Optimal throwing angle

Percentage Accurate: 67.5% → 99.4%

Time: 8.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))

C

double code(double v, double H) {
	return atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((v / sqrt(((v * v) - ((2.0d0 * 9.8d0) * h)))))
end function

Java

public static double code(double v, double H) {
	return Math.atan((v / Math.sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Python

def code(v, H):
	return math.atan((v / math.sqrt(((v * v) - ((2.0 * 9.8) * H)))))

Julia

function code(v, H)
	return atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
end

MATLAB

function tmp = code(v, H)
	tmp = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
end

Wolfram

code[v_, H_] := N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 67.5% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.95 \cdot 10^{+117}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+156)
   (atan -1.0)
   (if (<= v 1.95e+117) (atan (/ v (sqrt (fma v v (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+156) {
		tmp = atan(-1.0);
	} else if (v <= 1.95e+117) {
		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 <= -2e+156)
		tmp = atan(-1.0);
	elseif (v <= 1.95e+117)
		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, -2e+156], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.95e+117], 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 < -2e156

   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 -2e156 < v < 1.94999999999999995e117

   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 1.94999999999999995e117 < v

   1. Initial program 20.6%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 100.0%

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

Alternative 2: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9 \cdot 10^{-126}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.31 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9e-126)
   (atan -1.0)
   (if (<= v 1.31e-17)
     (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 <= -9e-126) {
		tmp = atan(-1.0);
	} else if (v <= 1.31e-17) {
		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 <= -9e-126)
		tmp = atan(-1.0);
	elseif (v <= 1.31e-17)
		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, -9e-126], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.31e-17], 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 < -9.0000000000000005e-126

   1. Initial program 63.6%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 84.5%

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

   if -9.0000000000000005e-126 < v < 1.31e-17

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 99.6%

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

   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 88.5

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

   7. Simplified 88.5%

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

   if 1.31e-17 < v

   1. Initial program 54.8%

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

   3. Taylor expanded in 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 86.6

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

   5. Simplified 86.6%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -9 \cdot 10^{-126}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.31 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9 \cdot 10^{-126}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.31 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9e-126)
   (atan -1.0)
   (if (<= v 1.31e-17)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9e-126) {
		tmp = atan(-1.0);
	} else if (v <= 1.31e-17) {
		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 <= (-9d-126)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.31d-17) 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 <= -9e-126) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.31e-17) {
		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 <= -9e-126:
		tmp = math.atan(-1.0)
	elif v <= 1.31e-17:
		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 <= -9e-126)
		tmp = atan(-1.0);
	elseif (v <= 1.31e-17)
		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 <= -9e-126)
		tmp = atan(-1.0);
	elseif (v <= 1.31e-17)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9e-126], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.31e-17], 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 < -9.0000000000000005e-126

   1. Initial program 63.6%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 84.5%

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

   if -9.0000000000000005e-126 < v < 1.31e-17

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 99.6%

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

   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 88.5

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

   7. Simplified 88.5%

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

   if 1.31e-17 < v

   1. Initial program 54.8%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 86.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -9 \cdot 10^{-126}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.31 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.12 \cdot 10^{-151}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v \cdot v}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.12e-151)
   (atan -1.0)
   (if (<= v 8e-136)
     (atan (* -0.10204081632653061 (/ (* v v) H)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.12e-151) {
		tmp = atan(-1.0);
	} else if (v <= 8e-136) {
		tmp = atan((-0.10204081632653061 * ((v * 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 <= (-1.12d-151)) then
        tmp = atan((-1.0d0))
    else if (v <= 8d-136) then
        tmp = atan(((-0.10204081632653061d0) * ((v * 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 <= -1.12e-151) {
		tmp = Math.atan(-1.0);
	} else if (v <= 8e-136) {
		tmp = Math.atan((-0.10204081632653061 * ((v * v) / H)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.12e-151:
		tmp = math.atan(-1.0)
	elif v <= 8e-136:
		tmp = math.atan((-0.10204081632653061 * ((v * v) / H)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.12e-151)
		tmp = atan(-1.0);
	elseif (v <= 8e-136)
		tmp = atan(Float64(-0.10204081632653061 * Float64(Float64(v * v) / H)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.12e-151)
		tmp = atan(-1.0);
	elseif (v <= 8e-136)
		tmp = atan((-0.10204081632653061 * ((v * v) / H)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.12e-151], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 8e-136], N[ArcTan[N[(-0.10204081632653061 * N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.11999999999999994e-151

   1. Initial program 64.8%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 81.8%

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

   if -1.11999999999999994e-151 < v < 8.00000000000000001e-136

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 35.6

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

   5. Simplified 35.6%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 35.6

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

   8. Simplified 35.6%

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

   if 8.00000000000000001e-136 < v

   1. Initial program 61.7%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 80.1%

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

Alternative 5: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 5e-296) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 5e-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 <= 5d-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 <= 5e-296) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 5e-296:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 5e-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 <= 5e-296)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 5e-296], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 5.0000000000000003e-296

   1. Initial program 69.7%

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

   3. Taylor expanded in v around -inf

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

   5. Simplified 70.8%

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

   if 5.0000000000000003e-296 < v

   1. Initial program 67.1%

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

   3. Taylor expanded in v around inf

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

   5. Simplified 69.4%

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

Alternative 6: 35.0% 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 68.5%

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

3. Taylor expanded in v around -inf

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

5. Simplified 38.2%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))