Optimal throwing angle

Percentage Accurate: 66.9% → 99.5%

Time: 9.9s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+106}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 1e+106) (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 <= 1e+106) {
		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 <= 1d+106) 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 <= 1e+106) {
		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 <= 1e+106:
		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 <= 1e+106)
		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 <= 1e+106)
		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, 1e+106], 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 < 1.00000000000000009e106

   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 1.00000000000000009e106 < v

   1. Initial program 25.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 25.6%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{1}\right) \]
   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 2: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.1e-19)
   (atan (* v (/ 1.0 (- (* 9.8 (/ H v)) v))))
   (if (<= v 2e-50)
     (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 <= -1.1e-19) {
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else if (v <= 2e-50) {
		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 <= (-1.1d-19)) then
        tmp = atan((v * (1.0d0 / ((9.8d0 * (h / v)) - v))))
    else if (v <= 2d-50) 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 <= -1.1e-19) {
		tmp = Math.atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else if (v <= 2e-50) {
		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 <= -1.1e-19:
		tmp = math.atan((v * (1.0 / ((9.8 * (H / v)) - v))))
	elif v <= 2e-50:
		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 <= -1.1e-19)
		tmp = atan(Float64(v * Float64(1.0 / Float64(Float64(9.8 * Float64(H / v)) - v))));
	elseif (v <= 2e-50)
		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 <= -1.1e-19)
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	elseif (v <= 2e-50)
		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, -1.1e-19], N[ArcTan[N[(v * N[(1.0 / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e-50], 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 < -1.0999999999999999e-19

   1. Initial program 48.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 48.1%

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

   3. Simplified 48.1%

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

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   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. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 90.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 90.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{0 - v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v \cdot 1 + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(0 - v\right) - v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right) \]

      4. neg-sub0 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(0 - v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot v\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right) \cdot v\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{v \cdot v}\right), v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), v\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 90.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), v\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 90.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(0 - v\right) - -9.8 \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}}\right) \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}{v}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)} \cdot v\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}\right), v\right)\right) \]

   11. Applied egg-rr 90.9%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\left(0 - v\right) + 9.8 \cdot \frac{H}{v}} \cdot v\right)} \]

   if -1.0999999999999999e-19 < v < 2.00000000000000002e-50

   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 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 85.4%

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

   7. Simplified 85.4%

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

   if 2.00000000000000002e-50 < v

   1. Initial program 48.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   6. 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 93.5%

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

   7. Simplified 93.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.35 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 2.45 \cdot 10^{-49}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.35e+20)
   (atan (* v (/ 1.0 (- (* 9.8 (/ H v)) v))))
   (if (<= v 2.45e-49)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (* v (+ 1.0 (* (/ H (* v v)) -9.8))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.35e+20) {
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else if (v <= 2.45e-49) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} 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 <= (-2.35d+20)) then
        tmp = atan((v * (1.0d0 / ((9.8d0 * (h / v)) - v))))
    else if (v <= 2.45d-49) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    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 <= -2.35e+20) {
		tmp = Math.atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	} else if (v <= 2.45e-49) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.35e+20:
		tmp = math.atan((v * (1.0 / ((9.8 * (H / v)) - v))))
	elif v <= 2.45e-49:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	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 <= -2.35e+20)
		tmp = atan(Float64(v * Float64(1.0 / Float64(Float64(9.8 * Float64(H / v)) - v))));
	elseif (v <= 2.45e-49)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	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 <= -2.35e+20)
		tmp = atan((v * (1.0 / ((9.8 * (H / v)) - v))));
	elseif (v <= 2.45e-49)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v * (1.0 + ((H / (v * v)) * -9.8)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.35e+20], N[ArcTan[N[(v * N[(1.0 / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2.45e-49], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $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 < -2.35e20

   1. Initial program 43.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   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. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 94.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 94.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{0 - v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v \cdot 1 + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(0 - v\right) - v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right) \]

      4. neg-sub0 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(0 - v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot v\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right) \cdot v\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{v \cdot v}\right), v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), v\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 94.3%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), v\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 94.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(0 - v\right) - -9.8 \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}}\right) \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}{v}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)} \cdot v\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}\right), v\right)\right) \]

   11. Applied egg-rr 94.3%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\left(0 - v\right) + 9.8 \cdot \frac{H}{v}} \cdot v\right)} \]

   if -2.35e20 < v < 2.4500000000000001e-49

   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 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in H around inf

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

      1. /-lowering-/.f64 83.3%

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

   10. Simplified 83.3%

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

   if 2.4500000000000001e-49 < v

   1. Initial program 48.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   6. 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 93.5%

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

   7. Simplified 93.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.35 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 2.45 \cdot 10^{-49}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.75 \cdot 10^{-109}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H} \cdot \left(v \cdot -0.10204081632653061\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e-106)
   (atan -1.0)
   (if (<= v 1.75e-109)
     (atan (* (/ v H) (* v -0.10204081632653061)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-106) {
		tmp = atan(-1.0);
	} else if (v <= 1.75e-109) {
		tmp = atan(((v / H) * (v * -0.10204081632653061)));
	} 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-106)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.75d-109) then
        tmp = atan(((v / h) * (v * (-0.10204081632653061d0))))
    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-106) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.75e-109) {
		tmp = Math.atan(((v / H) * (v * -0.10204081632653061)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-106:
		tmp = math.atan(-1.0)
	elif v <= 1.75e-109:
		tmp = math.atan(((v / H) * (v * -0.10204081632653061)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-106)
		tmp = atan(-1.0);
	elseif (v <= 1.75e-109)
		tmp = atan(Float64(Float64(v / H) * Float64(v * -0.10204081632653061)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e-106)
		tmp = atan(-1.0);
	elseif (v <= 1.75e-109)
		tmp = atan(((v / H) * (v * -0.10204081632653061)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-106], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.75e-109], N[ArcTan[N[(N[(v / H), $MachinePrecision] * N[(v * -0.10204081632653061), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.99999999999999983e-106

