Optimal throwing angle

Percentage Accurate: 67.4% → 99.5%

Time: 8.4s

Alternatives: 6

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: 67.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left({v}^{2} + H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan -1.0)
   (if (<= v 5.8e+94)
     (atan (* v (pow (+ (pow v 2.0) (* H -19.6)) -0.5)))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan(-1.0);
	} else if (v <= 5.8e+94) {
		tmp = atan((v * pow((pow(v, 2.0) + (H * -19.6)), -0.5)));
	} 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+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.8d+94) then
        tmp = atan((v * (((v ** 2.0d0) + (h * (-19.6d0))) ** (-0.5d0))))
    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+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.8e+94) {
		tmp = Math.atan((v * Math.pow((Math.pow(v, 2.0) + (H * -19.6)), -0.5)));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+154:
		tmp = math.atan(-1.0)
	elif v <= 5.8e+94:
		tmp = math.atan((v * math.pow((math.pow(v, 2.0) + (H * -19.6)), -0.5)))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.8e+94)
		tmp = atan(Float64(v * (Float64((v ^ 2.0) + Float64(H * -19.6)) ^ -0.5)));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.8e+94)
		tmp = atan((v * (((v ^ 2.0) + (H * -19.6)) ^ -0.5)));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.8e+94], N[ArcTan[N[(v * N[Power[N[(N[Power[v, 2.0], $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   1. Initial program 3.0%

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

      1. sqr-neg 3.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 3.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -2.00000000000000007e154 < v < 5.7999999999999997e94

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{0.5}}} \cdot v\right) \]

      4. pow-flip 99.8%

         \[\leadsto \tan^{-1} \left(\color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{\left(-0.5\right)}} \cdot v\right) \]

      5. sub-neg 99.8%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(v \cdot v + \left(-19.6 \cdot H\right)\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      6. +-commutative 99.8%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(\left(-19.6 \cdot H\right) + v \cdot v\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      7. *-commutative 99.8%

         \[\leadsto \tan^{-1} \left({\left(\left(-\color{blue}{H \cdot 19.6}\right) + v \cdot v\right)}^{\left(-0.5\right)} \cdot v\right) \]

      8. distribute-rgt-neg-in 99.8%

         \[\leadsto \tan^{-1} \left({\left(\color{blue}{H \cdot \left(-19.6\right)} + v \cdot v\right)}^{\left(-0.5\right)} \cdot v\right) \]

      9. fma-define 99.8%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(\mathsf{fma}\left(H, -19.6, v \cdot v\right)\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      10. metadata-eval 99.8%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, \color{blue}{-19.6}, v \cdot v\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      11. pow2 99.8%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      12. metadata-eval 99.8%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{-0.5}} \cdot v\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.5} \cdot v\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 99.8%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(H \cdot -19.6 + {v}^{2}\right)}}^{-0.5} \cdot v\right) \]

      2. +-commutative 99.8%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left({v}^{2} + H \cdot -19.6\right)}}^{-0.5} \cdot v\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left({v}^{2} + H \cdot -19.6\right)}}^{-0.5} \cdot v\right) \]

   if 5.7999999999999997e94 < v

   1. Initial program 22.8%

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

      1. sqr-neg 22.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 22.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 22.8%

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

   3. Simplified 22.8%

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

   5. Taylor expanded in H around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left({v}^{2} + H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan -1.0)
   (if (<= v 5.8e+94)
     (atan (/ v (sqrt (- (* v v) (* H 19.6)))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan(-1.0);
	} else if (v <= 5.8e+94) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} 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+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.8d+94) then
        tmp = atan((v / sqrt(((v * v) - (h * 19.6d0)))))
    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+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.8e+94) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+154:
		tmp = math.atan(-1.0)
	elif v <= 5.8e+94:
		tmp = math.atan((v / math.sqrt(((v * v) - (H * 19.6)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.8e+94)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.8e+94)
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.8e+94], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   1. Initial program 3.0%

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

      1. sqr-neg 3.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 3.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -2.00000000000000007e154 < v < 5.7999999999999997e94

   1. Initial program 99.7%

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

      1. sqr-neg 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 5.7999999999999997e94 < v

   1. Initial program 22.8%

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

      1. sqr-neg 22.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 22.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 22.8%

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

   3. Simplified 22.8%

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

   5. Taylor expanded in H around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.8 \cdot 10^{+94}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e-33)
   (atan -1.0)
   (if (<= v 1.9e-12)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e-33) {
		tmp = atan(-1.0);
	} else if (v <= 1.9e-12) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} 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 <= (-1d-33)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.9d-12) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    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 <= -1e-33) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.9e-12) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e-33:
		tmp = math.atan(-1.0)
	elif v <= 1.9e-12:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e-33)
		tmp = atan(-1.0);
	elseif (v <= 1.9e-12)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1e-33)
		tmp = atan(-1.0);
	elseif (v <= 1.9e-12)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e-33], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.9e-12], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.0000000000000001e-33

