Optimal throwing angle

Percentage Accurate: 67.4% → 99.7%

Time: 13.2s

Alternatives: 11

Speedup: 1.9×

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: 67.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(v \cdot v + H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 1e+127)
     (atan (* v (pow (+ (* v v) (* H -19.6)) -0.5)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1e+127) {
		tmp = atan((v * pow(((v * v) + (H * -19.6)), -0.5)));
	} 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+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 1d+127) then
        tmp = atan((v * (((v * v) + (h * (-19.6d0))) ** (-0.5d0))))
    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+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1e+127) {
		tmp = Math.atan((v * Math.pow(((v * v) + (H * -19.6)), -0.5)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 1e+127:
		tmp = math.atan((v * math.pow(((v * v) + (H * -19.6)), -0.5)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+127)
		tmp = atan(Float64(v * (Float64(Float64(v * v) + Float64(H * -19.6)) ^ -0.5)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+127)
		tmp = atan((v * (((v * v) + (H * -19.6)) ^ -0.5)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+127], N[ArcTan[N[(v * N[Power[N[(N[(v * v), $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.00000000000000004e154

   1. Initial program 3.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 3.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

   if -5.00000000000000004e154 < v < 9.99999999999999955e126

   1. Initial program 99.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}}{v}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}} \cdot v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}}\right), v\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{{\left(v \cdot v + H \cdot \frac{-98}{5}\right)}^{\frac{1}{2}}}\right), v\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left({\left(v \cdot v + H \cdot \frac{-98}{5}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(v \cdot v + H \cdot \frac{-98}{5}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

      10. metadata-eval 99.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right), \frac{-1}{2}\right), v\right)\right) \]

   6. Applied egg-rr 99.8%

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

   if 9.99999999999999955e126 < v

   1. Initial program 32.2%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 32.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 32.2%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(v \cdot v + H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 2e+127) (atan (/ v (sqrt (+ (* v v) (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+127) {
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	} 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+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+127) then
        tmp = atan((v / sqrt(((v * v) + (h * (-19.6d0))))))
    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+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+127) {
		tmp = Math.atan((v / Math.sqrt(((v * v) + (H * -19.6)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+127:
		tmp = math.atan((v / math.sqrt(((v * v) + (H * -19.6)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+127)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) + Float64(H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+127)
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+127], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.00000000000000004e154

   1. Initial program 3.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 3.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

   if -5.00000000000000004e154 < v < 1.99999999999999991e127

   1. Initial program 99.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   if 1.99999999999999991e127 < v

   1. Initial program 32.2%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 32.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 32.2%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

Alternative 3: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.2 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{{\left(\frac{-0.05102040816326531}{H}\right)}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.2e-18)
   (atan (/ v (- (* H (+ (/ 9.8 v) (* (/ H (* v (* v v))) 48.02))) v)))
   (if (<= v 2e-26)
     (atan (/ v (pow (/ -0.05102040816326531 H) -0.5)))
     (atan (/ v (* v (+ 1.0 (* (/ H (* v v)) -9.8))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.2e-18) {
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 2e-26) {
		tmp = atan((v / pow((-0.05102040816326531 / H), -0.5)));
	} else {
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5.2d-18)) then
        tmp = atan((v / ((h * ((9.8d0 / v) + ((h / (v * (v * v))) * 48.02d0))) - v)))
    else if (v <= 2d-26) then
        tmp = atan((v / (((-0.05102040816326531d0) / h) ** (-0.5d0))))
    else
        tmp = atan((v / (v * (1.0d0 + ((h / (v * v)) * (-9.8d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5.2e-18) {
		tmp = Math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 2e-26) {
		tmp = Math.atan((v / Math.pow((-0.05102040816326531 / H), -0.5)));
	} else {
		tmp = Math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.2e-18:
		tmp = math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)))
	elif v <= 2e-26:
		tmp = math.atan((v / math.pow((-0.05102040816326531 / H), -0.5)))
	else:
		tmp = math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.2e-18)
		tmp = atan(Float64(v / Float64(Float64(H * Float64(Float64(9.8 / v) + Float64(Float64(H / Float64(v * Float64(v * v))) * 48.02))) - v)));
	elseif (v <= 2e-26)
		tmp = atan(Float64(v / (Float64(-0.05102040816326531 / H) ^ -0.5)));
	else
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / Float64(v * v)) * -9.8)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5.2e-18)
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	elseif (v <= 2e-26)
		tmp = atan((v / ((-0.05102040816326531 / H) ^ -0.5)));
	else
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.2e-18], N[ArcTan[N[(v / N[(N[(H * N[(N[(9.8 / v), $MachinePrecision] + N[(N[(H / N[(v * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 48.02), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e-26], N[ArcTan[N[(v / N[Power[N[(-0.05102040816326531 / H), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.2000000000000001e-18

