Optimal throwing angle

Percentage Accurate: 66.7% → 98.9%

Time: 10.1s

Alternatives: 8

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: 66.7% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.9 \cdot 10^{+49}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 1.9e+49) (atan (/ v (sqrt (+ (* v v) (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 1.9e+49) {
		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 <= (-1d+155)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.9d+49) 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 <= -1e+155) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.9e+49) {
		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 <= -1e+155:
		tmp = math.atan(-1.0)
	elif v <= 1.9e+49:
		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 <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 1.9e+49)
		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 <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 1.9e+49)
		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, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.9e+49], 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 < -1.00000000000000001e155

   1. Initial program 3.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. 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 -1.00000000000000001e155 < v < 1.8999999999999999e49

   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 1.8999999999999999e49 < v

   1. Initial program 36.9%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 100.0%

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

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.2 \cdot 10^{-39}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-80}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(\frac{H}{-0.05102040816326531}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.2e-39)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (if (<= v 1.55e-80)
     (atan (* v (pow (/ H -0.05102040816326531) -0.5)))
     (atan (/ v (+ v (/ (* H -9.8) v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.2e-39) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 1.55e-80) {
		tmp = atan((v * pow((H / -0.05102040816326531), -0.5)));
	} 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 <= (-6.2d-39)) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else if (v <= 1.55d-80) then
        tmp = atan((v * ((h / (-0.05102040816326531d0)) ** (-0.5d0))))
    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 <= -6.2e-39) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 1.55e-80) {
		tmp = Math.atan((v * Math.pow((H / -0.05102040816326531), -0.5)));
	} else {
		tmp = Math.atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -6.2e-39:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	elif v <= 1.55e-80:
		tmp = math.atan((v * math.pow((H / -0.05102040816326531), -0.5)))
	else:
		tmp = math.atan((v / (v + ((H * -9.8) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.2e-39)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	elseif (v <= 1.55e-80)
		tmp = atan(Float64(v * (Float64(H / -0.05102040816326531) ^ -0.5)));
	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 <= -6.2e-39)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	elseif (v <= 1.55e-80)
		tmp = atan((v * ((H / -0.05102040816326531) ^ -0.5)));
	else
		tmp = atan((v / (v + ((H * -9.8) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.2e-39], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.55e-80], N[ArcTan[N[(v * N[Power[N[(H / -0.05102040816326531), $MachinePrecision], -0.5], $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 3 regimes

2. if v < -6.1999999999999994e-39

   1. Initial program 49.2%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   7. Simplified 78.6%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right) \cdot \left(0 - v\right)}}\right) \]

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

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

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

      16. /-lowering-/.f64 93.5%

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

   10. Simplified 93.5%

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

   if -6.1999999999999994e-39 < v < 1.55000000000000008e-80

   1. Initial program 99.5%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 99.5%

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

   7. Taylor expanded in v around 0

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

      1. /-lowering-/.f64 92.1%

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

   9. Simplified 92.1%

      \[\leadsto \tan^{-1} \left(\frac{v}{{\color{blue}{\left(\frac{-0.05102040816326531}{H}\right)}}^{-0.5}}\right) \]
   10. Step-by-step derivation

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow-flip N/A

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

       5. clear-num N/A

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

       6. inv-pow N/A

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

       7. div-inv N/A

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

       8. metadata-eval N/A

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

       9. pow-pow N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval N/A

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

       14. div-inv N/A

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

       15. /-lowering-/.f64 92.1%

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

   11. Applied egg-rr 92.1%

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

   if 1.55000000000000008e-80 < v

   1. Initial program 55.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in 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 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.2 \cdot 10^{-39}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 1.55 \cdot 10^{-80}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(\frac{H}{-0.05102040816326531}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.95 \cdot 10^{-38}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.95e-38)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (if (<= v 1.3e-77)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (/ v (+ v (/ (* H -9.8) v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.95e-38) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 1.3e-77) {
		tmp = atan((v / sqrt((H * -19.6))));
	} 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 <= (-2.95d-38)) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else if (v <= 1.3d-77) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    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 <= -2.95e-38) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 1.3e-77) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.95e-38:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	elif v <= 1.3e-77:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((v / (v + ((H * -9.8) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.95e-38)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	elseif (v <= 1.3e-77)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	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 <= -2.95e-38)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	elseif (v <= 1.3e-77)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v / (v + ((H * -9.8) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.95e-38], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.3e-77], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $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 3 regimes

2. if v < -2.94999999999999991e-38

   1. Initial program 49.2%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   7. Simplified 78.6%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right) \cdot \left(0 - v\right)}}\right) \]

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

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

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

      16. /-lowering-/.f64 93.5%

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

   10. Simplified 93.5%

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

   if -2.94999999999999991e-38 < v < 1.3000000000000001e-77

   1. Initial program 99.5%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in v around 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 92.0%

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

   7. Simplified 92.0%

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

   if 1.3000000000000001e-77 < v

   1. Initial program 55.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in 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 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.95 \cdot 10^{-38}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-146}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-104}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H} \cdot \left(v \cdot -0.10204081632653061\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e-146)
   (atan -1.0)
   (if (<= v 1.4e-104)
     (atan (* (/ v H) (* v -0.10204081632653061)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-146) {
		tmp = atan(-1.0);
	} else if (v <= 1.4e-104) {
		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 <= (-2d-146)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.4d-104) 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 <= -2e-146) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.4e-104) {
		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 <= -2e-146:
		tmp = math.atan(-1.0)
	elif v <= 1.4e-104:
		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 <= -2e-146)
		tmp = atan(-1.0);
	elseif (v <= 1.4e-104)
		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 <= -2e-146)
		tmp = atan(-1.0);
	elseif (v <= 1.4e-104)
		tmp = atan(((v / H) * (v * -0.10204081632653061)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-146], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.4e-104], 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 < -2.00000000000000005e-146

