
}

Python

def code(v):
	return (1.0 - (v * v)) * (math.sqrt(2.0) * 0.25)

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (sqrt(2.0) * 0.25);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $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-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot 0.25 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (* (sqrt 2.0) 0.25))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*r* 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 98.9%

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

5. Taylor expanded in v around 0 98.9%

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

6. Final simplification 98.9%

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

Reproduce

herbie shell --seed 2023301 
(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))))