   1. Initial program 55.4%

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

      1. sqr-neg 55.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 55.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in v around -inf 90.7%

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

   if -1.0000000000000001e-33 < v < 1.89999999999999998e-12

   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. sqr-neg 99.5%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 99.5%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

         \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{0.5}}} \cdot v\right) \]

      4. pow-flip 99.6%

         \[\leadsto \tan^{-1} \left(\color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{\left(-0.5\right)}} \cdot v\right) \]

      5. sub-neg 99.6%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(v \cdot v + \left(-19.6 \cdot H\right)\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      6. +-commutative 99.6%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(\left(-19.6 \cdot H\right) + v \cdot v\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      7. *-commutative 99.6%

         \[\leadsto \tan^{-1} \left({\left(\left(-\color{blue}{H \cdot 19.6}\right) + v \cdot v\right)}^{\left(-0.5\right)} \cdot v\right) \]

      8. distribute-rgt-neg-in 99.6%

         \[\leadsto \tan^{-1} \left({\left(\color{blue}{H \cdot \left(-19.6\right)} + v \cdot v\right)}^{\left(-0.5\right)} \cdot v\right) \]

      9. fma-define 99.6%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(\mathsf{fma}\left(H, -19.6, v \cdot v\right)\right)}}^{\left(-0.5\right)} \cdot v\right) \]

      10. metadata-eval 99.6%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, \color{blue}{-19.6}, v \cdot v\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      11. pow2 99.6%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      12. metadata-eval 99.6%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{-0.5}} \cdot v\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.5} \cdot v\right)} \]

   7. Taylor expanded in H around inf 88.4%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left(-19.6 \cdot H\right)}}^{-0.5} \cdot v\right) \]
   8. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \tan^{-1} \left({\color{blue}{\left(H \cdot -19.6\right)}}^{-0.5} \cdot v\right) \]

   9. Simplified 88.4%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left(H \cdot -19.6\right)}}^{-0.5} \cdot v\right) \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 88.1%

          \[\leadsto \tan^{-1} \left(\color{blue}{\left(\sqrt{{\left(H \cdot -19.6\right)}^{-0.5}} \cdot \sqrt{{\left(H \cdot -19.6\right)}^{-0.5}}\right)} \cdot v\right) \]

       2. sqrt-unprod 88.4%

          \[\leadsto \tan^{-1} \left(\color{blue}{\sqrt{{\left(H \cdot -19.6\right)}^{-0.5} \cdot {\left(H \cdot -19.6\right)}^{-0.5}}} \cdot v\right) \]

       3. pow-prod-up 88.4%

          \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{{\left(H \cdot -19.6\right)}^{\left(-0.5 + -0.5\right)}}} \cdot v\right) \]

       4. metadata-eval 88.4%

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

   11. Applied egg-rr 88.4%

       \[\leadsto \tan^{-1} \left(\color{blue}{\sqrt{{\left(H \cdot -19.6\right)}^{-1}}} \cdot v\right) \]
   12. Step-by-step derivation

       1. unpow-1 88.4%

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

       2. *-commutative 88.4%

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

       3. associate-/r* 88.4%

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

       4. metadata-eval 88.5%

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

   13. Simplified 88.5%

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

   if 1.89999999999999998e-12 < v

   1. Initial program 45.1%

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

      1. sqr-neg 45.1%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 45.1%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 45.1%

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

   3. Simplified 45.1%

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

   5. Taylor expanded in H around 0 92.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.3 \cdot 10^{-173}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.3e-173) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.3e-173) {
		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 <= (-2.3d-173)) 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 <= -2.3e-173) {
		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 <= -2.3e-173:
		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 <= -2.3e-173)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.3e-173)
		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, -2.3e-173], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.29999999999999988e-173

   1. Initial program 66.7%

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

      1. sqr-neg 66.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 66.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in v around -inf 74.2%

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

   if -2.29999999999999988e-173 < v

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 67.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in H around 0 66.0%

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

Alternative 5: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.2 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5.2e-296) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.2e-296) {
		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 <= (-5.2d-296)) 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 <= -5.2e-296) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.2e-296:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.2e-296)
		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 <= -5.2e-296)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.2e-296], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -5.2000000000000001e-296

   1. Initial program 70.0%

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

      1. sqr-neg 70.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 70.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in v around -inf 67.2%

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

   if -5.2000000000000001e-296 < v

   1. Initial program 63.7%

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

      1. sqr-neg 63.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      2. sqr-neg 63.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

      3. metadata-eval 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in v around inf 64.1%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.6% 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 66.9%

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

   1. sqr-neg 66.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

   2. sqr-neg 66.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} - \left(2 \cdot 9.8\right) \cdot H}}\right) \]

   3. metadata-eval 66.9%

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

3. Simplified 66.9%

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

5. Taylor expanded in v around -inf 35.0%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))