Optimal throwing angle

Percentage Accurate: 66.6% → 99.5%

Time: 10.5s

Alternatives: 11

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))

C

double code(double v, double H) {
	return atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((v / sqrt(((v * v) - ((2.0d0 * 9.8d0) * h)))))
end function

Java

public static double code(double v, double H) {
	return Math.atan((v / Math.sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Python

def code(v, H):
	return math.atan((v / math.sqrt(((v * v) - ((2.0 * 9.8) * H)))))

Julia

function code(v, H)
	return atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
end

MATLAB

function tmp = code(v, H)
	tmp = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
end

Wolfram

code[v_, H_] := N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))

C

double code(double v, double H) {
	return atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((v / sqrt(((v * v) - ((2.0d0 * 9.8d0) * h)))))
end function

Java

public static double code(double v, double H) {
	return Math.atan((v / Math.sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Python

def code(v, H):
	return math.atan((v / math.sqrt(((v * v) - ((2.0 * 9.8) * H)))))

Julia

function code(v, H)
	return atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
end

MATLAB

function tmp = code(v, H)
	tmp = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
end

Wolfram

code[v_, H_] := N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+114}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4e+114)
   (- 0.0 (atan (/ (/ v (+ 1.0 (/ -9.8 (/ (* v v) H)))) v)))
   (if (<= v 2e+112) (atan (/ v (sqrt (+ (* v v) (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4e+114) {
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 2e+112) {
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-4d+114)) then
        tmp = 0.0d0 - atan(((v / (1.0d0 + ((-9.8d0) / ((v * v) / h)))) / v))
    else if (v <= 2d+112) then
        tmp = atan((v / sqrt(((v * v) + (h * (-19.6d0))))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -4e+114) {
		tmp = 0.0 - Math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 2e+112) {
		tmp = Math.atan((v / Math.sqrt(((v * v) + (H * -19.6)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4e+114:
		tmp = 0.0 - math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v))
	elif v <= 2e+112:
		tmp = math.atan((v / math.sqrt(((v * v) + (H * -19.6)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4e+114)
		tmp = Float64(0.0 - atan(Float64(Float64(v / Float64(1.0 + Float64(-9.8 / Float64(Float64(v * v) / H)))) / v)));
	elseif (v <= 2e+112)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) + Float64(H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4e+114)
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	elseif (v <= 2e+112)
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4e+114], N[(0.0 - N[ArcTan[N[(N[(v / N[(1.0 + N[(-9.8 / N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2e+112], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4e114

   1. Initial program 21.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 21.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 98.4%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}\right) \]

      2. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{\mathsf{neg}\left(v\right)}\right) \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{neg}\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

      4. atan-neg N/A

         \[\leadsto \mathsf{neg}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right)\right)\right), v\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{1}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{\frac{-49}{5}}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \left(\frac{v \cdot v}{H}\right)\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\left(v \cdot v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

      15. *-lowering-*.f64 98.4%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

   9. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)} \]

   if -4e114 < v < 1.9999999999999999e112

   1. Initial program 99.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   if 1.9999999999999999e112 < v

   1. Initial program 17.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 17.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 17.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

