Optimal throwing angle

Percentage Accurate: 68.1% → 99.6%

Time: 10.0s

Alternatives: 10

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: 68.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-310} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\tan^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (/ v (sqrt (- (* v v) (* 19.6 H))))))
   (if (or (<= t_0 -2e-310) (not (<= t_0 0.0)))
     (atan t_0)
     (atan (/ v (fabs (fma -9.8 (/ H v) v)))))))

C

double code(double v, double H) {
	double t_0 = v / sqrt(((v * v) - (19.6 * H)));
	double tmp;
	if ((t_0 <= -2e-310) || !(t_0 <= 0.0)) {
		tmp = atan(t_0);
	} else {
		tmp = atan((v / fabs(fma(-9.8, (H / v), v))));
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H))))
	tmp = 0.0
	if ((t_0 <= -2e-310) || !(t_0 <= 0.0))
		tmp = atan(t_0);
	else
		tmp = atan(Float64(v / abs(fma(-9.8, Float64(H / v), v))));
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-310], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[ArcTan[t$95$0], $MachinePrecision], N[ArcTan[N[(v / N[Abs[N[(-9.8 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < -1.999999999999994e-310 or -0.0 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))

   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 -1.999999999999994e-310 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < -0.0

   1. Initial program 14.8%

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

      1. sqr-neg 14.8%

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

      2. sqr-neg 14.8%

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

      3. metadata-eval 14.8%

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

   3. Simplified 14.8%

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

   5. Taylor expanded in H around 0 45.4%

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

      1. add-sqr-sqrt 37.2%

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

      2. sqrt-unprod 14.8%

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

      3. pow2 14.8%

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

      4. +-commutative 14.8%

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

      5. fma-define 14.8%

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

   7. Applied egg-rr 14.8%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 14.8%

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

      2. rem-sqrt-square 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq -2 \cdot 10^{-310} \lor \neg \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq 0\right):\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{1}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}}{\frac{1}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= (* 19.6 H) -1e-303)
   (atan (/ (/ 1.0 (hypot v (sqrt (* H -19.6)))) (/ 1.0 v)))
   (atan (/ v (fabs (fma -9.8 (/ H v) v))))))

C

double code(double v, double H) {
	double tmp;
	if ((19.6 * H) <= -1e-303) {
		tmp = atan(((1.0 / hypot(v, sqrt((H * -19.6)))) / (1.0 / v)));
	} else {
		tmp = atan((v / fabs(fma(-9.8, (H / v), v))));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (Float64(19.6 * H) <= -1e-303)
		tmp = atan(Float64(Float64(1.0 / hypot(v, sqrt(Float64(H * -19.6)))) / Float64(1.0 / v)));
	else
		tmp = atan(Float64(v / abs(fma(-9.8, Float64(H / v), v))));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[N[(19.6 * H), $MachinePrecision], -1e-303], N[ArcTan[N[(N[(1.0 / N[Sqrt[v ^ 2 + N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(1.0 / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[Abs[N[(-9.8 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H) < -9.99999999999999931e-304

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. sqr-neg 79.4%

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

      3. metadata-eval 79.4%

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

   3. Simplified 79.4%

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

      1. add-cbrt-cube 54.6%

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

      2. pow1/3 51.7%

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

      3. add-sqr-sqrt 51.7%

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

      4. pow1 51.7%

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

      5. pow1/2 51.7%

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

      6. pow-prod-up 51.7%

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

      7. sub-neg 51.7%

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

      8. +-commutative 51.7%

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

      9. *-commutative 51.7%

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

      10. distribute-rgt-neg-in 51.7%

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

      11. fma-define 51.7%

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

      12. metadata-eval 51.7%

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

      13. pow2 51.7%

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

      14. metadata-eval 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow1/3 54.7%

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

   8. Simplified 54.7%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.25} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right)} \]
   11. Step-by-step derivation

       1. metadata-eval 79.2%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{\left(-0.25\right)}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       2. pow-flip 79.0%

          \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       3. add-sqr-sqrt 78.8%

          \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}} \cdot \sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       4. sqrt-unprod 79.0%

