Optimal throwing angle

Percentage Accurate: 66.9% → 98.9%

Time: 12.4s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.66 \cdot 10^{+46}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan -1.0)
   (if (<= v 1.66e+46)
     (atan (/ v (sqrt (+ (* v v) (* H -19.6)))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1.66e+46) {
		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+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.66d+46) 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+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.66e+46) {
		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+154:
		tmp = math.atan(-1.0)
	elif v <= 1.66e+46:
		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+154)
		tmp = atan(-1.0);
	elseif (v <= 1.66e+46)
		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+154)
		tmp = atan(-1.0);
	elseif (v <= 1.66e+46)
		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+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.66e+46], 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 < -1.00000000000000004e154

   1. Initial program 3.1%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 100.0%

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

   if -1.00000000000000004e154 < v < 1.66e46

   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 1.66e46 < v

   1. Initial program 39.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 39.3%

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

   3. Simplified 39.3%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 100.0%

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

Alternative 2: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.4 \cdot 10^{-92}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot \frac{96.04}{v \cdot v}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right)\\ \mathbf{elif}\;v \leq 2.35 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.4e-92)
   (atan
    (/
     v
     (- (/ (* H (/ 96.04 (* v v))) (/ (- 9.8 (* H (/ 48.02 (* v v)))) v)) v)))
   (if (<= v 2.35e-30)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.4e-92) {
		tmp = atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	} else if (v <= 2.35e-30) {
		tmp = atan((v / sqrt((H * -19.6))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / 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 <= (-2.4d-92)) then
        tmp = atan((v / (((h * (96.04d0 / (v * v))) / ((9.8d0 - (h * (48.02d0 / (v * v)))) / v)) - v)))
    else if (v <= 2.35d-30) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.4e-92) {
		tmp = Math.atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	} else if (v <= 2.35e-30) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.4e-92:
		tmp = math.atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)))
	elif v <= 2.35e-30:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.4e-92)
		tmp = atan(Float64(v / Float64(Float64(Float64(H * Float64(96.04 / Float64(v * v))) / Float64(Float64(9.8 - Float64(H * Float64(48.02 / Float64(v * v)))) / v)) - v)));
	elseif (v <= 2.35e-30)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.4e-92)
		tmp = atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	elseif (v <= 2.35e-30)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.4e-92], N[ArcTan[N[(v / N[(N[(N[(H * N[(96.04 / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(9.8 - N[(H * N[(48.02 / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2.35e-30], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.4000000000000001e-92

   1. Initial program 52.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]
   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 74.5%

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

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

      13. *-commutative N/A

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

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \left({v}^{3}\right)\right)\right)\right), v\right)\right)\right) \]

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 93.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right), v\right)\right)\right) \]

   10. Simplified 93.8%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{H \cdot \left(\frac{9.8}{v} + \frac{H \cdot 48.02}{v \cdot \left(v \cdot v\right)}\right) - v}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 93.7%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{\frac{96.04 - \frac{H \cdot 48.02}{v \cdot \left(v \cdot v\right)} \cdot \frac{H \cdot 48.02}{v}}{v \cdot v} \cdot H}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}}} - v}\right) \]

   13. Taylor expanded in H around 0

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2401}{25} \cdot \frac{H}{{v}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2401}{25} \cdot H}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{H \cdot \frac{2401}{25}}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \frac{\frac{2401}{25}}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \frac{\frac{2401}{25} \cdot 1}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \left(\frac{2401}{25} \cdot \frac{1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{2401}{25} \cdot \frac{1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{\frac{2401}{25} \cdot 1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{\frac{2401}{25}}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \left({v}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \left(v \cdot v\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       11. *-lowering-*.f64 93.8%