   1. Initial program 51.5%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 51.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{-1 \cdot \left(H \cdot \frac{-9.8}{v}\right) - v \cdot \left(1 + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)}}\right) \]

   8. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(H \cdot \left(\frac{2401}{50} \cdot \frac{H}{{v}^{3}} + \frac{49}{5} \cdot \frac{1}{v}\right) - v\right)}\right)\right) \]

   10. Simplified 95.3%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}}\right) \]

   if -5.2000000000000001e-18 < v < 2.0000000000000001e-26

   1. Initial program 99.6%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{1}{\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}}}\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{{\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{-1}}\right)\right)\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\frac{-1}{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{v \cdot v + H \cdot \frac{-98}{5}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(v \cdot v + H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      14. metadata-eval 99.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in v around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\color{blue}{\left(\frac{\frac{-5}{98}}{H}\right)}, \frac{-1}{2}\right)\right)\right) \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}}{H}\right), \frac{-1}{2}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{{\left(\sqrt{\frac{-5}{98}}\right)}^{2}}{H}\right), \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left({\left(\sqrt{\frac{-5}{98}}\right)}^{2}\right), H\right), \frac{-1}{2}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{-5}{98}} \cdot \sqrt{\frac{-5}{98}}\right), H\right), \frac{-1}{2}\right)\right)\right) \]

      5. rem-square-sqrt 91.9%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\frac{-5}{98}, H\right), \frac{-1}{2}\right)\right)\right) \]

   9. Simplified 91.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{{\color{blue}{\left(\frac{-0.05102040816326531}{H}\right)}}^{-0.5}}\right) \]

   if 2.0000000000000001e-26 < v

   1. Initial program 60.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{1}{\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}}}\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{{\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{-1}}\right)\right)\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\frac{-1}{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{v \cdot v + H \cdot \frac{-98}{5}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(v \cdot v + H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      14. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 90.8%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

   9. Simplified 90.8%

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

Alternative 4: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}\right)\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.5e-5)
   (atan (/ v (- (* H (+ (/ 9.8 v) (* (/ H (* v (* v v))) 48.02))) v)))
   (if (<= v 5.2e-40)
     (atan (* v (pow (* H -19.6) -0.5)))
     (atan (/ v (* v (+ 1.0 (* (/ H (* v v)) -9.8))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-5) {
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 5.2e-40) {
		tmp = atan((v * pow((H * -19.6), -0.5)));
	} else {
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-9.5d-5)) then
        tmp = atan((v / ((h * ((9.8d0 / v) + ((h / (v * (v * v))) * 48.02d0))) - v)))
    else if (v <= 5.2d-40) then
        tmp = atan((v * ((h * (-19.6d0)) ** (-0.5d0))))
    else
        tmp = atan((v / (v * (1.0d0 + ((h / (v * v)) * (-9.8d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -9.5e-5) {
		tmp = Math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 5.2e-40) {
		tmp = Math.atan((v * Math.pow((H * -19.6), -0.5)));
	} else {
		tmp = Math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.5e-5:
		tmp = math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)))
	elif v <= 5.2e-40:
		tmp = math.atan((v * math.pow((H * -19.6), -0.5)))
	else:
		tmp = math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.5e-5)
		tmp = atan(Float64(v / Float64(Float64(H * Float64(Float64(9.8 / v) + Float64(Float64(H / Float64(v * Float64(v * v))) * 48.02))) - v)));
	elseif (v <= 5.2e-40)
		tmp = atan(Float64(v * (Float64(H * -19.6) ^ -0.5)));
	else
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / Float64(v * v)) * -9.8)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.5e-5)
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	elseif (v <= 5.2e-40)
		tmp = atan((v * ((H * -19.6) ^ -0.5)));
	else
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.5e-5], N[ArcTan[N[(v / N[(N[(H * N[(N[(9.8 / v), $MachinePrecision] + N[(N[(H / N[(v * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 48.02), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5.2e-40], N[ArcTan[N[(v * N[Power[N[(H * -19.6), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -9.5000000000000005e-5