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

   7. Simplified 100.0%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+114}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.8e-16)
   (- 0.0 (atan (/ (/ v (+ 1.0 (/ -9.8 (/ (* v v) H)))) v)))
   (if (<= v 6.8e-29)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (+ 1.0 (* H (/ (+ 9.8 (* (/ H (* v v)) 144.06)) (* v v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.8e-16) {
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 6.8e-29) {
		tmp = atan((v / sqrt((H * -19.6))));
	} else {
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-3.8d-16)) then
        tmp = 0.0d0 - atan(((v / (1.0d0 + ((-9.8d0) / ((v * v) / h)))) / v))
    else if (v <= 6.8d-29) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    else
        tmp = atan((1.0d0 + (h * ((9.8d0 + ((h / (v * v)) * 144.06d0)) / (v * v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.8e-16) {
		tmp = 0.0 - Math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 6.8e-29) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.8e-16:
		tmp = 0.0 - math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v))
	elif v <= 6.8e-29:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.8e-16)
		tmp = Float64(0.0 - atan(Float64(Float64(v / Float64(1.0 + Float64(-9.8 / Float64(Float64(v * v) / H)))) / v)));
	elseif (v <= 6.8e-29)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	else
		tmp = atan(Float64(1.0 + Float64(H * Float64(Float64(9.8 + Float64(Float64(H / Float64(v * v)) * 144.06)) / Float64(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.8e-16)
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	elseif (v <= 6.8e-29)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.8e-16], N[(0.0 - N[ArcTan[N[(N[(v / N[(1.0 + N[(-9.8 / N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 6.8e-29], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 + N[(H * N[(N[(9.8 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * 144.06), $MachinePrecision]), $MachinePrecision] / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.80000000000000012e-16

   1. Initial program 49.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 49.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 49.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 93.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 93.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}\right) \]

      2. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{\mathsf{neg}\left(v\right)}\right) \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{neg}\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

      4. atan-neg N/A

         \[\leadsto \mathsf{neg}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right)\right)\right), v\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{1}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{\frac{-49}{5}}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \left(\frac{v \cdot v}{H}\right)\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\left(v \cdot v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

      15. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

   9. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)} \]

   if -3.80000000000000012e-16 < v < 6.79999999999999945e-29

   1. Initial program 99.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-98}{5} \cdot H\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 88.7%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right)\right)\right)\right) \]

   7. Simplified 88.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{-19.6 \cdot H}}}\right) \]

   if 6.79999999999999945e-29 < v

   1. Initial program 52.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 52.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(1 + H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \left(H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{7203}{50} \cdot \frac{H}{{v}^{4}}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{49}{5} \cdot \frac{1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5} \cdot 1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5}}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left({v}^{2}\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left(v \cdot v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(\frac{7203}{50} \cdot H\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(H \cdot \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right) \]

      21. pow-sqr N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{2} \cdot {v}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left({v}^{2}\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left(v \cdot v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      25. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 91.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(1 + H \cdot \left(\frac{9.8}{v \cdot v} + \frac{H \cdot 144.06}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \color{blue}{\left(\frac{\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\left(\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{H}{{v}^{2}} \cdot \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left(v \cdot v\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right) \]

   10. Simplified 91.2%

       \[\leadsto \tan^{-1} \left(1 + H \cdot \color{blue}{\frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.6 \cdot 10^{-16}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 3.15 \cdot 10^{-28}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.6e-16)
   (- 0.0 (atan (/ (/ v (+ 1.0 (/ -9.8 (/ (* v v) H)))) v)))
   (if (<= v 3.15e-28)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (+ 1.0 (* H (/ (+ 9.8 (* (/ H (* v v)) 144.06)) (* v v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.6e-16) {
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 3.15e-28) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.6d-16)) then
        tmp = 0.0d0 - atan(((v / (1.0d0 + ((-9.8d0) / ((v * v) / h)))) / v))
    else if (v <= 3.15d-28) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((1.0d0 + (h * ((9.8d0 + ((h / (v * v)) * 144.06d0)) / (v * v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.6e-16) {
		tmp = 0.0 - Math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else if (v <= 3.15e-28) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.6e-16:
		tmp = 0.0 - math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v))
	elif v <= 3.15e-28:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.6e-16)
		tmp = Float64(0.0 - atan(Float64(Float64(v / Float64(1.0 + Float64(-9.8 / Float64(Float64(v * v) / H)))) / v)));
	elseif (v <= 3.15e-28)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(1.0 + Float64(H * Float64(Float64(9.8 + Float64(Float64(H / Float64(v * v)) * 144.06)) / Float64(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.6e-16)
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	elseif (v <= 3.15e-28)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.6e-16], N[(0.0 - N[ArcTan[N[(N[(v / N[(1.0 + N[(-9.8 / N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3.15e-28], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 + N[(H * N[(N[(9.8 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * 144.06), $MachinePrecision]), $MachinePrecision] / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.60000000000000011e-16

