Optimal throwing angle

Percentage Accurate: 68.4% → 99.6%

Time: 7.7s

Alternatives: 8

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: 68.4% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\tan^{-1} \left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-0.5} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 2e+121)
     (atan (* (pow (fma -19.6 H (* v v)) -0.5) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 2e+121) {
		tmp = atan((pow(fma(-19.6, H, (v * v)), -0.5) * v));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 2e+121)
		tmp = atan(Float64((fma(-19.6, H, Float64(v * v)) ^ -0.5) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+121], N[ArcTan[N[(N[Power[N[(-19.6 * H + N[(v * v), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   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 -1.00000000000000001e155 < v < 2.00000000000000007e121

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   if 2.00000000000000007e121 < v

   1. Initial program 14.5%

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

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(H, -19.6, v \cdot v\right)}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 2e+121)
     (atan (* (sqrt (/ 1.0 (fma H -19.6 (* v v)))) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 2e+121) {
		tmp = atan((sqrt((1.0 / fma(H, -19.6, (v * v)))) * v));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 2e+121)
		tmp = atan(Float64(sqrt(Float64(1.0 / fma(H, -19.6, Float64(v * v)))) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+121], N[ArcTan[N[(N[Sqrt[N[(1.0 / N[(H * -19.6 + N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   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 -1.00000000000000001e155 < v < 2.00000000000000007e121

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{0 - v}}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right) \]

      4. div-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)} - \frac{v}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right)} \]

      5. frac-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)}\right)} \]

      6. sqr-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}}\right) \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

   4. Applied rewrites 93.4%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) - \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) \cdot v}{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right)} \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-atan.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 2.00000000000000007e121 < v

   1. Initial program 14.5%

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

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 2e+121)
     (atan (* (sqrt (/ 1.0 (fma v v (* -19.6 H)))) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 2e+121) {
		tmp = atan((sqrt((1.0 / fma(v, v, (-19.6 * H)))) * v));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 2e+121)
		tmp = atan(Float64(sqrt(Float64(1.0 / fma(v, v, Float64(-19.6 * H)))) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+121], N[ArcTan[N[(N[Sqrt[N[(1.0 / N[(v * v + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   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 -1.00000000000000001e155 < v < 2.00000000000000007e121

   1. Initial program 99.2%

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

   3. Taylor expanded in v around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.00000000000000007e121 < v

   1. Initial program 14.5%

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

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 2e+121) (atan (/ v (sqrt (fma v v (* -19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 2e+121) {
		tmp = atan((v / sqrt(fma(v, v, (-19.6 * H)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 2e+121)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(-19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+121], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   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 -1.00000000000000001e155 < v < 2.00000000000000007e121

   1. Initial program 99.2%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   if 2.00000000000000007e121 < v

   1. Initial program 14.5%

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

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

Alternative 5: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-9.8}{v}, H, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-70)
   (atan -1.0)
   (if (<= v 2e-60)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan (/ v (fma (/ -9.8 v) H v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-70) {
		tmp = atan(-1.0);
	} else if (v <= 2e-60) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan((v / fma((-9.8 / v), H, v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.6e-70)
		tmp = atan(-1.0);
	elseif (v <= 2e-60)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(Float64(v / fma(Float64(-9.8 / v), H, v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.6e-70], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e-60], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(N[(-9.8 / v), $MachinePrecision] * H + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.6000000000000002e-70

   1. Initial program 67.3%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 86.8%

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

   if -3.6000000000000002e-70 < v < 1.9999999999999999e-60

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{0 - v}}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right) \]

      4. div-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)} - \frac{v}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right)} \]

      5. frac-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)}\right)} \]

      6. sqr-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}}\right) \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

   4. Applied rewrites 87.3%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) - \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) \cdot v}{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right)} \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-atan.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 87.6%

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

   if 1.9999999999999999e-60 < v

   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 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. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 92.7

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

   5. Applied rewrites 92.7%

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

Alternative 6: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(9.8, \frac{H}{v \cdot v}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-70)
   (atan -1.0)
   (if (<= v 2e-60)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan (fma 9.8 (/ H (* v v)) 1.0)))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-70) {
		tmp = atan(-1.0);
	} else if (v <= 2e-60) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan(fma(9.8, (H / (v * v)), 1.0));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.6e-70)
		tmp = atan(-1.0);
	elseif (v <= 2e-60)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(fma(9.8, Float64(H / Float64(v * v)), 1.0));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.6e-70], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e-60], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.6000000000000002e-70

   1. Initial program 67.3%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 86.8%

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

   if -3.6000000000000002e-70 < v < 1.9999999999999999e-60

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \tan^{-1} \left(\frac{\color{blue}{0 - v}}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right) \]

      4. div-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)} - \frac{v}{\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}\right)} \]

      5. frac-sub N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)}\right)} \]

      6. sqr-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}}\right) \]

      7. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right)\right)}\right) \]

      8. remove-double-neg N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}} \cdot \sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H} \cdot \color{blue}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{0 \cdot \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) - \left(\mathsf{neg}\left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right) \cdot v}{\color{blue}{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right) \]

   4. Applied rewrites 87.3%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{0 \cdot \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) - \left(-\sqrt{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right) \cdot v}{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}\right)} \]

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. lower-atan.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 87.6%

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

   if 1.9999999999999999e-60 < v

   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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.3

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

   5. Applied rewrites 92.3%

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

Alternative 7: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-279}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5e-279) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-279) {
		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-279)) 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-279) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-279:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-279)
		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-279)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-279], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -4.99999999999999969e-279

   1. Initial program 75.5%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 67.8%

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

   if -4.99999999999999969e-279 < 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. Applied rewrites 64.3%

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

Alternative 8: 34.3% 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 71.1%

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

3. Taylor expanded in v around -inf

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

5. Applied rewrites 32.8%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))