Optimal throwing angle

Percentage Accurate: 67.9% → 99.6%

Time: 6.5s

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: 67.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{-1}{\frac{0.05102040816326531}{H}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan (/ v (- (fma (/ H v) -9.8 v))))
   (if (<= v 2e+111)
     (atan (/ v (sqrt (fma v v (/ -1.0 (/ 0.05102040816326531 H))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan((v / -fma((H / v), -9.8, v)));
	} else if (v <= 2e+111) {
		tmp = atan((v / sqrt(fma(v, v, (-1.0 / (0.05102040816326531 / H))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(Float64(v / Float64(-fma(Float64(H / v), -9.8, v))));
	elseif (v <= 2e+111)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(-1.0 / Float64(0.05102040816326531 / H))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[N[(v / (-N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e+111], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(-1.0 / N[(0.05102040816326531 / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. *-inverses N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if -2.00000000000000007e154 < v < 1.99999999999999991e111

   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. 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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-sub0 N/A

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

      6. flip-- N/A

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

      7. metadata-eval N/A

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

      8. swap-sqr N/A

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

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \left(\color{blue}{\frac{98}{5}} \cdot \left(2 \cdot \frac{49}{5}\right)\right) \cdot \left(H \cdot H\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \left(\frac{98}{5} \cdot \color{blue}{\frac{98}{5}}\right) \cdot \left(H \cdot H\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \color{blue}{\frac{9604}{25}} \cdot \left(H \cdot H\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \color{blue}{\left(\frac{-98}{5} \cdot \frac{-98}{5}\right)} \cdot \left(H \cdot H\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      13. swap-sqr N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \color{blue}{\left(\frac{-98}{5} \cdot H\right) \cdot \left(\frac{-98}{5} \cdot H\right)}}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \color{blue}{\left(\frac{-98}{5} \cdot H\right)} \cdot \left(\frac{-98}{5} \cdot H\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{0 - \left(\frac{-98}{5} \cdot H\right) \cdot \color{blue}{\left(\frac{-98}{5} \cdot H\right)}}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      16. neg-sub0 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \frac{\color{blue}{\mathsf{neg}\left(\left(\frac{-98}{5} \cdot H\right) \cdot \left(\frac{-98}{5} \cdot H\right)\right)}}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{\frac{\mathsf{neg}\left(\left(\frac{-98}{5} \cdot H\right) \cdot \left(\frac{-98}{5} \cdot H\right)\right)}{0 + \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)}}\right) \]

   6. Applied rewrites 82.5%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{\frac{-384.16 \cdot \left(H \cdot H\right)}{0 + 19.6 \cdot H}}\right)}}\right) \]

   8. Applied rewrites 99.8%

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

   if 1.99999999999999991e111 < v

   1. Initial program 25.1%

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

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(9.8 \cdot 2\right) \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan (/ v (- (fma (/ H v) -9.8 v))))
   (if (<= v 2e+111)
     (atan (/ v (sqrt (- (* v v) (* (* 9.8 2.0) H)))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan((v / -fma((H / v), -9.8, v)));
	} else if (v <= 2e+111) {
		tmp = atan((v / sqrt(((v * v) - ((9.8 * 2.0) * H)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(Float64(v / Float64(-fma(Float64(H / v), -9.8, v))));
	elseif (v <= 2e+111)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(9.8 * 2.0) * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[N[(v / (-N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e+111], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(9.8 * 2.0), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. *-inverses N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if -2.00000000000000007e154 < v < 1.99999999999999991e111

   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

   if 1.99999999999999991e111 < v

   1. Initial program 25.1%

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

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(9.8 \cdot 2\right) \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan (/ v (- (fma (/ H v) -9.8 v))))
   (if (<= v 2e+111) (atan (/ v (sqrt (fma v v (* -19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan((v / -fma((H / v), -9.8, v)));
	} else if (v <= 2e+111) {
		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 <= -2e+154)
		tmp = atan(Float64(v / Float64(-fma(Float64(H / v), -9.8, v))));
	elseif (v <= 2e+111)
		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, -2e+154], N[ArcTan[N[(v / (-N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e+111], 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 < -2.00000000000000007e154

