Optimal throwing angle

Percentage Accurate: 67.9% → 99.1%

Time: 11.4s

Alternatives: 8

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: 67.9% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.15 \cdot 10^{+139}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(\mathsf{fma}\left(v, v, H \cdot -19.6\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.15e+139)
   (atan -1.0)
   (if (<= v 2e+73) (atan (* v (pow (fma v v (* H -19.6)) -0.5))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.15e+139) {
		tmp = atan(-1.0);
	} else if (v <= 2e+73) {
		tmp = atan((v * pow(fma(v, v, (H * -19.6)), -0.5)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.15e+139)
		tmp = atan(-1.0);
	elseif (v <= 2e+73)
		tmp = atan(Float64(v * (fma(v, v, Float64(H * -19.6)) ^ -0.5)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.15e+139], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+73], N[ArcTan[N[(v * N[Power[N[(v * v + N[(H * -19.6), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.1499999999999999e139

   1. Initial program 5.0%

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

      1. sqr-neg 5.0%

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

      2. sqr-neg 5.0%

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

      3. metadata-eval 5.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -2.1499999999999999e139 < v < 1.99999999999999997e73

   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. add-cbrt-cube 75.3%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt[3]{\left(\sqrt{v \cdot v - 19.6 \cdot H} \cdot \sqrt{v \cdot v - 19.6 \cdot H}\right) \cdot \sqrt{v \cdot v - 19.6 \cdot H}}}}\right) \]

      2. pow1/3 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{{\left(\left(\sqrt{v \cdot v - 19.6 \cdot H} \cdot \sqrt{v \cdot v - 19.6 \cdot H}\right) \cdot \sqrt{v \cdot v - 19.6 \cdot H}\right)}^{0.3333333333333333}}}\right) \]

      3. add-sqr-sqrt 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left(\color{blue}{\left(v \cdot v - 19.6 \cdot H\right)} \cdot \sqrt{v \cdot v - 19.6 \cdot H}\right)}^{0.3333333333333333}}\right) \]

      4. pow1 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left(\color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{1}} \cdot \sqrt{v \cdot v - 19.6 \cdot H}\right)}^{0.3333333333333333}}\right) \]

      5. pow1/2 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(v \cdot v - 19.6 \cdot H\right)}^{1} \cdot \color{blue}{{\left(v \cdot v - 19.6 \cdot H\right)}^{0.5}}\right)}^{0.3333333333333333}}\right) \]

      6. pow-prod-up 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\color{blue}{\left({\left(v \cdot v - 19.6 \cdot H\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}}\right) \]

      7. sub-neg 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\color{blue}{\left(v \cdot v + \left(-19.6 \cdot H\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      8. +-commutative 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\color{blue}{\left(\left(-19.6 \cdot H\right) + v \cdot v\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      9. *-commutative 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(\left(-\color{blue}{H \cdot 19.6}\right) + v \cdot v\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      10. distribute-rgt-neg-in 71.2%

          \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(\color{blue}{H \cdot \left(-19.6\right)} + v \cdot v\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      11. fma-define 71.2%

          \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\color{blue}{\left(\mathsf{fma}\left(H, -19.6, v \cdot v\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      12. metadata-eval 71.2%

          \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(\mathsf{fma}\left(H, \color{blue}{-19.6}, v \cdot v\right)\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      13. pow2 71.2%

          \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}}\right) \]

      14. metadata-eval 71.2%

          \[\leadsto \tan^{-1} \left(\frac{v}{{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}\right) \]

   6. Applied egg-rr 71.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}\right) \]
   7. Step-by-step derivation

      1. unpow1/3 75.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{1.5}}}}\right) \]

   8. Simplified 75.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{1.5}}}}\right) \]
   9. Step-by-step derivation

      1. pow1/3 71.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}\right) \]

      2. pow-pow 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      3. metadata-eval 99.7%