   1. Initial program 51.5%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 51.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{-1 \cdot \left(H \cdot \frac{-9.8}{v}\right) - v \cdot \left(1 + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)}}\right) \]

   8. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(H \cdot \left(\frac{2401}{50} \cdot \frac{H}{{v}^{3}} + \frac{49}{5} \cdot \frac{1}{v}\right) - v\right)}\right)\right) \]

   10. Simplified 95.3%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}}\right) \]

   if -9.5000000000000005e-5 < v < 5.2000000000000003e-40

   1. Initial program 99.6%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}}{v}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}} \cdot v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\sqrt{v \cdot v + H \cdot \frac{-98}{5}}}\right), v\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{{\left(v \cdot v + H \cdot \frac{-98}{5}\right)}^{\frac{1}{2}}}\right), v\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left({\left(v \cdot v + H \cdot \frac{-98}{5}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(v \cdot v + H \cdot \frac{-98}{5}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), v\right)\right) \]

      10. metadata-eval 99.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right), \frac{-1}{2}\right), v\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in v around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(\frac{-98}{5} \cdot H\right)}, \frac{-1}{2}\right), v\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 91.9%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right), \frac{-1}{2}\right), v\right)\right) \]

   9. Simplified 91.9%

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

   if 5.2000000000000003e-40 < v

   1. Initial program 60.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{1}{\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}}}\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{{\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{-1}}\right)\right)\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\frac{-1}{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{v \cdot v + H \cdot \frac{-98}{5}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(v \cdot v + H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      14. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 90.8%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

   9. Simplified 90.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}\right)\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}\right)\\ \mathbf{elif}\;v \leq 1.08 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.4e-16)
   (atan (/ v (- (* H (+ (/ 9.8 v) (* (/ H (* v (* v v))) 48.02))) v)))
   (if (<= v 1.08e-35)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (/ v (* v (+ 1.0 (* (/ H (* v v)) -9.8))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.4e-16) {
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 1.08e-35) {
		tmp = atan((v / sqrt((H * -19.6))));
	} else {
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-4.4d-16)) then
        tmp = atan((v / ((h * ((9.8d0 / v) + ((h / (v * (v * v))) * 48.02d0))) - v)))
    else if (v <= 1.08d-35) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    else
        tmp = atan((v / (v * (1.0d0 + ((h / (v * v)) * (-9.8d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -4.4e-16) {
		tmp = Math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	} else if (v <= 1.08e-35) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4.4e-16:
		tmp = math.atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)))
	elif v <= 1.08e-35:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.4e-16)
		tmp = atan(Float64(v / Float64(Float64(H * Float64(Float64(9.8 / v) + Float64(Float64(H / Float64(v * Float64(v * v))) * 48.02))) - v)));
	elseif (v <= 1.08e-35)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	else
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / Float64(v * v)) * -9.8)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4.4e-16)
		tmp = atan((v / ((H * ((9.8 / v) + ((H / (v * (v * v))) * 48.02))) - v)));
	elseif (v <= 1.08e-35)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.4e-16], N[ArcTan[N[(v / N[(N[(H * N[(N[(9.8 / v), $MachinePrecision] + N[(N[(H / N[(v * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 48.02), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.08e-35], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.40000000000000001e-16

   1. Initial program 51.5%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 51.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 51.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 83.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{-1 \cdot \left(H \cdot \frac{-9.8}{v}\right) - v \cdot \left(1 + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right)}}\right) \]

   8. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(H \cdot \left(\frac{2401}{50} \cdot \frac{H}{{v}^{3}} + \frac{49}{5} \cdot \frac{1}{v}\right) - v\right)}\right)\right) \]

   10. Simplified 95.3%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}}\right) \]

   if -4.40000000000000001e-16 < v < 1.08000000000000003e-35

   1. Initial program 99.6%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{-98}{5} \cdot H\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\frac{-98}{5}, H\right)\right)\right)\right) \]

   7. Simplified 91.8%

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

   if 1.08000000000000003e-35 < v

   1. Initial program 60.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 60.1%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{\frac{1}{\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}}}\right)\right)\right) \]

      3. inv-pow N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{{\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{-1}}\right)\right)\right) \]