   1. Initial program 49.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 49.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 49.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 93.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 93.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}\right) \]

      2. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{\mathsf{neg}\left(v\right)}\right) \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{neg}\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

      4. atan-neg N/A

         \[\leadsto \mathsf{neg}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right)\right)\right), v\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{1}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{\frac{-49}{5}}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \left(\frac{v \cdot v}{H}\right)\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\left(v \cdot v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

      15. *-lowering-*.f64 93.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

   9. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)} \]

   if -1.60000000000000011e-16 < v < 3.1499999999999999e-28

   1. Initial program 99.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{-98}{5} \cdot H + {v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{-98}{5} \cdot H\right), \left({v}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right), \left({v}^{2}\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right), \left(v \cdot v\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 99.5%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right), \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{-19.6 \cdot H + v \cdot v}}\right)} \]

   8. Taylor expanded in H around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{\frac{-5}{98}}{H}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 88.6%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\frac{-5}{98}, H\right)\right)\right)\right) \]

   10. Simplified 88.6%

       \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\color{blue}{\frac{-0.05102040816326531}{H}}}\right) \]

   if 3.1499999999999999e-28 < v

   1. Initial program 52.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 52.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 52.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(1 + H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \left(H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{7203}{50} \cdot \frac{H}{{v}^{4}}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{49}{5} \cdot \frac{1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5} \cdot 1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5}}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left({v}^{2}\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left(v \cdot v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(\frac{7203}{50} \cdot H\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(H \cdot \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right) \]

      21. pow-sqr N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{2} \cdot {v}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left({v}^{2}\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left(v \cdot v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      25. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 91.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(1 + H \cdot \left(\frac{9.8}{v \cdot v} + \frac{H \cdot 144.06}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \color{blue}{\left(\frac{\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\left(\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{H}{{v}^{2}} \cdot \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left(v \cdot v\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right) \]

   10. Simplified 91.2%

       \[\leadsto \tan^{-1} \left(1 + H \cdot \color{blue}{\frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.6 \cdot 10^{-16}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{elif}\;v \leq 3.15 \cdot 10^{-28}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 7.5 \cdot 10^{-187}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - \frac{-9.8 \cdot \frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 7.5e-187)
   (atan (/ v (* v (- -1.0 (/ (* -9.8 (/ H v)) v)))))
   (atan (+ 1.0 (* H (/ (+ 9.8 (* (/ H (* v v)) 144.06)) (* v v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= 7.5e-187) {
		tmp = atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	} else {
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= 7.5d-187) then
        tmp = atan((v / (v * ((-1.0d0) - (((-9.8d0) * (h / v)) / v)))))
    else
        tmp = atan((1.0d0 + (h * ((9.8d0 + ((h / (v * v)) * 144.06d0)) / (v * v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 7.5e-187) {
		tmp = Math.atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	} else {
		tmp = Math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 7.5e-187:
		tmp = math.atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))))
	else:
		tmp = math.atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 7.5e-187)
		tmp = atan(Float64(v / Float64(v * Float64(-1.0 - Float64(Float64(-9.8 * Float64(H / v)) / v)))));
	else
		tmp = atan(Float64(1.0 + Float64(H * Float64(Float64(9.8 + Float64(Float64(H / Float64(v * v)) * 144.06)) / Float64(v * v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= 7.5e-187)
		tmp = atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	else
		tmp = atan((1.0 + (H * ((9.8 + ((H / (v * v)) * 144.06)) / (v * v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 7.5e-187], N[ArcTan[N[(v / N[(v * N[(-1.0 - N[(N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(1.0 + N[(H * N[(N[(9.8 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * 144.06), $MachinePrecision]), $MachinePrecision] / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 7.5000000000000004e-187

