Optimal throwing angle

Percentage Accurate: 68.0% → 99.6%

Time: 8.5s

Alternatives: 7

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: 68.0% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+125}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan -1.0)
   (if (<= v 3e+125)
     (atan (* v (sqrt (/ 1.0 (fma v v (* H -19.6))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan(-1.0);
	} else if (v <= 3e+125) {
		tmp = atan((v * sqrt((1.0 / fma(v, v, (H * -19.6))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+154)
		tmp = atan(-1.0);
	elseif (v <= 3e+125)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / fma(v, v, Float64(H * -19.6))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3e+125], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / N[(v * v + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000004e154

   1. Initial program 5.5%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000004e154 < v < 3.00000000000000015e125

   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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.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 rewrites 99.8%

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

   if 3.00000000000000015e125 < v

   1. Initial program 18.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+125}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+125}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.4e+154)
   (atan -1.0)
   (if (<= v 3e+125) (atan (/ v (sqrt (fma v v (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.4e+154) {
		tmp = atan(-1.0);
	} else if (v <= 3e+125) {
		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 <= -1.4e+154)
		tmp = atan(-1.0);
	elseif (v <= 3e+125)
		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, -1.4e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3e+125], 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 < -1.4e154

   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. Applied rewrites 100.0%

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

   if -1.4e154 < v < 3.00000000000000015e125

   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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   if 3.00000000000000015e125 < v

   1. Initial program 18.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 100.0%

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

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.5 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.6e-76)
   (atan -1.0)
   (if (<= v 3.5e-82)
     (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 <= -6.6e-76) {
		tmp = atan(-1.0);
	} else if (v <= 3.5e-82) {
		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 <= -6.6e-76)
		tmp = atan(-1.0);
	elseif (v <= 3.5e-82)
		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, -6.6e-76], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3.5e-82], 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 < -6.59999999999999967e-76

   1. Initial program 54.9%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 92.7%

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

   if -6.59999999999999967e-76 < v < 3.4999999999999999e-82

   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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.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 rewrites 99.7%

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

   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. lower-/.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 97.4

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

   7. Applied rewrites 97.4%

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

   if 3.4999999999999999e-82 < v

   1. Initial program 57.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. 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. lower-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. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.5 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.7 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.6e-76)
   (atan -1.0)
   (if (<= v 3.7e-82)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.6e-76) {
		tmp = atan(-1.0);
	} else if (v <= 3.7e-82) {
		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 <= (-6.6d-76)) then
        tmp = atan((-1.0d0))
    else if (v <= 3.7d-82) 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 <= -6.6e-76) {
		tmp = Math.atan(-1.0);
	} else if (v <= 3.7e-82) {
		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 <= -6.6e-76:
		tmp = math.atan(-1.0)
	elif v <= 3.7e-82:
		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 <= -6.6e-76)
		tmp = atan(-1.0);
	elseif (v <= 3.7e-82)
		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 <= -6.6e-76)
		tmp = atan(-1.0);
	elseif (v <= 3.7e-82)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.6e-76], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3.7e-82], 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 < -6.59999999999999967e-76

   1. Initial program 54.9%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 92.7%

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

   if -6.59999999999999967e-76 < v < 3.7000000000000001e-82

   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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.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 rewrites 99.7%

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

   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. lower-/.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 97.4

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

   7. Applied rewrites 97.4%

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

   if 3.7000000000000001e-82 < v

   1. Initial program 57.4%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 88.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.7 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.5 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.1 \cdot 10^{-102}:\\ \;\;\;\;\tan^{-1} \left(\left(v \cdot -0.10204081632653061\right) \cdot \frac{v}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.5e-185)
   (atan -1.0)
   (if (<= v 4.1e-102)
     (atan (* (* v -0.10204081632653061) (/ v H)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.5e-185) {
		tmp = atan(-1.0);
	} else if (v <= 4.1e-102) {
		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 <= (-1.5d-185)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.1d-102) 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 <= -1.5e-185) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.1e-102) {
		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 <= -1.5e-185:
		tmp = math.atan(-1.0)
	elif v <= 4.1e-102:
		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 <= -1.5e-185)
		tmp = atan(-1.0);
	elseif (v <= 4.1e-102)
		tmp = atan(Float64(Float64(v * -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 <= -1.5e-185)
		tmp = atan(-1.0);
	elseif (v <= 4.1e-102)
		tmp = atan(((v * -0.10204081632653061) * (v / H)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.5e-185], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.1e-102], N[ArcTan[N[(N[(v * -0.10204081632653061), $MachinePrecision] * N[(v / H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.50000000000000015e-185

   1. Initial program 61.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 79.9%

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

   if -1.50000000000000015e-185 < v < 4.1000000000000003e-102

   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. lower-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. lower-/.f64 29.9

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

   5. Applied rewrites 29.9%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 29.8

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

   8. Applied rewrites 29.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 29.9

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

   10. Applied rewrites 29.9%

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

   if 4.1000000000000003e-102 < v

   1. Initial program 58.6%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 87.3%

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

Alternative 6: 67.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d-310)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if v < -1.999999999999994e-310

   1. Initial program 69.0%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 65.4%

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

   if -1.999999999999994e-310 < v

   1. Initial program 67.8%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 68.9%

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

Alternative 7: 34.2% 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 68.4%

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

3. Taylor expanded in v around -inf

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

5. Applied rewrites 33.1%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))