      4. sqrt-pow1 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\frac{-1}{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left({\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{v \cdot v - H \cdot \frac{-98}{5}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - \left(H \cdot \frac{-98}{5}\right) \cdot \left(H \cdot \frac{-98}{5}\right)}{v \cdot v - H \cdot \frac{-98}{5}}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      9. flip-+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\left(\frac{1}{v \cdot v + H \cdot \frac{-98}{5}}\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(v \cdot v + H \cdot \frac{-98}{5}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(v \cdot v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right) \]

      14. metadata-eval 60.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   6. Applied egg-rr 60.1%

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

   7. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 90.8%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right) \]

   9. Simplified 90.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \left(\frac{9.8}{v} + \frac{H}{v \cdot \left(v \cdot v\right)} \cdot 48.02\right) - v}\right)\\ \mathbf{elif}\;v \leq 1.08 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.3 \cdot 10^{-113}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.4 \cdot 10^{-133}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{0.10204081632653061}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.3e-113)
   (atan -1.0)
   (if (<= v 3.4e-133)
     (atan (* v (/ 0.10204081632653061 (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.3e-113) {
		tmp = atan(-1.0);
	} else if (v <= 3.4e-133) {
		tmp = atan((v * (0.10204081632653061 / (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 <= (-4.3d-113)) then
        tmp = atan((-1.0d0))
    else if (v <= 3.4d-133) then
        tmp = atan((v * (0.10204081632653061d0 / (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 <= -4.3e-113) {
		tmp = Math.atan(-1.0);
	} else if (v <= 3.4e-133) {
		tmp = Math.atan((v * (0.10204081632653061 / (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4.3e-113:
		tmp = math.atan(-1.0)
	elif v <= 3.4e-133:
		tmp = math.atan((v * (0.10204081632653061 / (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.3e-113)
		tmp = atan(-1.0);
	elseif (v <= 3.4e-133)
		tmp = atan(Float64(v * Float64(0.10204081632653061 / Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4.3e-113)
		tmp = atan(-1.0);
	elseif (v <= 3.4e-133)
		tmp = atan((v * (0.10204081632653061 / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.3e-113], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3.4e-133], N[ArcTan[N[(v * N[(0.10204081632653061 / N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.3e-113

   1. Initial program 59.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 59.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 83.7%

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

   if -4.3e-113 < v < 3.40000000000000006e-133

   1. Initial program 99.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + 1\right)\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v \cdot 1\right)\right)\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v\right)\right)\right)\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) + \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) - v\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right), v\right)\right)\right) \]

   7. Simplified 23.8%

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

   8. Taylor expanded in v around 0

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(\frac{5}{49} \cdot \frac{{v}^{2}}{H}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{{v}^{2}}{H} \cdot \frac{5}{49}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{{v}^{2}}{H}\right), \frac{5}{49}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({v}^{2}\right), H\right), \frac{5}{49}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(v \cdot v\right), H\right), \frac{5}{49}\right)\right) \]

      5. *-lowering-*.f64 23.5%

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(v, v\right), H\right), \frac{5}{49}\right)\right) \]

   10. Simplified 23.5%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\left(v \cdot \frac{v}{H}\right) \cdot \frac{5}{49}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \left(\frac{v}{H} \cdot \frac{5}{49}\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \left(\frac{v}{H} \cdot \frac{\frac{-5}{49}}{-1}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \left(\frac{v}{H} \cdot \frac{\mathsf{neg}\left(\frac{5}{49}\right)}{-1}\right)\right)\right) \]

       5. times-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \frac{v \cdot \left(\mathsf{neg}\left(\frac{5}{49}\right)\right)}{H \cdot -1}\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(v \cdot \frac{5}{49}\right)}{H \cdot -1}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(v \cdot \frac{5}{49}\right)}{-1 \cdot H}\right)\right) \]

       8. neg-mul-1 N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \frac{\mathsf{neg}\left(v \cdot \frac{5}{49}\right)}{\mathsf{neg}\left(H\right)}\right)\right) \]

       9. frac-2neg N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(v \cdot \frac{v \cdot \frac{5}{49}}{H}\right)\right) \]

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

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{v \cdot \frac{5}{49}}{H}\right)\right)\right) \]

       11. clear-num N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{1}{\frac{H}{v \cdot \frac{5}{49}}}\right)\right)\right) \]

