Falkner and Boettcher, Appendix B, 2

Percentage Accurate: 100.0% → 99.5%

Time: 4.0s

Alternatives: 4

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2}}{4} \cdot \left(1 + {v}^{2} \cdot -2.5\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (* (/ (sqrt 2.0) 4.0) (+ 1.0 (* (pow v 2.0) -2.5))))

C

double code(double v) {
	return (sqrt(2.0) / 4.0) * (1.0 + (pow(v, 2.0) * -2.5));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = (sqrt(2.0d0) / 4.0d0) * (1.0d0 + ((v ** 2.0d0) * (-2.5d0)))
end function

Java

public static double code(double v) {
	return (Math.sqrt(2.0) / 4.0) * (1.0 + (Math.pow(v, 2.0) * -2.5));
}

Python

def code(v):
	return (math.sqrt(2.0) / 4.0) * (1.0 + (math.pow(v, 2.0) * -2.5))

Julia

function code(v)
	return Float64(Float64(sqrt(2.0) / 4.0) * Float64(1.0 + Float64((v ^ 2.0) * -2.5)))
end

MATLAB

function tmp = code(v)
	tmp = (sqrt(2.0) / 4.0) * (1.0 + ((v ^ 2.0) * -2.5));
end

Wolfram

code[v_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[(1.0 + N[(N[Power[v, 2.0], $MachinePrecision] * -2.5), $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%

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

   2. associate-*r* 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 100.0%

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

   1. *-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-*r* 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

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

   1. associate-*l/ 99.6%

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

   2. associate-/l* 99.6%

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

   3. *-un-lft-identity 99.6%

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

   4. pow2 99.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{4}{1 - \color{blue}{{v}^{2}}}} \]

6. Applied egg-rr 99.6%

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

   1. add-log-exp 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. log-prod 99.6%

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

   4. metadata-eval 99.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{0} + \log \left(e^{\frac{4}{1 - {v}^{2}}}\right)} \]

   5. add-log-exp 99.6%

      \[\leadsto \frac{\sqrt{2}}{0 + \color{blue}{\frac{4}{1 - {v}^{2}}}} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{0 + \frac{4}{1 - {v}^{2}}}} \]
9. Step-by-step derivation

   1. +-lft-identity 99.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{4}{1 - {v}^{2}}}} \]

   2. metadata-eval 99.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{\frac{-4}{-1}}}{1 - {v}^{2}}} \]

   3. associate-/r* 99.6%

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

   4. neg-mul-1 99.6%

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

   5. sub-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. distribute-neg-in 99.6%

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

   8. remove-double-neg 99.6%

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

   9. sub-neg 99.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{-4}{\color{blue}{{v}^{2} - 1}}} \]

   10. unpow2 99.6%

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

   11. fma-neg 99.6%

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

   12. metadata-eval 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*l* 100.0%

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

   2. associate-*r* 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.0% 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-*l* 100.0%

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

   2. associate-*r* 100.0%

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

   3. cancel-sign-sub-inv 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in v around 0 99.6%

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

5. Taylor expanded in v around 0 99.6%

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

6. Final simplification 99.6%

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

Reproduce

herbie shell --seed 2023339 
(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))))