Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 100.0%

Time: 11.7s

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (*
  (/
   (pow (pow (* 2.0 (- 2.0 (fma 3.0 (pow v 2.0) 1.0))) 1.5) 0.3333333333333333)
   4.0)
  (- 1.0 (* v v))))

C

double code(double v) {
	return (pow(pow((2.0 * (2.0 - fma(3.0, pow(v, 2.0), 1.0))), 1.5), 0.3333333333333333) / 4.0) * (1.0 - (v * v));
}

Julia

function code(v)
	return Float64(Float64(((Float64(2.0 * Float64(2.0 - fma(3.0, (v ^ 2.0), 1.0))) ^ 1.5) ^ 0.3333333333333333) / 4.0) * Float64(1.0 - Float64(v * v)))
end

Wolfram

code[v_] := N[(N[(N[Power[N[Power[N[(2.0 * N[(2.0 - N[(3.0 * N[Power[v, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. log1p-undefine 99.9%

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

   4. add-exp-log 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

   7. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. sqrt-unprod 100.0%

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

   3. associate--r- 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. metadata-eval 100.0%

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

8. Simplified 100.0%

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

   1. add-cbrt-cube 98.4%

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

   2. pow1/3 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\left(\sqrt{2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)} \cdot \sqrt{2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)}\right) \cdot \sqrt{2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)}\right)}^{0.3333333333333333}}}{4} \cdot \left(1 - v \cdot v\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(\color{blue}{\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)} \cdot \sqrt{2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)}\right)}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{{\left(\color{blue}{{\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)}\right)}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{{\left({\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{0.5}}\right)}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{{\color{blue}{\left({\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left({\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{4} \cdot \left(1 - v \cdot v\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left({\left(2 \cdot \left(2 - \mathsf{fma}\left(3, {v}^{2}, 1\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{4} \cdot \left(1 - v \cdot v\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. log1p-undefine 99.9%

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

   4. add-exp-log 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

   7. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. sqrt-unprod 100.0%

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

   3. associate--r- 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. metadata-eval 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (1.0d0 - (v * v)) * (sqrt((2.0d0 + ((v ** 2.0d0) * (-6.0d0)))) / 4.0d0)
end function

Java

public static double code(double v) {
	return (1.0 - (v * v)) * (Math.sqrt((2.0 + (Math.pow(v, 2.0) * -6.0))) / 4.0);
}

Python

def code(v):
	return (1.0 - (v * v)) * (math.sqrt((2.0 + (math.pow(v, 2.0) * -6.0))) / 4.0)

Julia

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

MATLAB

function tmp = code(v)
	tmp = (1.0 - (v * v)) * (sqrt((2.0 + ((v ^ 2.0) * -6.0))) / 4.0);
end

Wolfram

code[v_] := N[(N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 99.9%

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

   3. log1p-undefine 99.9%

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

   4. add-exp-log 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

   7. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

   2. sqrt-unprod 100.0%

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

   3. associate--r- 100.0%

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

6. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. metadata-eval 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in v around 0 100.0%

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

    1. *-un-lft-identity 100.0%

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

    2. distribute-lft-in 100.0%

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

    3. metadata-eval 100.0%

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

    4. associate-*r* 100.0%

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

    5. metadata-eval 100.0%

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

11. Applied egg-rr 100.0%

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

    1. *-lft-identity 100.0%

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

    2. *-commutative 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

    \[\leadsto \left(1 - v \cdot v\right) \cdot \frac{\sqrt{2 + {v}^{2} \cdot -6}}{4} \]
15. Add Preprocessing

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 99.9%

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

3. Taylor expanded in v around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. associate-*r* 98.9%

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

   3. distribute-rgt-out 98.9%

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

   4. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in v around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. associate-*r* 99.0%

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

   3. distribute-rgt-out 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

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 99.9%

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

3. Taylor expanded in v around 0 97.9%

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

4. Final simplification 97.9%

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

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 99.9%

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

3. Taylor expanded in v around 0 97.9%

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

   1. add-sqr-sqrt 96.4%

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

   2. pow2 96.4%

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

   3. *-commutative 96.4%

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

   4. metadata-eval 96.4%

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

   5. div-inv 96.4%

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

   6. add-sqr-sqrt 96.4%

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

   7. sqrt-unprod 96.4%

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

   8. frac-times 96.4%

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

   9. rem-square-sqrt 96.4%

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

   10. metadata-eval 96.4%

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

   11. metadata-eval 96.4%

       \[\leadsto {\left(\sqrt{\sqrt{\color{blue}{0.125}} \cdot \left(1 - v \cdot v\right)}\right)}^{2} \]

   12. pow2 96.4%

       \[\leadsto {\left(\sqrt{\sqrt{0.125} \cdot \left(1 - \color{blue}{{v}^{2}}\right)}\right)}^{2} \]

5. Applied egg-rr 96.4%

   \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.125} \cdot \left(1 - {v}^{2}\right)}\right)}^{2}} \]

6. Taylor expanded in v around 0 97.8%

   \[\leadsto \color{blue}{\sqrt{0.125}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))))