Falkner and Boettcher, Appendix B, 1

Percentage Accurate: 99.1% → 99.1%

Time: 15.7s

Alternatives: 7

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: 99.1% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}}}\right)\right) \]

6. Applied egg-rr 99.2%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. frac-2neg N/A

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

   2. distribute-frac-neg N/A

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

   3. acos-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)} \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)}\right) \]

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

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

   7. /-lowering-/.f64 N/A

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

   8. sub-neg N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

   14. metadata-eval N/A

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

   15. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -5\right)\right), \left(0 - \left(v \cdot v - 1\right)\right)\right)\right)\right) \]

   16. associate--r- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -5\right)\right), \left(\left(0 - v \cdot v\right) + 1\right)\right)\right)\right) \]

   17. neg-sub0 N/A

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

   18. +-commutative N/A

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

   19. sub-neg N/A

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

   20. --lowering--.f64 N/A

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

   21. *-lowering-*.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 99.1% 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\frac{\pi \cdot \left(\pi \cdot \pi\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}{\frac{\pi \cdot \pi}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\pi}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}\right)}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{8} - {\sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)}^{3}}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right) \cdot \left(\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{1 + \left(v \cdot v\right) \cdot -5}{v \cdot v + -1}\right)\right)}}}\right)\right) \]

6. Applied egg-rr 99.2%

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

7. Taylor expanded in v around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

   9. *-commutative N/A

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

   10. *-lowering-*.f64 98.7%

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

9. Simplified 98.7%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. frac-2neg N/A

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

   2. distribute-frac-neg N/A

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

   3. acos-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) - \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)} \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{neg}\left(\left(v \cdot v - 1\right)\right)}\right)}\right) \]

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

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

   7. /-lowering-/.f64 N/A

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

   8. sub-neg N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

   14. metadata-eval N/A

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

   15. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -5\right)\right), \left(0 - \left(v \cdot v - 1\right)\right)\right)\right)\right) \]

   16. associate--r- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{acos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), -5\right)\right), \left(\left(0 - v \cdot v\right) + 1\right)\right)\right)\right) \]

   17. neg-sub0 N/A

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

   18. +-commutative N/A

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

   19. sub-neg N/A

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

   20. --lowering--.f64 N/A

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

   21. *-lowering-*.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 98.7%

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

7. Simplified 98.7%

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

Alternative 6: 98.5% 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 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 7: 97.8% 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 99.2%

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
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 98.1%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))