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

   15. Simplified 93.8%

       \[\leadsto \tan^{-1} \left(\frac{v}{\frac{\color{blue}{H \cdot \frac{96.04}{v \cdot v}}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right) \]

   if -2.4000000000000001e-92 < v < 2.34999999999999985e-30

   1. Initial program 99.6%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0

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

      1. *-lowering-*.f64 95.4%

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

   7. Simplified 95.4%

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

   if 2.34999999999999985e-30 < v

   1. Initial program 45.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 45.7%

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

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 96.7%

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

   7. Simplified 96.7%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 97.9%

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

   9. Applied egg-rr 97.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.4 \cdot 10^{-92}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot \frac{96.04}{v \cdot v}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right)\\ \mathbf{elif}\;v \leq 2.35 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.5 \cdot 10^{-129}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot \frac{96.04}{v \cdot v}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + -9.8 \cdot \frac{H}{v \cdot v}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.5e-129)
   (atan
    (/
     v
     (- (/ (* H (/ 96.04 (* v v))) (/ (- 9.8 (* H (/ 48.02 (* v v)))) v)) v)))
   (atan (/ v (* v (+ 1.0 (* -9.8 (/ H (* v v)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.5e-129) {
		tmp = atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	} else {
		tmp = atan((v / (v * (1.0 + (-9.8 * (H / (v * 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 <= (-2.5d-129)) then
        tmp = atan((v / (((h * (96.04d0 / (v * v))) / ((9.8d0 - (h * (48.02d0 / (v * v)))) / v)) - v)))
    else
        tmp = atan((v / (v * (1.0d0 + ((-9.8d0) * (h / (v * v)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.5e-129) {
		tmp = Math.atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	} else {
		tmp = Math.atan((v / (v * (1.0 + (-9.8 * (H / (v * v)))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.5e-129:
		tmp = math.atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)))
	else:
		tmp = math.atan((v / (v * (1.0 + (-9.8 * (H / (v * v)))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.5e-129)
		tmp = atan(Float64(v / Float64(Float64(Float64(H * Float64(96.04 / Float64(v * v))) / Float64(Float64(9.8 - Float64(H * Float64(48.02 / Float64(v * v)))) / v)) - v)));
	else
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.5e-129)
		tmp = atan((v / (((H * (96.04 / (v * v))) / ((9.8 - (H * (48.02 / (v * v)))) / v)) - v)));
	else
		tmp = atan((v / (v * (1.0 + (-9.8 * (H / (v * v)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.5e-129], N[ArcTan[N[(v / N[(N[(N[(H * N[(96.04 / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(9.8 - N[(H * N[(48.02 / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.50000000000000014e-129

   1. Initial program 55.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 55.4%

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

   3. Simplified 55.4%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 70.6%

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

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

      13. *-commutative N/A

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

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \left({v}^{3}\right)\right)\right)\right), v\right)\right)\right) \]

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 88.9%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right), v\right)\right)\right) \]

   10. Simplified 88.9%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{H \cdot \left(\frac{9.8}{v} + \frac{H \cdot 48.02}{v \cdot \left(v \cdot v\right)}\right) - v}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. flip-+ N/A

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

       3. associate-*l/ N/A

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

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

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

   12. Applied egg-rr 88.7%

       \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{\frac{96.04 - \frac{H \cdot 48.02}{v \cdot \left(v \cdot v\right)} \cdot \frac{H \cdot 48.02}{v}}{v \cdot v} \cdot H}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}}} - v}\right) \]

   13. Taylor expanded in H around 0

       \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2401}{25} \cdot \frac{H}{{v}^{2}}\right)}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]
   14. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{2401}{25} \cdot H}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{H \cdot \frac{2401}{25}}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \frac{\frac{2401}{25}}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \frac{\frac{2401}{25} \cdot 1}{{v}^{2}}\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(H \cdot \left(\frac{2401}{25} \cdot \frac{1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{2401}{25} \cdot \frac{1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{\frac{2401}{25} \cdot 1}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \left(\frac{\frac{2401}{25}}{{v}^{2}}\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \left({v}^{2}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \left(v \cdot v\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