          \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       5. pow-prod-up 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\left(0.25 + 0.25\right)}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       6. metadata-eval 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{0.5}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       7. fma-undefine 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\color{blue}{\left(H \cdot -19.6 + {v}^{2}\right)}}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       8. +-commutative 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\color{blue}{\left({v}^{2} + H \cdot -19.6\right)}}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       9. unpow2 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(\color{blue}{v \cdot v} + H \cdot -19.6\right)}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       10. add-sqr-sqrt 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(v \cdot v + \color{blue}{\sqrt{H \cdot -19.6} \cdot \sqrt{H \cdot -19.6}}\right)}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       11. pow1/2 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{\sqrt{v \cdot v + \sqrt{H \cdot -19.6} \cdot \sqrt{H \cdot -19.6}}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       12. hypot-undefine 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       13. add-sqr-sqrt 78.9%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}} \cdot \sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}}\right) \]

       14. sqrt-unprod 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}}\right) \]

       15. pow-prod-up 79.0%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\left(0.25 + 0.25\right)}}}}\right) \]

   12. Applied egg-rr 99.6%

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

   if -9.99999999999999931e-304 < (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)

   1. Initial program 56.4%

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

      1. sqr-neg 56.4%

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

      2. sqr-neg 56.4%

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

      3. metadata-eval 56.4%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in H around 0 45.9%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-unprod 56.4%

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

      3. pow2 56.4%

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

      4. +-commutative 56.4%

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

      5. fma-define 56.4%

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

   7. Applied egg-rr 56.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 56.4%

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

      2. rem-sqrt-square 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{1}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}}{\frac{1}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= (* 19.6 H) -1e-303)
   (atan (/ 1.0 (/ (hypot v (sqrt (* H -19.6))) v)))
   (atan (/ v (fabs (fma -9.8 (/ H v) v))))))

C

double code(double v, double H) {
	double tmp;
	if ((19.6 * H) <= -1e-303) {
		tmp = atan((1.0 / (hypot(v, sqrt((H * -19.6))) / v)));
	} else {
		tmp = atan((v / fabs(fma(-9.8, (H / v), v))));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (Float64(19.6 * H) <= -1e-303)
		tmp = atan(Float64(1.0 / Float64(hypot(v, sqrt(Float64(H * -19.6))) / v)));
	else
		tmp = atan(Float64(v / abs(fma(-9.8, Float64(H / v), v))));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[N[(19.6 * H), $MachinePrecision], -1e-303], N[ArcTan[N[(1.0 / N[(N[Sqrt[v ^ 2 + N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[Abs[N[(-9.8 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H) < -9.99999999999999931e-304

   1. Initial program 79.4%

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

      1. sqr-neg 79.4%

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

      2. sqr-neg 79.4%

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

      3. metadata-eval 79.4%

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

   3. Simplified 79.4%

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

      1. add-cbrt-cube 54.6%

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

      2. pow1/3 51.7%

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

      3. add-sqr-sqrt 51.7%

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

      4. pow1 51.7%

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

      5. pow1/2 51.7%

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

      6. pow-prod-up 51.7%

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

      7. sub-neg 51.7%

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

      8. +-commutative 51.7%

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

      9. *-commutative 51.7%

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

      10. distribute-rgt-neg-in 51.7%

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

      11. fma-define 51.7%

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

      12. metadata-eval 51.7%

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

      13. pow2 51.7%

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

      14. metadata-eval 51.7%

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

   6. Applied egg-rr 51.7%

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

      1. unpow1/3 54.7%

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

   8. Simplified 54.7%

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

   10. Applied egg-rr 79.2%

       \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.25} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right)} \]
   11. Step-by-step derivation

       1. metadata-eval 79.2%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{\left(-0.25\right)}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       2. pow-flip 79.0%

          \[\leadsto \tan^{-1} \left(\color{blue}{\frac{1}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       3. add-sqr-sqrt 78.8%

          \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}} \cdot \sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       4. sqrt-unprod 79.0%

          \[\leadsto \tan^{-1} \left(\frac{1}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       5. pow-prod-up 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\left(0.25 + 0.25\right)}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       6. metadata-eval 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{0.5}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       7. fma-undefine 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\color{blue}{\left(H \cdot -19.6 + {v}^{2}\right)}}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       8. +-commutative 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\color{blue}{\left({v}^{2} + H \cdot -19.6\right)}}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       9. unpow2 79.1%