   1. Initial program 69.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 69.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 69.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 66.0%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 66.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{H}{v}}{v} \cdot \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{H}{v} \cdot \frac{-49}{5}}{v}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{H}{v} \cdot \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{H}{v}\right), \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      5. /-lowering-/.f64 66.0%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(H, v\right), \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   9. Applied egg-rr 66.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \color{blue}{\frac{\frac{H}{v} \cdot -9.8}{v}}\right) \cdot \left(0 - v\right)}\right) \]

   if 7.5000000000000004e-187 < v

   1. Initial program 61.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 61.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(1 + H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \left(H \cdot \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{4}} + \frac{49}{5} \cdot \frac{1}{{v}^{2}}\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{7203}{50} \cdot \frac{H}{{v}^{4}}\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \left(\frac{49}{5} \cdot \frac{1}{{v}^{2}} + \left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{49}{5} \cdot \frac{1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5} \cdot 1}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\left(\frac{\frac{49}{5}}{{v}^{2}}\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left({v}^{2}\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \left(v \cdot v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\left(\frac{7203}{50} \cdot \frac{1}{{v}^{4}}\right) \cdot H\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot 1}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50}}{{v}^{4}} \cdot H\right)\right)\right)\right)\right) \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \left(\frac{\frac{7203}{50} \cdot H}{{v}^{4}}\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(\frac{7203}{50} \cdot H\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\left(H \cdot \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{4}\right)\right)\right)\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right) \]

      21. pow-sqr N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \left({v}^{2} \cdot {v}^{2}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left({v}^{2}\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\left(v \cdot v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left({v}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      25. unpow2 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(v, v\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{7203}{50}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(v \cdot v\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 75.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(1 + H \cdot \left(\frac{9.8}{v \cdot v} + \frac{H \cdot 144.06}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \color{blue}{\left(\frac{\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}}{{v}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\left(\frac{49}{5} + \frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{7203}{50} \cdot \frac{H}{{v}^{2}}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \left(\frac{H}{{v}^{2}} \cdot \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left({v}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \left(v \cdot v\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{7203}{50}\right)\right), \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right) \]

   10. Simplified 77.9%

       \[\leadsto \tan^{-1} \left(1 + H \cdot \color{blue}{\frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 7.5 \cdot 10^{-187}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - \frac{-9.8 \cdot \frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(1 + H \cdot \frac{9.8 + \frac{H}{v \cdot v} \cdot 144.06}{v \cdot v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-277}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.5e-277)
   (- 0.0 (atan (/ (/ v (+ 1.0 (/ -9.8 (/ (* v v) H)))) v)))
   (atan (/ v (+ v (/ (* -9.8 H) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-277) {
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else {
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-9.5d-277)) then
        tmp = 0.0d0 - atan(((v / (1.0d0 + ((-9.8d0) / ((v * v) / h)))) / v))
    else
        tmp = atan((v / (v + (((-9.8d0) * h) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-277) {
		tmp = 0.0 - Math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	} else {
		tmp = Math.atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.5e-277:
		tmp = 0.0 - math.atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v))
	else:
		tmp = math.atan((v / (v + ((-9.8 * H) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.5e-277)
		tmp = Float64(0.0 - atan(Float64(Float64(v / Float64(1.0 + Float64(-9.8 / Float64(Float64(v * v) / H)))) / v)));
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(-9.8 * H) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.5e-277)
		tmp = 0.0 - atan(((v / (1.0 + (-9.8 / ((v * v) / H)))) / v));
	else
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.5e-277], N[(0.0 - N[ArcTan[N[(N[(v / N[(1.0 + N[(-9.8 / N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(-9.8 * H), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.5e-277

   1. Initial program 65.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 65.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 65.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 72.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 72.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot \left(\mathsf{neg}\left(v\right)\right)}\right) \]

      2. associate-/r* N/A

         \[\leadsto \tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{\mathsf{neg}\left(v\right)}\right) \]

      3. distribute-frac-neg2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{neg}\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

      4. atan-neg N/A

         \[\leadsto \mathsf{neg}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\tan^{-1} \left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\left(\frac{\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}}{v}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\frac{v}{1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}}\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \left(1 + \frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right), v\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right)\right)\right), v\right)\right)\right) \]