       11. *-lowering-*.f64 89.1%

           \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{25}, \mathsf{*.f64}\left(v, v\right)\right)\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{49}{5}, \mathsf{*.f64}\left(H, \mathsf{/.f64}\left(\frac{2401}{50}, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right), v\right)\right)\right) \]

   15. Simplified 89.1%

       \[\leadsto \tan^{-1} \left(\frac{v}{\frac{\color{blue}{H \cdot \frac{96.04}{v \cdot v}}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right) \]

   if -2.50000000000000014e-129 < v

   1. Initial program 68.4%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in v around inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(v \cdot \left(1 + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)}\right)\right) \]
   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 70.5%

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

   7. Simplified 70.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.5 \cdot 10^{-129}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot \frac{96.04}{v \cdot v}}{\frac{9.8 - H \cdot \frac{48.02}{v \cdot v}}{v}} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + -9.8 \cdot \frac{H}{v \cdot v}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.5e-101)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (if (<= v 5e-149) (atan (/ v (/ (* H 9.8) v))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 5e-149) {
		tmp = atan((v / ((H * 9.8) / v)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-3.5d-101)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else if (v <= 5d-149) then
        tmp = atan((v / ((h * 9.8d0) / v)))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 5e-149) {
		tmp = Math.atan((v / ((H * 9.8) / v)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.5e-101:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	elif v <= 5e-149:
		tmp = math.atan((v / ((H * 9.8) / v)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.5e-101)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))));
	elseif (v <= 5e-149)
		tmp = atan(Float64(v / Float64(Float64(H * 9.8) / v)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.5e-101)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	elseif (v <= 5e-149)
		tmp = atan((v / ((H * 9.8) / v)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.5e-101], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5e-149], N[ArcTan[N[(v / N[(N[(H * 9.8), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.49999999999999994e-101

   1. Initial program 52.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in v around -inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.1%

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

   7. Simplified 93.1%

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

   if -3.49999999999999994e-101 < v < 4.99999999999999968e-149

   1. Initial program 99.6%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around -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 0.5%

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

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

      13. *-commutative N/A

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

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \left({v}^{3}\right)\right)\right)\right), v\right)\right)\right) \]

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 30.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right), v\right)\right)\right) \]

   10. Simplified 30.7%

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

   11. Taylor expanded in H around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 33.2%

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

   13. Simplified 33.2%

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

   14. Taylor expanded in H around inf

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 33.2%

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

   16. Simplified 33.2%

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

   if 4.99999999999999968e-149 < v

   1. Initial program 54.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 85.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 6 \cdot 10^{-149}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.5e-101)
   (atan -1.0)
   (if (<= v 6e-149) (atan (/ v (/ (* H 9.8) v))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = atan(-1.0);
	} else if (v <= 6e-149) {
		tmp = atan((v / ((H * 9.8) / v)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-3.5d-101)) then
        tmp = atan((-1.0d0))
    else if (v <= 6d-149) then
        tmp = atan((v / ((h * 9.8d0) / v)))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = Math.atan(-1.0);
	} else if (v <= 6e-149) {
		tmp = Math.atan((v / ((H * 9.8) / v)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.5e-101:
		tmp = math.atan(-1.0)
	elif v <= 6e-149:
		tmp = math.atan((v / ((H * 9.8) / v)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.5e-101)
		tmp = atan(-1.0);
	elseif (v <= 6e-149)
		tmp = atan(Float64(v / Float64(Float64(H * 9.8) / v)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.5e-101)
		tmp = atan(-1.0);
	elseif (v <= 6e-149)
		tmp = atan((v / ((H * 9.8) / v)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.5e-101], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 6e-149], N[ArcTan[N[(v / N[(N[(H * 9.8), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.49999999999999994e-101

   1. Initial program 52.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 52.9%

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

   3. Simplified 52.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 92.8%

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

   if -3.49999999999999994e-101 < v < 6.0000000000000003e-149

   1. Initial program 99.6%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around -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 0.5%

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

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

      13. *-commutative N/A

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

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \left({v}^{3}\right)\right)\right)\right), v\right)\right)\right) \]

