Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.2% → 98.8%

Time: 18.5s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))

C

double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - (5.0d0 * (v * v))) / ((v * v) - 1.0d0)))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(5.0 * Float64(v * v))) / Float64(Float64(v * v) - 1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(5.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (v)
 :precision binary64
 (acos (+ (* 4.0 (+ (* v v) (pow v 4.0))) -1.0)))

C

double code(double v) {
	return acos(((4.0 * ((v * v) + pow(v, 4.0))) + -1.0));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((4.0d0 * ((v * v) + (v ** 4.0d0))) + (-1.0d0)))
end function

Java

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

Python

def code(v):
	return math.acos(((4.0 * ((v * v) + math.pow(v, 4.0))) + -1.0))

Julia

function code(v)
	return acos(Float64(Float64(4.0 * Float64(Float64(v * v) + (v ^ 4.0))) + -1.0))
end

MATLAB

function tmp = code(v)
	tmp = acos(((4.0 * ((v * v) + (v ^ 4.0))) + -1.0));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(4.0 * N[(N[(v * v), $MachinePrecision] + N[Power[v, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Taylor expanded in v around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. unpow2 99.1%

      \[\leadsto \cos^{-1} \left(\left(4 \cdot \color{blue}{\left(v \cdot v\right)} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right) \]

   3. distribute-lft-out 99.1%

      \[\leadsto \cos^{-1} \left(\color{blue}{4 \cdot \left(v \cdot v + {v}^{4}\right)} + \left(-1\right)\right) \]

   4. metadata-eval 99.1%

      \[\leadsto \cos^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + \color{blue}{-1}\right) \]

4. Simplified 99.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right)} \]

5. Final simplification 99.1%

   \[\leadsto \cos^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right) \]

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v + -1}\right) \end{array} \]

FPCore

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* (* v v) 5.0)) (+ (* v v) -1.0))))

C

double code(double v) {
	return acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((1.0d0 - ((v * v) * 5.0d0)) / ((v * v) + (-1.0d0))))
end function

Java

public static double code(double v) {
	return Math.acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)));
}

Python

def code(v):
	return math.acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)))

Julia

function code(v)
	return acos(Float64(Float64(1.0 - Float64(Float64(v * v) * 5.0)) / Float64(Float64(v * v) + -1.0)))
end

MATLAB

function tmp = code(v)
	tmp = acos(((1.0 - ((v * v) * 5.0)) / ((v * v) + -1.0)));
end

Wolfram

code[v_] := N[ArcCos[N[(N[(1.0 - N[(N[(v * v), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] / N[(N[(v * v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Final simplification 99.0%

   \[\leadsto \cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v + -1}\right) \]

Alternative 3: 98.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (v) :precision binary64 (acos (+ -1.0 (* v (* 4.0 v)))))

C

double code(double v) {
	return acos((-1.0 + (v * (4.0 * v))));
}

Fortran

real(8) function code(v)
    real(8), intent (in) :: v
    code = acos(((-1.0d0) + (v * (4.0d0 * v))))
end function

Java

public static double code(double v) {
	return Math.acos((-1.0 + (v * (4.0 * v))));
}

Python

def code(v):
	return math.acos((-1.0 + (v * (4.0 * v))))

Julia

function code(v)
	return acos(Float64(-1.0 + Float64(v * Float64(4.0 * v))))
end

MATLAB

function tmp = code(v)
	tmp = acos((-1.0 + (v * (4.0 * v))));
end

Wolfram

code[v_] := N[ArcCos[N[(-1.0 + N[(v * N[(4.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Taylor expanded in v around 0 99.1%

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

   1. sub-neg 99.1%

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

   2. unpow2 99.1%

      \[\leadsto \cos^{-1} \left(\left(4 \cdot \color{blue}{\left(v \cdot v\right)} + 4 \cdot {v}^{4}\right) + \left(-1\right)\right) \]

   3. distribute-lft-out 99.1%

      \[\leadsto \cos^{-1} \left(\color{blue}{4 \cdot \left(v \cdot v + {v}^{4}\right)} + \left(-1\right)\right) \]

   4. metadata-eval 99.1%

      \[\leadsto \cos^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) + \color{blue}{-1}\right) \]

4. Simplified 99.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) + -1\right)} \]

5. Taylor expanded in v around 0 99.0%

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

   1. unpow2 99.0%

      \[\leadsto \cos^{-1} \left(4 \cdot \color{blue}{\left(v \cdot v\right)} + -1\right) \]

   2. *-commutative 99.0%

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(v \cdot v\right) \cdot 4} + -1\right) \]

   3. associate-*l* 99.0%

      \[\leadsto \cos^{-1} \left(\color{blue}{v \cdot \left(v \cdot 4\right)} + -1\right) \]

   4. *-commutative 99.0%

      \[\leadsto \cos^{-1} \left(v \cdot \color{blue}{\left(4 \cdot v\right)} + -1\right) \]

7. Simplified 99.0%

   \[\leadsto \cos^{-1} \left(\color{blue}{v \cdot \left(4 \cdot v\right)} + -1\right) \]

8. Final simplification 99.0%

   \[\leadsto \cos^{-1} \left(-1 + v \cdot \left(4 \cdot v\right)\right) \]

Alternative 4: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} -1 \end{array} \]

FPCore

(FPCore (v) :precision binary64 (acos -1.0))

C

double code(double v) {
	return acos(-1.0);
}

Fortran

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

Java

public static double code(double v) {
	return Math.acos(-1.0);
}

Python

def code(v):
	return math.acos(-1.0)

Julia

function code(v)
	return acos(-1.0)
end

MATLAB

function tmp = code(v)
	tmp = acos(-1.0);
end

Wolfram

code[v_] := N[ArcCos[-1.0], $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]

2. Taylor expanded in v around 0 98.2%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]

3. Final simplification 98.2%

   \[\leadsto \cos^{-1} -1 \]

Reproduce

herbie shell --seed 2023223 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))