      11. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{1}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

      12. un-div-inv N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{\frac{-49}{5}}{\frac{v \cdot v}{H}}\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \left(\frac{v \cdot v}{H}\right)\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\left(v \cdot v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

      15. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-49}{5}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right)\right)\right)\right), v\right)\right)\right) \]

   9. Applied egg-rr 72.7%

      \[\leadsto \color{blue}{-\tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)} \]

   if -9.5e-277 < v

   1. Initial program 67.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 67.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 67.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{\frac{-49}{5} \cdot H}{v}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(\frac{-49}{5} \cdot H\right), v\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(H \cdot \frac{-49}{5}\right), v\right)\right)\right)\right) \]

      17. *-lowering-*.f64 68.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), v\right)\right)\right)\right) \]

   7. Simplified 68.8%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{H \cdot -9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-277}:\\ \;\;\;\;0 - \tan^{-1} \left(\frac{\frac{v}{1 + \frac{-9.8}{\frac{v \cdot v}{H}}}}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - \frac{-9.8 \cdot \frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-280)
   (atan (/ v (* v (- -1.0 (/ (* -9.8 (/ H v)) v)))))
   (atan (/ v (+ v (/ (* -9.8 H) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-280) {
		tmp = atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	} else {
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d-280)) then
        tmp = atan((v / (v * ((-1.0d0) - (((-9.8d0) * (h / v)) / v)))))
    else
        tmp = atan((v / (v + (((-9.8d0) * h) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e-280) {
		tmp = Math.atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	} else {
		tmp = Math.atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-280:
		tmp = math.atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))))
	else:
		tmp = math.atan((v / (v + ((-9.8 * H) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-280)
		tmp = atan(Float64(v / Float64(v * Float64(-1.0 - Float64(Float64(-9.8 * Float64(H / v)) / v)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(-9.8 * H) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e-280)
		tmp = atan((v / (v * (-1.0 - ((-9.8 * (H / v)) / v)))));
	else
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-280], N[ArcTan[N[(v / N[(v * N[(-1.0 - N[(N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(-9.8 * H), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.9999999999999999e-280

   1. Initial program 66.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 72.4%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{H}{v}}{v} \cdot \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{H}{v} \cdot \frac{-49}{5}}{v}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{H}{v} \cdot \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{H}{v}\right), \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      5. /-lowering-/.f64 72.5%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(H, v\right), \frac{-49}{5}\right), v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   9. Applied egg-rr 72.5%

      \[\leadsto \tan^{-1} \left(\frac{v}{\left(1 + \color{blue}{\frac{\frac{H}{v} \cdot -9.8}{v}}\right) \cdot \left(0 - v\right)}\right) \]

   if -1.9999999999999999e-280 < v

   1. Initial program 66.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{\frac{-49}{5} \cdot H}{v}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(\frac{-49}{5} \cdot H\right), v\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(H \cdot \frac{-49}{5}\right), v\right)\right)\right)\right) \]

      17. *-lowering-*.f64 69.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), v\right)\right)\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{H \cdot -9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - \frac{-9.8 \cdot \frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - -9.8 \cdot \frac{H}{v \cdot v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-280)
   (atan (/ v (* v (- -1.0 (* -9.8 (/ H (* v v)))))))
   (atan (/ v (+ v (/ (* -9.8 H) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-280) {
		tmp = atan((v / (v * (-1.0 - (-9.8 * (H / (v * v)))))));
	} else {
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d-280)) then
        tmp = atan((v / (v * ((-1.0d0) - ((-9.8d0) * (h / (v * v)))))))
    else
        tmp = atan((v / (v + (((-9.8d0) * h) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e-280) {
		tmp = Math.atan((v / (v * (-1.0 - (-9.8 * (H / (v * v)))))));
	} else {
		tmp = Math.atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-280:
		tmp = math.atan((v / (v * (-1.0 - (-9.8 * (H / (v * v)))))))
	else:
		tmp = math.atan((v / (v + ((-9.8 * H) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-280)
		tmp = atan(Float64(v / Float64(v * Float64(-1.0 - Float64(-9.8 * Float64(H / Float64(v * v)))))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(-9.8 * H) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e-280)
		tmp = atan((v / (v * (-1.0 - (-9.8 * (H / (v * v)))))));
	else
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-280], N[ArcTan[N[(v / N[(v * N[(-1.0 - N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(-9.8 * H), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.9999999999999999e-280