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. *-inverses N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if -2.00000000000000007e154 < v < 1.99999999999999991e111

   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. 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 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.99999999999999991e111 < v

   1. Initial program 25.1%

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

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

Alternative 4: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.3 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \mathbf{elif}\;v \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-9.8}{v}, H, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.3e-70)
   (atan (/ v (- (fma (/ H v) -9.8 v))))
   (if (<= v 4.6e-85)
     (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 <= -5.3e-70) {
		tmp = atan((v / -fma((H / v), -9.8, v)));
	} else if (v <= 4.6e-85) {
		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 <= -5.3e-70)
		tmp = atan(Float64(v / Float64(-fma(Float64(H / v), -9.8, v))));
	elseif (v <= 4.6e-85)
		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, -5.3e-70], N[ArcTan[N[(v / (-N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 4.6e-85], 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 < -5.29999999999999983e-70

   1. Initial program 49.8%

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

   3. Taylor expanded in v around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. *-inverses N/A

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

      16. *-lft-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   7. Applied rewrites 92.5%

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

   if -5.29999999999999983e-70 < v < 4.6000000000000001e-85

   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. 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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 97.2%

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

   if 4.6000000000000001e-85 < v

   1. Initial program 57.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. 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 89.5

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

   5. Applied rewrites 89.5%

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

Alternative 5: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.3 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-9.8}{v}, H, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.3e-70)
   (atan -1.0)
   (if (<= v 4.6e-85)
     (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 <= -5.3e-70) {
		tmp = atan(-1.0);
	} else if (v <= 4.6e-85) {
		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 <= -5.3e-70)
		tmp = atan(-1.0);
	elseif (v <= 4.6e-85)
		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, -5.3e-70], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.6e-85], 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 < -5.29999999999999983e-70

   1. Initial program 49.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 91.8%

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

   if -5.29999999999999983e-70 < v < 4.6000000000000001e-85

   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. 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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 97.2%

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

   if 4.6000000000000001e-85 < v

   1. Initial program 57.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. 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 89.5

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

   5. Applied rewrites 89.5%

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

Alternative 6: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.3 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.6 \cdot 10^{-85}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.3e-70)
   (atan -1.0)
   (if (<= v 4.6e-85)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.3e-70) {
		tmp = atan(-1.0);
	} else if (v <= 4.6e-85) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5.3d-70)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.6d-85) then
        tmp = atan((sqrt(((-0.05102040816326531d0) / h)) * v))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5.3e-70) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.6e-85) {
		tmp = Math.atan((Math.sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.3e-70:
		tmp = math.atan(-1.0)
	elif v <= 4.6e-85:
		tmp = math.atan((math.sqrt((-0.05102040816326531 / H)) * v))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.3e-70)
		tmp = atan(-1.0);
	elseif (v <= 4.6e-85)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5.3e-70)
		tmp = atan(-1.0);
	elseif (v <= 4.6e-85)
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.3e-70], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.6e-85], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.29999999999999983e-70

   1. Initial program 49.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 91.8%

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

   if -5.29999999999999983e-70 < v < 4.6000000000000001e-85

   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. 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. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 97.2%

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

   if 4.6000000000000001e-85 < v

   1. Initial program 57.7%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 89.0%

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

Alternative 7: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.5 \cdot 10^{-304}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -6.5e-304) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.5e-304) {
		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 <= (-6.5d-304)) 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 <= -6.5e-304) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -6.5e-304:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.5e-304)
		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 <= -6.5e-304)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.5e-304], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -6.50000000000000011e-304

   1. Initial program 63.3%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 69.0%

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

   if -6.50000000000000011e-304 < v

   1. Initial program 70.2%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 64.5%

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

Alternative 8: 34.8% 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 66.9%

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

3. Taylor expanded in v around -inf

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

5. Applied rewrites 33.9%

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

Reproduce

herbie shell --seed 2024285 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))