   1. Initial program 59.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 59.1%

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

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 79.8%

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

   if -2.00000000000000005e-146 < v < 1.4e-104

   1. Initial program 99.5%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   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 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in H around -inf

      \[\leadsto \mathsf{atan.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{25}{2401} \cdot \frac{{v}^{4}}{H} + \frac{5}{49} \cdot {v}^{2}}{H}\right)}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{25}{2401} \cdot \frac{{v}^{4}}{H} + \frac{5}{49} \cdot {v}^{2}}{H}\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{\frac{25}{2401} \cdot \frac{{v}^{4}}{H} + \frac{5}{49} \cdot {v}^{2}}{\mathsf{neg}\left(H\right)}\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{atan.f64}\left(\left(\frac{\frac{25}{2401} \cdot \frac{{v}^{4}}{H} + \frac{5}{49} \cdot {v}^{2}}{-1 \cdot H}\right)\right) \]

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

         \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(\left(\frac{25}{2401} \cdot \frac{{v}^{4}}{H} + \frac{5}{49} \cdot {v}^{2}\right), \left(-1 \cdot H\right)\right)\right) \]

   10. Simplified 25.7%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(v \cdot \left(v \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{0.010412328196584756}{H} + \left(v \cdot v\right) \cdot 0.10204081632653061}{-H}\right)} \]
   11. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{\left(v \cdot \left(v \cdot \left(v \cdot v\right)\right)\right) \cdot \frac{\frac{25}{2401}}{H} + \left(v \cdot v\right) \cdot \frac{5}{49}}{-1 \cdot H}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H}\right) + \left(v \cdot v\right) \cdot \frac{5}{49}}{-1 \cdot H}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H}\right) + v \cdot \left(v \cdot \frac{5}{49}\right)}{-1 \cdot H}\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}\right)}{-1 \cdot H}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}\right)}{H \cdot -1}\right)\right) \]

       6. times-frac N/A

          \[\leadsto \mathsf{atan.f64}\left(\left(\frac{v}{H} \cdot \frac{\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}}{-1}\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\left(\frac{v}{H}\right), \left(\frac{\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}}{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, H\right), \left(\frac{\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}}{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(v, H\right), \mathsf{/.f64}\left(\left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \frac{\frac{25}{2401}}{H} + v \cdot \frac{5}{49}\right), -1\right)\right)\right) \]

   12. Applied egg-rr 25.8%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{v}{H} \cdot \frac{\frac{v \cdot \left(v \cdot v\right)}{\frac{H}{0.010412328196584756}} + v \cdot 0.10204081632653061}{-1}\right)} \]

   13. Taylor expanded in v around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 25.8%

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

   15. Simplified 25.8%

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

   if 1.4e-104 < v

   1. Initial program 57.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. 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 57.4%

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

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 88.0%

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

Alternative 5: 72.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-273}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 2e-273)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (atan (/ v (+ v (/ (* H -9.8) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= 2e-273) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= 2d-273) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else
        tmp = atan((v / (v + ((h * (-9.8d0)) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 2e-273) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = Math.atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 2e-273:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	else:
		tmp = math.atan((v / (v + ((H * -9.8) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 2e-273)
		tmp = atan(Float64(v / 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-273)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	else
		tmp = atan((v / (v + ((H * -9.8) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 2e-273], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $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 < 2e-273

   1. Initial program 69.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   7. Simplified 48.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\left(\left(1 + \frac{H}{v \cdot v} \cdot -9.8\right) + \frac{\left(H \cdot H\right) \cdot -48.02}{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}\right) \cdot \left(0 - v\right)}}\right) \]

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

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

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

      16. /-lowering-/.f64 69.6%

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

   10. Simplified 69.6%

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

   if 2e-273 < v

   1. Initial program 64.8%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in 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 73.9%

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

   7. Simplified 73.9%

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

Alternative 6: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.35 \cdot 10^{-147}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.35e-147) (atan -1.0) (atan (/ v (+ v (/ (* H -9.8) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.35e-147) {
		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 <= (-2.35d-147)) 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 <= -2.35e-147) {
		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 <= -2.35e-147:
		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 <= -2.35e-147)
		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 <= -2.35e-147)
		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, -2.35e-147], 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 < -2.34999999999999994e-147

   1. Initial program 59.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 59.1%

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

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 79.8%

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

   if -2.34999999999999994e-147 < v

   1. Initial program 72.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 72.7%

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

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 65.6%

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

   7. Simplified 65.6%

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

Alternative 7: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.15 \cdot 10^{-303}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.15e-303) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.15e-303) {
		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 <= (-1.15d-303)) 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 <= -1.15e-303) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.15e-303:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.15e-303)
		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 <= -1.15e-303)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.15e-303], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.14999999999999998e-303

   1. Initial program 67.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 64.3%

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

   if -1.14999999999999998e-303 < v

   1. Initial program 67.2%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 68.5%

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

Alternative 8: 35.2% 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 67.3%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
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 67.3%

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

3. Simplified 67.3%

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

5. Taylor expanded in v around -inf

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

7. Simplified 33.1%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))