   1. Initial program 66.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 72.4%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 72.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]

   if -1.9999999999999999e-280 < v

   1. Initial program 66.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{\frac{-49}{5} \cdot H}{v}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(\frac{-49}{5} \cdot H\right), v\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(H \cdot \frac{-49}{5}\right), v\right)\right)\right)\right) \]

      17. *-lowering-*.f64 69.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), v\right)\right)\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{H \cdot -9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - -9.8 \cdot \frac{H}{v \cdot v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.2 \cdot 10^{-94}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-187}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(\frac{v}{H} \cdot 0.10204081632653061\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.2e-94)
   (atan -1.0)
   (if (<= v 4e-187) (atan (* v (* (/ v H) 0.10204081632653061))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.2e-94) {
		tmp = atan(-1.0);
	} else if (v <= 4e-187) {
		tmp = atan((v * ((v / H) * 0.10204081632653061)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5.2d-94)) then
        tmp = atan((-1.0d0))
    else if (v <= 4d-187) then
        tmp = atan((v * ((v / h) * 0.10204081632653061d0)))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5.2e-94) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4e-187) {
		tmp = Math.atan((v * ((v / H) * 0.10204081632653061)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.2e-94:
		tmp = math.atan(-1.0)
	elif v <= 4e-187:
		tmp = math.atan((v * ((v / H) * 0.10204081632653061)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.2e-94)
		tmp = atan(-1.0);
	elseif (v <= 4e-187)
		tmp = atan(Float64(v * Float64(Float64(v / H) * 0.10204081632653061)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5.2e-94)
		tmp = atan(-1.0);
	elseif (v <= 4e-187)
		tmp = atan((v * ((v / H) * 0.10204081632653061)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.2e-94], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4e-187], N[ArcTan[N[(v * N[(N[(v / H), $MachinePrecision] * 0.10204081632653061), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.19999999999999988e-94

   1. Initial program 57.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 57.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

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

   7. Simplified 84.3%

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

   if -5.19999999999999988e-94 < v < 4.0000000000000001e-187

   1. Initial program 99.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(-1 \cdot v\right) \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) \cdot \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(-1 \cdot v\right)\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \left(0 - v\right)\right)\right)\right) \]

      12. --lowering--.f64 20.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   7. Simplified 20.8%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) \cdot \left(0 - v\right)}}\right) \]

   8. Taylor expanded in H around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)}, \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\left(\frac{\frac{-49}{5} \cdot H}{{v}^{2}}\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{-49}{5} \cdot H\right), \left({v}^{2}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \frac{-49}{5}\right), \left({v}^{2}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), \left({v}^{2}\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), \left(v \cdot v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

      6. *-lowering-*.f64 20.8%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), \mathsf{*.f64}\left(v, v\right)\right), \mathsf{\_.f64}\left(0, v\right)\right)\right)\right) \]

   10. Simplified 20.8%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{H \cdot -9.8}{v \cdot v}} \cdot \left(0 - v\right)}\right) \]

   11. Taylor expanded in v around 0

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{5}{49} \cdot \frac{{v}^{2}}{H}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \tan^{-1} \left(\frac{\frac{5}{49} \cdot {v}^{2}}{H}\right) \]

       2. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{{v}^{2} \cdot \frac{5}{49}}{H}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \tan^{-1} \left({v}^{2} \cdot \frac{\frac{5}{49}}{H}\right) \]

       4. metadata-eval N/A

          \[\leadsto \tan^{-1} \left({v}^{2} \cdot \frac{\frac{5}{49} \cdot 1}{H}\right) \]

