Optimal throwing angle

Percentage Accurate: 67.3% → 99.7%

Time: 8.6s

Alternatives: 6

Speedup: 1.3×

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: 67.3% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.25 \cdot 10^{+112}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.2e+154)
   (atan -1.0)
   (if (<= v 1.25e+112) (atan (/ v (sqrt (fma v v (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.2e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1.25e+112) {
		tmp = atan((v / sqrt(fma(v, v, (H * -19.6)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.2e+154)
		tmp = atan(-1.0);
	elseif (v <= 1.25e+112)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.2e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.25e+112], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.20000000000000007e154

   1. Initial program 3.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   if -1.20000000000000007e154 < v < 1.25e112

   1. Initial program 99.8%

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

      1. sub-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)}}}\right) \]

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

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, \mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \mathsf{neg}\left(\color{blue}{H \cdot \left(2 \cdot \frac{49}{5}\right)}\right)\right)}}\right) \]

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

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)}\right)}}\right) \]

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

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)}\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{98}{5}}\right)\right)\right)}}\right) \]

      7. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   if 1.25e112 < v

   1. Initial program 21.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

Alternative 2: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 0.028:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.5e-9)
   (atan -1.0)
   (if (<= v 0.028)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (fma H (/ -9.8 v) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-9) {
		tmp = atan(-1.0);
	} else if (v <= 0.028) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / fma(H, (-9.8 / v), v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.5e-9)
		tmp = atan(-1.0);
	elseif (v <= 0.028)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / fma(H, Float64(-9.8 / v), v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.5e-9], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 0.028], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(H * N[(-9.8 / v), $MachinePrecision] + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -9.5000000000000007e-9

   1. Initial program 48.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 93.4%

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

   if -9.5000000000000007e-9 < v < 0.0280000000000000006

   1. Initial program 99.6%

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

      1. clear-num N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}{v}}\right)} \]

      2. associate-/r/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot v\right)} \]

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

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot v\right)} \]

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in v around 0

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

      1. rem-square-sqrt N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}}}{H}} \cdot v\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\sqrt{\frac{-5}{98}}\right)}^{2}}}{H}} \cdot v\right) \]

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

         \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{\frac{{\left(\sqrt{\frac{-5}{98}}\right)}^{2}}{H}}} \cdot v\right) \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}}}{H}} \cdot v\right) \]

      5. rem-square-sqrt 84.7

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

   7. Simplified 84.7%

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

   if 0.0280000000000000006 < v

   1. Initial program 47.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in H around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{-49}{5} \cdot \frac{H}{v}}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \mathsf{neg}\left(\frac{\color{blue}{\frac{49}{5}}}{v}\right), v\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \color{blue}{\frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}}, v\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{\color{blue}{\frac{-49}{5}}}{v}, v\right)}\right) \]

      14. /-lowering-/.f64 96.8

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

   5. Simplified 96.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 0.028:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 0.0054:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.05e-6)
   (atan -1.0)
   (if (<= v 0.0054)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.05e-6) {
		tmp = atan(-1.0);
	} else if (v <= 0.0054) {
		tmp = atan((v * sqrt((-0.05102040816326531 / 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 <= (-1.05d-6)) then
        tmp = atan((-1.0d0))
    else if (v <= 0.0054d0) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / 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 <= -1.05e-6) {
		tmp = Math.atan(-1.0);
	} else if (v <= 0.0054) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.05e-6:
		tmp = math.atan(-1.0)
	elif v <= 0.0054:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.05e-6)
		tmp = atan(-1.0);
	elseif (v <= 0.0054)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.05e-6)
		tmp = atan(-1.0);
	elseif (v <= 0.0054)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.05e-6], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 0.0054], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.0499999999999999e-6

   1. Initial program 48.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 93.4%

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

   if -1.0499999999999999e-6 < v < 0.0054000000000000003

   1. Initial program 99.6%

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

      1. clear-num N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}{v}}\right)} \]

      2. associate-/r/ N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot v\right)} \]

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

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot v\right)} \]

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in v around 0

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

      1. rem-square-sqrt N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}}}{H}} \cdot v\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\sqrt{\frac{-5}{98}}\right)}^{2}}}{H}} \cdot v\right) \]

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

         \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{\frac{{\left(\sqrt{\frac{-5}{98}}\right)}^{2}}{H}}} \cdot v\right) \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}}}{H}} \cdot v\right) \]

      5. rem-square-sqrt 84.7

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

   7. Simplified 84.7%

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

   if 0.0054000000000000003 < v

   1. Initial program 47.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 96.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.05 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 0.0054:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.4 \cdot 10^{-121}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-121}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.4e-121)
   (atan -1.0)
   (if (<= v 4.2e-121)
     (atan (* v (* -0.10204081632653061 (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.4e-121) {
		tmp = atan(-1.0);
	} else if (v <= 4.2e-121) {
		tmp = atan((v * (-0.10204081632653061 * (v / 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 <= (-3.4d-121)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.2d-121) then
        tmp = atan((v * ((-0.10204081632653061d0) * (v / 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 <= -3.4e-121) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.2e-121) {
		tmp = Math.atan((v * (-0.10204081632653061 * (v / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.4e-121:
		tmp = math.atan(-1.0)
	elif v <= 4.2e-121:
		tmp = math.atan((v * (-0.10204081632653061 * (v / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.4e-121)
		tmp = atan(-1.0);
	elseif (v <= 4.2e-121)
		tmp = atan(Float64(v * Float64(-0.10204081632653061 * Float64(v / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.4e-121)
		tmp = atan(-1.0);
	elseif (v <= 4.2e-121)
		tmp = atan((v * (-0.10204081632653061 * (v / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.4e-121], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.2e-121], N[ArcTan[N[(v * N[(-0.10204081632653061 * N[(v / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.40000000000000001e-121

   1. Initial program 55.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 85.7%

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

   if -3.40000000000000001e-121 < v < 4.1999999999999997e-121

   1. Initial program 99.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in H around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{-49}{5} \cdot \frac{H}{v}}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \mathsf{neg}\left(\frac{\color{blue}{\frac{49}{5}}}{v}\right), v\right)}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \color{blue}{\frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}}, v\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{\color{blue}{\frac{-49}{5}}}{v}, v\right)}\right) \]

      14. /-lowering-/.f64 16.6

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

   5. Simplified 16.6%

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

   6. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-5}{49} \cdot \frac{{v}^{2}}{H}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{{v}^{2}}{H} \cdot \frac{-5}{49}\right)} \]

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

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{{v}^{2}}{H} \cdot \frac{-5}{49}\right)} \]

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

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{{v}^{2}}{H}} \cdot \frac{-5}{49}\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 16.4

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

   8. Simplified 16.4%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{v \cdot v}{H} \cdot -0.10204081632653061\right)} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 16.6

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

   10. Applied egg-rr 16.6%

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

   if 4.1999999999999997e-121 < v

   1. Initial program 59.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 83.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.4 \cdot 10^{-121}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-121}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.65 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.65e-296) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.65e-296) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.65d-296)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.65e-296) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.65e-296:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.65e-296)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.65e-296)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.65e-296], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.65e-296

   1. Initial program 64.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Simplified 69.1%

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

   if -1.65e-296 < v

   1. Initial program 66.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 69.8%

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

Alternative 6: 34.5% accurate, 1.4× 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 65.4%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around -inf

   \[\leadsto \tan^{-1} \color{blue}{-1} \]
4. Step-by-step derivation

5. Simplified 34.1%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))