Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.3% → 99.2%

Time: 16.7s

Alternatives: 5

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: 99.3% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(v \cdot v - 1\right), \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \left(1 + \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(5 \cdot \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\left(v \cdot v\right) \cdot 5\right)\right)\right)\right)\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \left(\left(v \cdot v\right) \cdot \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(5\right)\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval 99.1%

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), -1\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -5\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.1%

   \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{\frac{v \cdot v + -1}{1 + \left(v \cdot v\right) \cdot -5}}\right)} \]
5. Add Preprocessing

Alternative 2: 99.3% 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 99.1%

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

3. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) - 1\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{acos.f64}\left(\left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right) + -1\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(-1 + {v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left({v}^{2} \cdot \left(4 + 4 \cdot {v}^{2}\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(\left(v \cdot v\right) \cdot \left(4 + 4 \cdot {v}^{2}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(v \cdot \left(v \cdot \left(4 + 4 \cdot {v}^{2}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(v \cdot \left(\left(4 + 4 \cdot {v}^{2}\right) \cdot v\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \left(\left(4 + 4 \cdot {v}^{2}\right) \cdot v\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \left(v \cdot \left(4 + 4 \cdot {v}^{2}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(4 + 4 \cdot {v}^{2}\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(4 + {v}^{2} \cdot 4\right)\right)\right)\right)\right) \]

   12. distribute-rgt1-in N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(\left({v}^{2} + 1\right) \cdot 4\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \left(4 \cdot \left({v}^{2} + 1\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(4, \left({v}^{2} + 1\right)\right)\right)\right)\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(4, \left(1 + {v}^{2}\right)\right)\right)\right)\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(1, \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(1, \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(4, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.1%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{\left(4 \cdot {v}^{2} - 1\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(4 \cdot {v}^{2} + -1\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\left(-1 + 4 \cdot {v}^{2}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left(4 \cdot {v}^{2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \left({v}^{2} \cdot 4\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left({v}^{2}\right), 4\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\left(v \cdot v\right), 4\right)\right)\right) \]

   8. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{acos.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), 4\right)\right)\right) \]

5. Simplified 98.7%

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

Alternative 5: 98.0% 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 99.1%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{acos.f64}\left(\color{blue}{-1}\right) \]
4. Step-by-step derivation

5. Simplified 98.0%

   \[\leadsto \cos^{-1} \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))