       5. associate-*r/ N/A

          \[\leadsto \tan^{-1} \left({v}^{2} \cdot \left(\frac{5}{49} \cdot \frac{1}{H}\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\left({v}^{2} \cdot \left(\frac{5}{49} \cdot \frac{1}{H}\right)\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\left({v}^{2} \cdot \frac{\frac{5}{49} \cdot 1}{H}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\left({v}^{2} \cdot \frac{\frac{5}{49}}{H}\right)\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{{v}^{2} \cdot \frac{5}{49}}{H}\right)\right) \]

       10. associate-*l/ N/A

           \[\leadsto \mathsf{atan.f64}\left(\left(\frac{{v}^{2}}{H} \cdot \frac{5}{49}\right)\right) \]

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

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{{v}^{2}}{H}\right), \frac{5}{49}\right)\right) \]

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

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({v}^{2}\right), H\right), \frac{5}{49}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(v \cdot v\right), H\right), \frac{5}{49}\right)\right) \]

       14. *-lowering-*.f64 20.8%

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right), \frac{5}{49}\right)\right) \]

   13. Simplified 20.8%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v \cdot v}{H} \cdot 0.10204081632653061\right)} \]
   14. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\left(v \cdot \frac{v}{H}\right) \cdot \frac{5}{49}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \left(\frac{v}{H} \cdot \frac{5}{49}\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{v}{H} \cdot \frac{5}{49}\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(\left(\frac{v}{H}\right), \frac{5}{49}\right)\right)\right) \]

       5. /-lowering-/.f64 20.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, H\right), \frac{5}{49}\right)\right)\right) \]

   15. Applied egg-rr 20.9%

       \[\leadsto \tan^{-1} \color{blue}{\left(v \cdot \left(\frac{v}{H} \cdot 0.10204081632653061\right)\right)} \]

   if 4.0000000000000001e-187 < v

   1. Initial program 61.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 61.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

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

   7. Simplified 76.3%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.02e-103) (atan -1.0) (atan (/ v (+ v (/ (* -9.8 H) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.02e-103) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.02d-103)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v / (v + (((-9.8d0) * h) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.02e-103) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + ((-9.8 * H) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.02e-103:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + ((-9.8 * H) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.02e-103)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(-9.8 * H) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.02e-103)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + ((-9.8 * H) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.02e-103], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(-9.8 * H), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.01999999999999998e-103

   1. Initial program 58.2%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 58.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

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

   7. Simplified 83.5%

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

   if -1.01999999999999998e-103 < v

   1. Initial program 72.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 72.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(\frac{\frac{-49}{5} \cdot H}{v}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(\frac{-49}{5} \cdot H\right), v\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\left(H \cdot \frac{-49}{5}\right), v\right)\right)\right)\right) \]

      17. *-lowering-*.f64 61.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{-49}{5}\right), v\right)\right)\right)\right) \]

   7. Simplified 61.6%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{H \cdot -9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{-9.8 \cdot H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5e-311) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-311) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5d-311)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5e-311) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-311:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-311)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e-311)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-311], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -5.00000000000023e-311

   1. Initial program 66.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

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

   7. Simplified 67.5%

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

   if -5.00000000000023e-311 < v

   1. Initial program 66.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

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

   7. Simplified 67.4%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (v H) :precision binary64 (atan -1.0))

C

double code(double v, double H) {
	return atan(-1.0);
}

Fortran

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

Java

public static double code(double v, double H) {
	return Math.atan(-1.0);
}

Python

def code(v, H):
	return math.atan(-1.0)

Julia

function code(v, H)
	return atan(-1.0)
end

MATLAB

function tmp = code(v, H)
	tmp = atan(-1.0);
end

Wolfram

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

Derivation

1. Initial program 66.5%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval 66.5%

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

3. Simplified 66.5%

   \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around -inf

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

7. Simplified 36.2%

   \[\leadsto \tan^{-1} \color{blue}{-1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))