          \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(\color{blue}{v \cdot v} + H \cdot -19.6\right)}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       10. add-sqr-sqrt 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{{\left(v \cdot v + \color{blue}{\sqrt{H \cdot -19.6} \cdot \sqrt{H \cdot -19.6}}\right)}^{0.5}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       11. pow1/2 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{\sqrt{v \cdot v + \sqrt{H \cdot -19.6} \cdot \sqrt{H \cdot -19.6}}}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       12. hypot-undefine 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}}} \cdot \frac{v}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}\right) \]

       13. add-sqr-sqrt 78.9%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}} \cdot \sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}}\right) \]

       14. sqrt-unprod 79.1%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{0.25}}}}\right) \]

       15. pow-prod-up 79.0%

           \[\leadsto \tan^{-1} \left(\frac{1}{\sqrt{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}} \cdot \frac{v}{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\left(0.25 + 0.25\right)}}}}\right) \]

   12. Applied egg-rr 99.0%

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

   if -9.99999999999999931e-304 < (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)

   1. Initial program 56.4%

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

      1. sqr-neg 56.4%

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

      2. sqr-neg 56.4%

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

      3. metadata-eval 56.4%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in H around 0 45.9%

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

      1. add-sqr-sqrt 44.7%

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

      2. sqrt-unprod 56.4%

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

      3. pow2 56.4%

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

      4. +-commutative 56.4%

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

      5. fma-define 56.4%

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

   7. Applied egg-rr 56.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 56.4%

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

      2. rem-sqrt-square 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-303}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.8 \cdot 10^{+100}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 5.8e+100) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 5.8e+100) {
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	} 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 <= 5.8d+100) then
        tmp = atan((v / sqrt(((v * v) - (19.6d0 * h)))))
    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 <= 5.8e+100) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (19.6 * H)))));
	} 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 <= 5.8e+100:
		tmp = math.atan((v / math.sqrt(((v * v) - (19.6 * H)))))
	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 <= 5.8e+100)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H)))));
	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 <= 5.8e+100)
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	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, 5.8e+100], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $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 99.9%

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

   if -5.00000000000000004e154 < v < 5.8000000000000001e100

   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 5.8000000000000001e100 < v

   1. Initial program 36.2%

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

      1. sqr-neg 36.2%

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

      2. sqr-neg 36.2%

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

      3. metadata-eval 36.2%

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

   3. Simplified 36.2%

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

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.7 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 7.6 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.7e-47)
   (atan -1.0)
   (if (<= v 7.6e-31)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.7e-47) {
		tmp = atan(-1.0);
	} else if (v <= 7.6e-31) {
		tmp = atan((v / sqrt((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 <= (-2.7d-47)) then
        tmp = atan((-1.0d0))
    else if (v <= 7.6d-31) then
        tmp = atan((v / sqrt((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 <= -2.7e-47) {
		tmp = Math.atan(-1.0);
	} else if (v <= 7.6e-31) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.7e-47:
		tmp = math.atan(-1.0)
	elif v <= 7.6e-31:
		tmp = math.atan((v / math.sqrt((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 <= -2.7e-47)
		tmp = atan(-1.0);
	elseif (v <= 7.6e-31)
		tmp = atan(Float64(v / sqrt(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 <= -2.7e-47)
		tmp = atan(-1.0);
	elseif (v <= 7.6e-31)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.7e-47], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 7.6e-31], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $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 < -2.6999999999999998e-47

   1. Initial program 48.8%

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

      1. sqr-neg 48.8%

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

      2. sqr-neg 48.8%

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

      3. metadata-eval 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in v around -inf 92.4%