   1. Initial program 58.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 58.1%

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

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 82.1%

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

   if -4.99999999999999983e-106 < v < 1.75e-109

   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. pow1/2 N/A

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

      2. flip-+ N/A

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

      3. fmm-def N/A

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

      4. *-commutative N/A

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

      5. clear-num N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   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. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 29.7%

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

   9. Simplified 29.7%

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

   10. Taylor expanded in v around 0

       \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(\frac{-5}{49} \cdot \frac{{v}^{2}}{H}\right)}\right) \]
   11. 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 29.7%

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

   12. Simplified 29.7%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. /-lowering-/.f64 29.9%

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

   14. Applied egg-rr 29.9%

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

   if 1.75e-109 < v

   1. Initial program 53.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 88.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.75 \cdot 10^{-109}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H} \cdot \left(v \cdot -0.10204081632653061\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-306)
   (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-306) {
		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-306)) 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-306) {
		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-306:
		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-306)
		tmp = atan(Float64(v * Float64(1.0 / Float64(Float64(9.8 * Float64(H / v)) - v))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(H * -9.8) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e-306)
		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-306], 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[(N[(H * -9.8), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.00000000000000006e-306

   1. Initial program 67.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 67.6%

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

   3. Simplified 67.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 70.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 70.5%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{0 - v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v \cdot 1 + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(0 - v\right) - v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right) \]

      4. neg-sub0 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(0 - v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot v\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right) \cdot v\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{v \cdot v}\right), v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), v\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 70.5%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), v\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 70.5%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(0 - v\right) - -9.8 \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}}\right) \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}{v}}\right)\right) \]

       2. associate-/r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)} \cdot v\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}\right), v\right)\right) \]

   11. Applied egg-rr 70.5%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\left(0 - v\right) + 9.8 \cdot \frac{H}{v}} \cdot v\right)} \]

   if -2.00000000000000006e-306 < v

   1. Initial program 63.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 63.7%

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

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 75.6%

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

   7. Simplified 75.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-306}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e-106) (atan -1.0) (atan (/ v (+ v (/ (* H -9.8) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-106) {
		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 <= (-5d-106)) 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 <= -5e-106) {
		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 <= -5e-106:
		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 <= -5e-106)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(H * -9.8) / v))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e-106)
		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, -5e-106], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(H * -9.8), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -4.99999999999999983e-106

   1. Initial program 58.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 58.1%

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

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 82.1%

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

   if -4.99999999999999983e-106 < v

   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 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. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. /-lowering-/.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

Alternative 7: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-106}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{9.8}{\frac{v}{H}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e-106) (atan -1.0) (atan (/ v (+ v (/ 9.8 (/ v H)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-106) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v / (v + (9.8 / (v / H)))));
	}
	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-106)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v / (v + (9.8d0 / (v / h)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5e-106) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + (9.8 / (v / H)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-106:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + (9.8 / (v / H)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-106)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(9.8 / Float64(v / H)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e-106)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + (9.8 / (v / H)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-106], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(9.8 / N[(v / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -4.99999999999999983e-106

   1. Initial program 58.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 58.1%

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

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 82.1%

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

   if -4.99999999999999983e-106 < v

   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

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)}\right)\right) \]
   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. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \left(\frac{H}{{v}^{2}} \cdot \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{H}{{v}^{2}}\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left({v}^{2}\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 11.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), \frac{-49}{5}\right)\right)\right)\right)\right)\right) \]

   7. Simplified 11.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{0 - v \cdot \left(1 + \frac{H}{v \cdot v} \cdot -9.8\right)}}\right) \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v \cdot 1 + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(0 - \left(v + v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \left(\left(0 - v\right) - v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right) \]

      4. neg-sub0 N/A

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\left(0 - v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(v \cdot \left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{H}{v \cdot v} \cdot \frac{-49}{5}\right) \cdot v\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\left(\frac{-49}{5} \cdot \frac{H}{v \cdot v}\right) \cdot v\right)\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \left(\frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \left(\frac{H}{v \cdot v} \cdot v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\left(\frac{H}{v \cdot v}\right), v\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \left(v \cdot v\right)\right), v\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 11.2%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, v\right), \mathsf{*.f64}\left(\frac{-49}{5}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(H, \mathsf{*.f64}\left(v, v\right)\right), v\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 11.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(0 - v\right) - -9.8 \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}}\right) \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{1}{\frac{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}{v}}\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)}{v}\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(0 - v\right) - \frac{-49}{5} \cdot \left(\frac{H}{v \cdot v} \cdot v\right)\right), v\right)\right)\right) \]

   11. Applied egg-rr 11.2%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{\left(0 - v\right) + 9.8 \cdot \frac{H}{v}}{v}}\right)} \]

   13. Applied egg-rr 68.2%

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

Alternative 8: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -4e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4e-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 <= -4e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.999999999999988e-310

   1. Initial program 67.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 64.3%

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

   if -3.999999999999988e-310 < v

   1. Initial program 63.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 63.7%

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

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 70.0%

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

Alternative 9: 34.8% 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 65.4%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
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 65.4%

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

3. Simplified 65.4%

   \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
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 29.6%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))