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 30.7%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right), v\right)\right)\right) \]

   10. Simplified 30.7%

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

   11. Taylor expanded in H around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 33.2%

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

   13. Simplified 33.2%

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

   14. Taylor expanded in H around inf

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 33.2%

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

   16. Simplified 33.2%

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

   if 6.0000000000000003e-149 < v

   1. Initial program 54.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 85.3%

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

Alternative 6: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{-208}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.4 \cdot 10^{-111}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-9.8 \cdot \frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.2e-208)
   (atan -1.0)
   (if (<= v 2.4e-111) (atan (/ v (* -9.8 (/ H v)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.2e-208) {
		tmp = atan(-1.0);
	} else if (v <= 2.4e-111) {
		tmp = atan((v / (-9.8 * (H / v))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.2d-208)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.4d-111) then
        tmp = atan((v / ((-9.8d0) * (h / v))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.2e-208) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.4e-111) {
		tmp = Math.atan((v / (-9.8 * (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.2e-208:
		tmp = math.atan(-1.0)
	elif v <= 2.4e-111:
		tmp = math.atan((v / (-9.8 * (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.2e-208)
		tmp = atan(-1.0);
	elseif (v <= 2.4e-111)
		tmp = atan(Float64(v / Float64(-9.8 * Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.2e-208)
		tmp = atan(-1.0);
	elseif (v <= 2.4e-111)
		tmp = atan((v / (-9.8 * (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.2e-208], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.4e-111], N[ArcTan[N[(v / N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.1999999999999999e-208

   1. Initial program 58.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 81.9%

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

   if -1.1999999999999999e-208 < v < 2.4000000000000001e-111

   1. Initial program 99.7%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in 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 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 39.3%

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

   10. Simplified 39.3%

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

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

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

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

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

       3. *-commutative N/A

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

       4. associate-/l* N/A

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

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

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

       6. /-lowering-/.f64 39.3%

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

   12. Applied egg-rr 39.3%

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

   if 2.4000000000000001e-111 < v

   1. Initial program 51.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 51.9%

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

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 90.2%

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

Alternative 7: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.22 \cdot 10^{-228}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{\frac{H}{\frac{v}{9.8}} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 1.22e-228)
   (atan (* v (/ 1.0 (- (/ H (/ v 9.8)) v))))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= 1.22e-228) {
		tmp = atan((v * (1.0 / ((H / (v / 9.8)) - v))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / 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 <= 1.22d-228) then
        tmp = atan((v * (1.0d0 / ((h / (v / 9.8d0)) - v))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 1.22e-228) {
		tmp = Math.atan((v * (1.0 / ((H / (v / 9.8)) - v))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 1.22e-228:
		tmp = math.atan((v * (1.0 / ((H / (v / 9.8)) - v))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 1.22e-228)
		tmp = atan(Float64(v * Float64(1.0 / Float64(Float64(H / Float64(v / 9.8)) - v))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= 1.22e-228)
		tmp = atan((v * (1.0 / ((H / (v / 9.8)) - v))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 1.22e-228], N[ArcTan[N[(v * N[(1.0 / N[(N[(H / N[(v / 9.8), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.2199999999999999e-228

   1. Initial program 65.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]
   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 54.8%

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

   8. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

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

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

      13. *-commutative N/A

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

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

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \left({v}^{3}\right)\right)\right)\right), v\right)\right)\right) \]

      15. cube-mult N/A

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

      16. unpow2 N/A

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(H, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{49}{5}, v\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(H, \frac{2401}{50}\right), \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, v\right)\right)\right)\right)\right), v\right)\right)\right) \]

   10. Simplified 77.8%

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

   11. Taylor expanded in H around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. associate-*r/ N/A

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

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

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. associate-*r/ N/A

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

       10. *-commutative N/A

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

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

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

       12. *-commutative N/A

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

       13. *-lowering-*.f64 78.6%

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

   13. Simplified 78.6%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

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

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

       5. associate-*l/ N/A

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

       6. associate-/r/ N/A

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

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

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

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

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

       9. /-lowering-/.f64 78.6%

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

   15. Applied egg-rr 78.6%

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

   if 1.2199999999999999e-228 < v

   1. Initial program 59.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in 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 77.6%