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

   if -2.6999999999999998e-47 < v < 7.5999999999999999e-31

   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 89.0%

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

      1. *-commutative 89.0%

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

   7. Simplified 89.0%

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

   if 7.5999999999999999e-31 < v

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in H around 0 87.1%

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

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.5 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.5e-46)
   (atan -1.0)
   (if (<= v 1.02e-29)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.5e-46) {
		tmp = atan(-1.0);
	} else if (v <= 1.02e-29) {
		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 <= (-5.5d-46)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.02d-29) 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 <= -5.5e-46) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.02e-29) {
		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 <= -5.5e-46:
		tmp = math.atan(-1.0)
	elif v <= 1.02e-29:
		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 <= -5.5e-46)
		tmp = atan(-1.0);
	elseif (v <= 1.02e-29)
		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 <= -5.5e-46)
		tmp = atan(-1.0);
	elseif (v <= 1.02e-29)
		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, -5.5e-46], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.02e-29], 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 < -5.49999999999999983e-46

   1. Initial program 48.8%

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

      1. sqr-neg 48.8%

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

      2. sqr-neg 48.8%

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

      3. metadata-eval 48.8%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in v around -inf 92.4%

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

   if -5.49999999999999983e-46 < v < 1.01999999999999994e-29

   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 99.5%

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

   6. Taylor expanded in v around 0 89.0%

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

   if 1.01999999999999994e-29 < v

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. metadata-eval 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in H around 0 87.1%

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

Alternative 7: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.65 \cdot 10^{-209}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.82 \cdot 10^{-131}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-9.8 \cdot \frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.65e-209)
   (atan -1.0)
   (if (<= v 1.82e-131) (atan (/ v (* -9.8 (/ H v)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.65e-209) {
		tmp = atan(-1.0);
	} else if (v <= 1.82e-131) {
		tmp = atan((v / (-9.8 * (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 <= (-1.65d-209)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.82d-131) then
        tmp = atan((v / ((-9.8d0) * (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 <= -1.65e-209) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.82e-131) {
		tmp = Math.atan((v / (-9.8 * (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.65e-209:
		tmp = math.atan(-1.0)
	elif v <= 1.82e-131:
		tmp = math.atan((v / (-9.8 * (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.65e-209)
		tmp = atan(-1.0);
	elseif (v <= 1.82e-131)
		tmp = atan(Float64(v / Float64(-9.8 * Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.65e-209)
		tmp = atan(-1.0);
	elseif (v <= 1.82e-131)
		tmp = atan((v / (-9.8 * (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.65e-209], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.82e-131], N[ArcTan[N[(v / N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.64999999999999987e-209

   1. Initial program 61.2%

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

      1. sqr-neg 61.2%

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

      2. sqr-neg 61.2%

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

      3. metadata-eval 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in v around -inf 75.4%

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

   if -1.64999999999999987e-209 < v < 1.8200000000000001e-131

   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 H around 0 28.8%

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

   6. Taylor expanded in v around 0 28.8%

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

   if 1.8200000000000001e-131 < v

   1. Initial program 71.4%

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

      1. sqr-neg 71.4%

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

      2. sqr-neg 71.4%

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

      3. metadata-eval 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in v around inf 75.4%

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

Alternative 8: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.5 \cdot 10^{-210}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.5e-210) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.5e-210) {
		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 <= (-6.5d-210)) 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 <= -6.5e-210) {
		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 <= -6.5e-210:
		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 <= -6.5e-210)
		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 <= -6.5e-210)
		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, -6.5e-210], 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 < -6.49999999999999961e-210

   1. Initial program 61.2%

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

      1. sqr-neg 61.2%

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

      2. sqr-neg 61.2%

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

      3. metadata-eval 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in v around -inf 75.4%

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

   if -6.49999999999999961e-210 < v

   1. Initial program 81.2%

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

      1. sqr-neg 81.2%

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

      2. sqr-neg 81.2%

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

      3. metadata-eval 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in H around 0 59.7%

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

Alternative 9: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.15 \cdot 10^{-300}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 1.15e-300) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 1.15e-300) {
		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 <= 1.15d-300) 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 <= 1.15e-300) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 1.15e-300:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 1.15e-300)
		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 <= 1.15e-300)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 1.15e-300], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.15e-300

   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 64.8%

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

   if 1.15e-300 < v

   1. Initial program 78.1%

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

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. metadata-eval 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in v around inf 58.4%

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

Alternative 10: 34.9% 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 72.2%

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

   1. sqr-neg 72.2%

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

   2. sqr-neg 72.2%

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

   3. metadata-eval 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in v around -inf 35.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))