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

      4. pow1/2 99.7%

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

      5. fma-undefine 99.7%

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

      6. +-commutative 99.7%

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

      7. unpow2 99.7%

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

      8. add-sqr-sqrt 80.0%

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

      9. hypot-undefine 80.0%

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

      10. add-sqr-sqrt 79.6%

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

      11. sqrt-unprod 80.0%

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

      12. sqr-neg 80.0%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 10.6%

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

      15. un-div-inv 10.6%

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

      16. *-commutative 10.6%

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

   10. Applied egg-rr 80.0%

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

       1. inv-pow 80.0%

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

       2. hypot-undefine 80.0%

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

       3. sqrt-pow2 79.9%

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

       4. add-sqr-sqrt 99.8%

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

       5. fma-define 99.8%

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

       6. metadata-eval 99.8%

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

   12. Applied egg-rr 99.8%

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

   if 1.99999999999999997e73 < v

   1. Initial program 36.5%

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

      1. sqr-neg 36.5%

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

      2. sqr-neg 36.5%

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

      3. metadata-eval 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.15 \cdot 10^{+139}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(\mathsf{fma}\left(v, v, H \cdot -19.6\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+156)
   (atan -1.0)
   (if (<= v 2e+73) (atan (/ v (sqrt (- (* v v) (* H 19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+156) {
		tmp = atan(-1.0);
	} else if (v <= 2e+73) {
		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 <= (-1d+156)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+73) 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 <= -1e+156) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+73) {
		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 <= -1e+156:
		tmp = math.atan(-1.0)
	elif v <= 2e+73:
		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 <= -1e+156)
		tmp = atan(-1.0);
	elseif (v <= 2e+73)
		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 <= -1e+156)
		tmp = atan(-1.0);
	elseif (v <= 2e+73)
		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, -1e+156], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+73], 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 < -9.9999999999999998e155

   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 -9.9999999999999998e155 < v < 1.99999999999999997e73

   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 1.99999999999999997e73 < v

   1. Initial program 36.5%

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

      1. sqr-neg 36.5%

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

      2. sqr-neg 36.5%

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

      3. metadata-eval 36.5%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.8 \cdot 10^{-247}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.8e-247)
   (atan (/ v (- (fma -9.8 (/ H v) v))))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.8e-247) {
		tmp = atan((v / -fma(-9.8, (H / v), v)));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.8e-247)
		tmp = atan(Float64(v / Float64(-fma(-9.8, Float64(H / v), v))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.8e-247], N[ArcTan[N[(v / (-N[(-9.8 * N[(H / v), $MachinePrecision] + v), $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 2 regimes

2. if v < -3.79999999999999988e-247

   1. Initial program 61.7%

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

      1. sqr-neg 61.7%

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

      2. sqr-neg 61.7%

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

      3. metadata-eval 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in H around 0 3.4%

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

      1. +-commutative 3.4%

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

      2. fma-define 3.4%

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

   7. Simplified 3.4%

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

      1. frac-2neg 3.4%

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

      2. neg-sub0 3.4%

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

      3. div-sub 3.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-prod 42.6%

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

      6. sqr-neg 42.6%

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

      7. sqrt-unprod 74.3%

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

      8. add-sqr-sqrt 74.7%

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

      9. frac-2neg 74.7%

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

   9. Applied egg-rr 74.7%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0}{-\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)} - \frac{v}{\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)}\right)} \]
   10. Step-by-step derivation

       1. div0 74.7%

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

       2. neg-sub0 74.7%

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

       3. distribute-neg-frac2 74.7%

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

   11. Simplified 74.7%

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

   if -3.79999999999999988e-247 < v

   1. Initial program 68.6%

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

      1. sqr-neg 68.6%

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

      2. sqr-neg 68.6%

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

      3. metadata-eval 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in H around 0 70.9%

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

Alternative 4: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-148}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-138}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \frac{9.8}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-148)
   (atan -1.0)
   (if (<= v 2.5e-138) (atan (/ v (* H (/ 9.8 v)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-148) {
		tmp = atan(-1.0);
	} else if (v <= 2.5e-138) {
		tmp = atan((v / (H * (9.8 / v))));
	} 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 <= (-3.6d-148)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.5d-138) then
        tmp = atan((v / (h * (9.8d0 / v))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-148) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.5e-138) {
		tmp = Math.atan((v / (H * (9.8 / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.6e-148:
		tmp = math.atan(-1.0)
	elif v <= 2.5e-138:
		tmp = math.atan((v / (H * (9.8 / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.6e-148)
		tmp = atan(-1.0);
	elseif (v <= 2.5e-138)
		tmp = atan(Float64(v / Float64(H * Float64(9.8 / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.6e-148)
		tmp = atan(-1.0);
	elseif (v <= 2.5e-138)
		tmp = atan((v / (H * (9.8 / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.6e-148], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.5e-138], N[ArcTan[N[(v / N[(H * N[(9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.5999999999999998e-148

   1. Initial program 56.5%

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

      1. sqr-neg 56.5%

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

      2. sqr-neg 56.5%

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

      3. metadata-eval 56.5%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in v around -inf 82.2%