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

   7. Simplified 77.6%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 78.5%

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

   9. Applied egg-rr 78.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.22 \cdot 10^{-228}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{\frac{H}{\frac{v}{9.8}} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.14 \cdot 10^{-266}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \frac{9.8}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.14e-266)
   (atan (/ v (- (* H (/ 9.8 v)) v)))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.14e-266) {
		tmp = atan((v / ((H * (9.8 / v)) - v)));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / 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 <= (-1.14d-266)) then
        tmp = atan((v / ((h * (9.8d0 / v)) - v)))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.14e-266) {
		tmp = Math.atan((v / ((H * (9.8 / v)) - v)));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.14e-266:
		tmp = math.atan((v / ((H * (9.8 / v)) - v)))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.14e-266)
		tmp = atan(Float64(v / Float64(Float64(H * Float64(9.8 / v)) - v)));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.14e-266)
		tmp = atan((v / ((H * (9.8 / v)) - v)));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.14e-266], N[ArcTan[N[(v / N[(N[(H * N[(9.8 / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.1400000000000001e-266

   1. Initial program 60.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 60.7%

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

   3. Simplified 60.7%

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

   5. Taylor expanded in v around -inf

      \[\leadsto \mathsf{atan.f64}\left(\mathsf{/.f64}\left(v, \color{blue}{\left(-1 \cdot \left(v \cdot \left(1 + \left(\frac{-2401}{50} \cdot \frac{{H}^{2}}{{v}^{4}} + \frac{-49}{5} \cdot \frac{H}{{v}^{2}}\right)\right)\right)\right)}\right)\right) \]
   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 62.2%

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

   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. *-lowering-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 80.9%

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

   10. Simplified 80.9%

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

   if -1.1400000000000001e-266 < 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 75.6%

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

   7. Simplified 75.6%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 76.3%

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

   9. Applied egg-rr 76.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.14 \cdot 10^{-266}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{H \cdot \frac{9.8}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.5e-101)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / 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 <= (-3.5d-101)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.5e-101:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.5e-101)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.5e-101)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.5e-101], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.49999999999999994e-101

   1. Initial program 52.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in v around -inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.1%

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

   7. Simplified 93.1%

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

   if -3.49999999999999994e-101 < v

   1. Initial program 69.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 69.6%

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

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 67.2%

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

   7. Simplified 67.2%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. /-lowering-/.f64 67.8%

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

   9. Applied egg-rr 67.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.5e-101)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (atan (/ v (+ v (/ (* H -9.8) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-3.5d-101)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else
        tmp = atan((v / (v + ((h * (-9.8d0)) / v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-101) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = Math.atan((v / (v + ((H * -9.8) / v))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.5e-101:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	else:
		tmp = math.atan((v / (v + ((H * -9.8) / v))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.5e-101)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(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 <= -3.5e-101)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	else
		tmp = atan((v / (v + ((H * -9.8) / v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.5e-101], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(H * -9.8), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.49999999999999994e-101

   1. Initial program 52.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in v around -inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.1%

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

   7. Simplified 93.1%

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

   if -3.49999999999999994e-101 < v

   1. Initial program 69.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 69.6%

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

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 67.2%

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

   7. Simplified 67.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-101}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H \cdot -9.8}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-310) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5d-310)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -4.999999999999985e-310

   1. Initial program 63.0%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 63.0%

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

   3. Simplified 63.0%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 73.8%

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

   if -4.999999999999985e-310 < v

   1. Initial program 62.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 71.2%

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

Alternative 12: 34.3% 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 62.8%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
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 62.8%

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

3. Simplified 62.8%

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

5. Taylor expanded in v around -inf

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

7. Simplified 39.5%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))