       12. associate-/r* N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{1}{\frac{\frac{H}{v}}{\frac{5}{49}}}\right)\right)\right) \]

       13. clear-num N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \left(\frac{\frac{5}{49}}{\frac{H}{v}}\right)\right)\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(\frac{5}{49}, \left(\frac{H}{v}\right)\right)\right)\right) \]

       15. /-lowering-/.f64 23.8%

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(v, \mathsf{/.f64}\left(\frac{5}{49}, \mathsf{/.f64}\left(H, v\right)\right)\right)\right) \]

   12. Applied egg-rr 23.8%

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

   if 3.40000000000000006e-133 < v

   1. Initial program 66.9%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 78.7%

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

Alternative 7: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + H \cdot \frac{-9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-291)
   (atan (* v (/ 1.0 (- (* 9.8 (/ H v)) v))))
   (atan (/ v (+ v (* H (/ -9.8 v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-291) {
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else {
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d-291)) then
        tmp = atan((v * (1.0d0 / ((9.8d0 * (h / v)) - v))))
    else
        tmp = atan((v / (v + (h * ((-9.8d0) / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e-291) {
		tmp = Math.atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else {
		tmp = Math.atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-291:
		tmp = math.atan((v * (1.0 / ((9.8 * (H / v)) - v))))
	else:
		tmp = math.atan((v / (v + (H * (-9.8 / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-291)
		tmp = atan(Float64(v * Float64(1.0 / Float64(Float64(9.8 * Float64(H / v)) - v))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(H * Float64(-9.8 / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e-291)
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	else
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-291], N[ArcTan[N[(v * N[(1.0 / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(H * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.99999999999999992e-291

   1. Initial program 68.6%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 68.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 68.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + 1\right)\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v \cdot 1\right)\right)\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v\right)\right)\right)\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) + \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) - v\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right), v\right)\right)\right) \]

   7. Simplified 68.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{-1 \cdot \left(H \cdot \frac{\frac{-49}{5}}{v}\right) - v}{v}}\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{-1 \cdot \left(H \cdot \frac{\frac{-49}{5}}{v}\right) - v} \cdot v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{-1 \cdot \left(H \cdot \frac{\frac{-49}{5}}{v}\right) - v}\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(-1 \cdot \left(H \cdot \frac{\frac{-49}{5}}{v}\right) - v\right)\right), v\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(-1 \cdot \left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right), v\right)\right), v\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(H \cdot \frac{\frac{-49}{5}}{v}\right)\right), v\right)\right), v\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{H \cdot \frac{-49}{5}}{v}\right)\right), v\right)\right), v\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{H \cdot \frac{-49}{5}}{\mathsf{neg}\left(v\right)}\right), v\right)\right), v\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{\frac{-49}{5} \cdot H}{\mathsf{neg}\left(v\right)}\right), v\right)\right), v\right)\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{\frac{-49}{5} \cdot H}{-1 \cdot v}\right), v\right)\right), v\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{\frac{-49}{5}}{-1} \cdot \frac{H}{v}\right), v\right)\right), v\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\frac{49}{5} \cdot \frac{H}{v}\right), v\right)\right), v\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\frac{-49}{5}\right)\right) \cdot \frac{H}{v}\right), v\right)\right), v\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{-49}{5}\right)\right), \left(\frac{H}{v}\right)\right), v\right)\right), v\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{49}{5}, \left(\frac{H}{v}\right)\right), v\right)\right), v\right)\right) \]

      16. /-lowering-/.f64 68.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{49}{5}, \mathsf{/.f64}\left(H, v\right)\right), v\right)\right), v\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if -1.99999999999999992e-291 < v

   1. Initial program 74.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 74.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\frac{-49}{5}}{v}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 68.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{-49}{5}, v\right)\right)\right)\right)\right) \]

   7. Simplified 68.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + H \cdot \frac{-9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + H \cdot \frac{-9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + H \cdot \frac{-9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-291)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (atan (/ v (+ v (* H (/ -9.8 v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-291) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d-291)) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else
        tmp = atan((v / (v + (h * ((-9.8d0) / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e-291) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = Math.atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-291:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	else:
		tmp = math.atan((v / (v + (H * (-9.8 / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-291)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	else
		tmp = atan(Float64(v / Float64(v + Float64(H * Float64(-9.8 / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e-291)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	else
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-291], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(H * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.99999999999999992e-291