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

   if -3.5999999999999998e-148 < v < 2.49999999999999994e-138

   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 H around 0 28.9%

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

      1. +-commutative 28.9%

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

      2. fma-define 28.9%

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

   7. Simplified 28.9%

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

   8. Taylor expanded in H around inf 28.9%

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

      1. associate-*r/ 28.9%

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

      2. frac-2neg 28.9%

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

      3. add-sqr-sqrt 13.5%

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

      4. sqrt-unprod 28.9%

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

      5. sqr-neg 28.9%

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

      6. sqrt-unprod 15.2%

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

      7. add-sqr-sqrt 29.2%

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

   10. Applied egg-rr 29.2%

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

       1. distribute-lft-neg-in 29.2%

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

       2. metadata-eval 29.2%

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

       3. *-commutative 29.2%

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

       4. *-lft-identity 29.2%

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

       5. times-frac 29.2%

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

       6. /-rgt-identity 29.2%

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

   12. Simplified 29.2%

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

   if 2.49999999999999994e-138 < v

   1. Initial program 57.4%

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

      1. sqr-neg 57.4%

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

      2. sqr-neg 57.4%

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

      3. metadata-eval 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in v around inf 82.8%

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

Alternative 5: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.6 \cdot 10^{-191}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.4 \cdot 10^{-130}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.6e-191)
   (atan -1.0)
   (if (<= v 5.4e-130)
     (atan (* -0.10204081632653061 (/ v (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.6e-191) {
		tmp = atan(-1.0);
	} else if (v <= 5.4e-130) {
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	} 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 <= (-2.6d-191)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.4d-130) then
        tmp = atan(((-0.10204081632653061d0) * (v / (h / v))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.6e-191) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.4e-130) {
		tmp = Math.atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.6e-191:
		tmp = math.atan(-1.0)
	elif v <= 5.4e-130:
		tmp = math.atan((-0.10204081632653061 * (v / (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.6e-191)
		tmp = atan(-1.0);
	elseif (v <= 5.4e-130)
		tmp = atan(Float64(-0.10204081632653061 * Float64(v / Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.6e-191)
		tmp = atan(-1.0);
	elseif (v <= 5.4e-130)
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.6e-191], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.4e-130], N[ArcTan[N[(-0.10204081632653061 * N[(v / N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.59999999999999986e-191

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in v around -inf 76.1%

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

   if -2.59999999999999986e-191 < v < 5.39999999999999982e-130

   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 H around 0 35.1%

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

      1. +-commutative 35.1%

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

      2. fma-define 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in H around inf 35.1%

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

      1. *-un-lft-identity 35.1%

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

      2. times-frac 35.1%

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

      3. metadata-eval 35.1%

         \[\leadsto \tan^{-1} \left(\color{blue}{-0.10204081632653061} \cdot \frac{v}{\frac{H}{v}}\right) \]

   10. Applied egg-rr 35.1%

       \[\leadsto \tan^{-1} \color{blue}{\left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)} \]

   if 5.39999999999999982e-130 < v

   1. Initial program 57.4%

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

      1. sqr-neg 57.4%

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

      2. sqr-neg 57.4%

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

      3. metadata-eval 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in v around inf 82.8%

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

Alternative 6: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.5 \cdot 10^{-191}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.5e-191) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.5e-191) {
		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 <= (-2.5d-191)) 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 <= -2.5e-191) {
		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 <= -2.5e-191:
		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 <= -2.5e-191)
		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 <= -2.5e-191)
		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, -2.5e-191], 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 < -2.5e-191

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in v around -inf 76.1%

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

   if -2.5e-191 < v

   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 H around 0 69.0%

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

Alternative 7: 66.9% 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 64.1%

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

      1. sqr-neg 64.1%

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

      2. sqr-neg 64.1%

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

      3. metadata-eval 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in v around -inf 68.3%

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

   if -1.999999999999994e-310 < v

   1. Initial program 66.2%

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

      1. sqr-neg 66.2%

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

      2. sqr-neg 66.2%

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

      3. metadata-eval 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in v around inf 66.3%

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

Alternative 8: 34.4% 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 65.0%

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

   1. sqr-neg 65.0%

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

   2. sqr-neg 65.0%

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

   3. metadata-eval 65.0%

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

3. Simplified 65.0%

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

5. Taylor expanded in v around -inf 38.5%

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

Reproduce

herbie shell --seed 2024102 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))