Optimal throwing angle

Percentage Accurate: 67.4% → 99.6%

Time: 8.3s

Alternatives: 7

Speedup: 1.9×

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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \mathsf{fma}\left(H, -19.6, \mathsf{fma}\left(H, -19.6, H \cdot 19.6\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 1e+127)
     (atan (/ v (sqrt (fma v v (fma H -19.6 (fma H -19.6 (* H 19.6)))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1e+127) {
		tmp = atan((v / sqrt(fma(v, v, fma(H, -19.6, fma(H, -19.6, (H * 19.6)))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+127)
		tmp = atan(Float64(v / sqrt(fma(v, v, fma(H, -19.6, fma(H, -19.6, Float64(H * 19.6)))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+127], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(H * -19.6 + N[(H * -19.6 + N[(H * 19.6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.00000000000000004e154

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -5.00000000000000004e154 < v < 9.99999999999999955e126

   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. prod-diff 99.7%

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

      2. *-commutative 99.7%

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

      3. fma-define 99.7%

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

      4. associate-+l+ 99.7%

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

      5. fma-define 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. fma-define 99.8%

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

      9. metadata-eval 99.8%

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

      10. *-commutative 99.8%

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

      11. fma-undefine 99.7%

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

      12. distribute-lft-neg-in 99.7%

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

      13. distribute-rgt-neg-in 99.7%

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

      14. fma-define 99.8%

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

      15. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 9.99999999999999955e126 < v

   1. Initial program 32.2%

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

      1. sqr-neg 32.2%

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

      2. sqr-neg 32.2%

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

      3. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \mathsf{fma}\left(H, -19.6, \mathsf{fma}\left(H, -19.6, H \cdot 19.6\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan -1.0)
   (if (<= v 2e+127)
     (atan (* v (sqrt (/ 1.0 (- (* v v) (* H 19.6))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+127) {
		tmp = atan((v * sqrt((1.0 / ((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 <= (-1d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+127) then
        tmp = atan((v * sqrt((1.0d0 / ((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 <= -1e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+127) {
		tmp = Math.atan((v * Math.sqrt((1.0 / ((v * v) - (H * 19.6))))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+127:
		tmp = math.atan((v * math.sqrt((1.0 / ((v * v) - (H * 19.6))))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+127)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / 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 <= -1e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+127)
		tmp = atan((v * sqrt((1.0 / ((v * v) - (H * 19.6))))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+127], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000004e154

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -1.00000000000000004e154 < v < 1.99999999999999991e127

   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. Taylor expanded in v around 0 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   6. Step-by-step derivation

      1. pow2 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.99999999999999991e127 < v

   1. Initial program 32.2%

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

      1. sqr-neg 32.2%

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

      2. sqr-neg 32.2%

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

      3. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 1e+127) (atan (/ v (sqrt (- (* v v) (* H 19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1e+127) {
		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 <= (-5d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 1d+127) 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 <= -5e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1e+127) {
		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 <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 1e+127:
		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 <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+127)
		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 <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+127)
		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, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+127], 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 < -5.00000000000000004e154

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -5.00000000000000004e154 < v < 9.99999999999999955e126

   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 9.99999999999999955e126 < v

   1. Initial program 32.2%

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

      1. sqr-neg 32.2%

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

      2. sqr-neg 32.2%

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

      3. metadata-eval 32.2%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.6 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.5e-16)
   (atan -1.0)
   (if (<= v 2.6e-36)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-16) {
		tmp = atan(-1.0);
	} else if (v <= 2.6e-36) {
		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 <= (-9.5d-16)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.6d-36) 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 <= -9.5e-16) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.6e-36) {
		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 <= -9.5e-16:
		tmp = math.atan(-1.0)
	elif v <= 2.6e-36:
		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 <= -9.5e-16)
		tmp = atan(-1.0);
	elseif (v <= 2.6e-36)
		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 <= -9.5e-16)
		tmp = atan(-1.0);
	elseif (v <= 2.6e-36)
		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, -9.5e-16], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.6e-36], 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 < -9.5000000000000005e-16

   1. Initial program 51.5%

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

      1. sqr-neg 51.5%

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

      2. sqr-neg 51.5%

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

      3. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in v around -inf 94.8%

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

   if -9.5000000000000005e-16 < v < 2.6e-36

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in v around 0 91.9%

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

   if 2.6e-36 < v

   1. Initial program 60.1%

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

      1. sqr-neg 60.1%

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

      2. sqr-neg 60.1%

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

      3. metadata-eval 60.1%

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

   3. Simplified 60.1%

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

   5. Taylor expanded in H around 0 90.8%

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

Alternative 5: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -8.5 \cdot 10^{-217}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -8.5e-217) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -8.5e-217) {
		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 <= (-8.5d-217)) 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 <= -8.5e-217) {
		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 <= -8.5e-217:
		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 <= -8.5e-217)
		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 <= -8.5e-217)
		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, -8.5e-217], 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 < -8.4999999999999994e-217

   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. sqr-neg 66.3%

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

      2. sqr-neg 66.3%

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

      3. metadata-eval 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in v around -inf 70.5%

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

   if -8.4999999999999994e-217 < v

   1. Initial program 76.1%

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

      1. sqr-neg 76.1%

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

      2. sqr-neg 76.1%

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

      3. metadata-eval 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in H around 0 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 6: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -2e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-310)
		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 <= -2e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.999999999999994e-310

   1. Initial program 69.3%

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

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in v around -inf 64.5%

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

   if -1.999999999999994e-310 < v

   1. Initial program 74.2%

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

      1. sqr-neg 74.2%

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

      2. sqr-neg 74.2%

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

      3. metadata-eval 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in v around inf 62.3%

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

Alternative 7: 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 71.8%

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

   1. sqr-neg 71.8%

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

   2. sqr-neg 71.8%

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

   3. metadata-eval 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in v around -inf 32.2%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))