   1. Initial program 68.6%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 68.6%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 68.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + 1\right)\right)\right)\right)\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v \cdot 1\right)\right)\right)\right)\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\mathsf{neg}\left(\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right) + v\right)\right)\right)\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) + \left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right) - v\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(v \cdot \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right), v\right)\right)\right) \]

   7. Simplified 68.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\left(-1 \cdot H\right) \cdot \frac{\frac{-49}{5}}{v}\right), v\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\left(-1 \cdot H\right) \cdot \frac{1}{\frac{v}{\frac{-49}{5}}}\right), v\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{-1 \cdot H}{\frac{v}{\frac{-49}{5}}}\right), v\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H \cdot -1}{\frac{v}{\frac{-49}{5}}}\right), v\right)\right)\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H \cdot -1}{v \cdot \frac{1}{\frac{-49}{5}}}\right), v\right)\right)\right) \]

      6. times-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H}{v} \cdot \frac{-1}{\frac{1}{\frac{-49}{5}}}\right), v\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H}{v} \cdot \frac{-1}{\frac{-5}{49}}\right), v\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H}{v} \cdot \frac{49}{5}\right), v\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(\frac{H}{v} \cdot \left(\mathsf{neg}\left(\frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{H}{v}\right), \left(\mathsf{neg}\left(\frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(H, v\right), \left(\mathsf{neg}\left(\frac{-49}{5}\right)\right)\right), v\right)\right)\right) \]

      12. metadata-eval 68.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(H, v\right), \frac{49}{5}\right), v\right)\right)\right) \]

   9. Applied egg-rr 68.2%

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

   if -1.99999999999999992e-291 < v

   1. Initial program 74.7%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 74.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\frac{-49}{5}}{v}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 68.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{-49}{5}, v\right)\right)\right)\right)\right) \]

   7. Simplified 68.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + H \cdot \frac{-9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + H \cdot \frac{-9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -8.5 \cdot 10^{-217}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + H \cdot \frac{-9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -8.5e-217) (atan -1.0) (atan (/ v (+ v (* H (/ -9.8 v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -8.5e-217) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-8.5d-217)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v / (v + (h * ((-9.8d0) / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -8.5e-217) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + (H * (-9.8 / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -8.5e-217:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + (H * (-9.8 / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -8.5e-217)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(H * Float64(-9.8 / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -8.5e-217)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + (H * (-9.8 / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -8.5e-217], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(H * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -8.4999999999999994e-217

   1. Initial program 66.3%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 66.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 70.5%

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

   if -8.4999999999999994e-217 < v

   1. Initial program 76.1%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 76.1%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 76.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in H around 0

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v + \frac{-49}{5} \cdot \frac{H}{v}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H}{v} \cdot \frac{-49}{5}\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + \frac{H \cdot \frac{-49}{5}}{v}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\frac{-49}{5}}{v}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(v + H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \left(H \cdot \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{49}{5} \cdot \frac{1}{v}\right)\right)\right)\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5} \cdot 1}{v}\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{\frac{49}{5}}{v}\right)\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\mathsf{neg}\left(\frac{49}{5}\right)}{v}\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \left(\frac{\frac{-49}{5}}{v}\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f64 66.0%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(v, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{-49}{5}, v\right)\right)\right)\right)\right) \]

   7. Simplified 66.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + H \cdot \frac{-9.8}{v}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if v < -1.999999999999994e-310

   1. Initial program 69.3%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 69.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 69.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
   6. Step-by-step derivation

   7. Simplified 64.5%

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

   if -1.999999999999994e-310 < v

   1. Initial program 74.2%

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

      1. atan-lowering-atan.f64 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval 74.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Simplified 62.3%

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

Alternative 11: 34.6% accurate, 2.1× 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 71.8%

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

   1. atan-lowering-atan.f64 N/A

      \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\sqrt{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\left(v \cdot v + \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(v \cdot v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\mathsf{neg}\left(H \cdot \left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(H \cdot \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \left(\mathsf{neg}\left(\frac{98}{5}\right)\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval 71.8%

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(H, \frac{-98}{5}\right)\right)\right)\right)\right) \]

3. Simplified 71.8%

   \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around -inf

   \[\leadsto \mathsf{atan.f64}\left(\color{blue}{-1}\right) \]
6. Step-by-step derivation

7. Simplified 32.2%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))