Optimal throwing angle

Percentage Accurate: 68.1% → 99.4%

Time: 11.0s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot v + H \cdot 19.6\\ \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.4 \cdot 10^{+124}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\left(v \cdot v + H \cdot -19.6\right) \cdot \left(t\_0 \cdot \frac{1}{t\_0}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (+ (* v v) (* H 19.6))))
   (if (<= v -5e+154)
     (atan -1.0)
     (if (<= v 4.4e+124)
       (atan (/ v (sqrt (* (+ (* v v) (* H -19.6)) (* t_0 (/ 1.0 t_0))))))
       (atan 1.0)))))

C

double code(double v, double H) {
	double t_0 = (v * v) + (H * 19.6);
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 4.4e+124) {
		tmp = atan((v / sqrt((((v * v) + (H * -19.6)) * (t_0 * (1.0 / t_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) :: t_0
    real(8) :: tmp
    t_0 = (v * v) + (h * 19.6d0)
    if (v <= (-5d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.4d+124) then
        tmp = atan((v / sqrt((((v * v) + (h * (-19.6d0))) * (t_0 * (1.0d0 / t_0))))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = (v * v) + (H * 19.6);
	double tmp;
	if (v <= -5e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.4e+124) {
		tmp = Math.atan((v / Math.sqrt((((v * v) + (H * -19.6)) * (t_0 * (1.0 / t_0))))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = (v * v) + (H * 19.6)
	tmp = 0
	if v <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 4.4e+124:
		tmp = math.atan((v / math.sqrt((((v * v) + (H * -19.6)) * (t_0 * (1.0 / t_0))))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	t_0 = Float64(Float64(v * v) + Float64(H * 19.6))
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 4.4e+124)
		tmp = atan(Float64(v / sqrt(Float64(Float64(Float64(v * v) + Float64(H * -19.6)) * Float64(t_0 * Float64(1.0 / t_0))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = (v * v) + (H * 19.6);
	tmp = 0.0;
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 4.4e+124)
		tmp = atan((v / sqrt((((v * v) + (H * -19.6)) * (t_0 * (1.0 / t_0))))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] + N[(H * 19.6), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.4e+124], N[ArcTan[N[(v / N[Sqrt[N[(N[(N[(v * v), $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.00000000000000004e154

   1. Initial program 3.1%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 100.0%

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

   if -5.00000000000000004e154 < v < 4.4000000000000002e124

   1. Initial program 99.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 99.8%

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

   3. Simplified 99.8%

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

      1. flip-+ N/A

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

      2. fmm-def N/A

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

      3. *-commutative N/A

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

      4. div-inv N/A

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

      5. difference-of-squares N/A

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

      6. fmm-def N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.8%

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

   if 4.4000000000000002e124 < v

   1. Initial program 32.0%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 32.0%

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

   3. Simplified 32.0%

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

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 5e+120) (atan (/ v (sqrt (+ (* v v) (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 5e+120) {
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 5d+120) then
        tmp = atan((v / sqrt(((v * v) + (h * (-19.6d0))))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5e+120) {
		tmp = Math.atan((v / Math.sqrt(((v * v) + (H * -19.6)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 5e+120:
		tmp = math.atan((v / math.sqrt(((v * v) + (H * -19.6)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 5e+120)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) + Float64(H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 5e+120)
		tmp = atan((v / sqrt(((v * v) + (H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5e+120], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.00000000000000004e154

   1. Initial program 3.1%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf

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

   7. Simplified 100.0%

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

   if -5.00000000000000004e154 < v < 5.00000000000000019e120

   1. Initial program 99.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 99.8%

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

   3. Simplified 99.8%

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

   if 5.00000000000000019e120 < v

   1. Initial program 32.0%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 32.0%

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

   3. Simplified 32.0%

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

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

Alternative 3: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{-63}:\\ \;\;\;\;\tan^{-1} \left(-1 + H \cdot \left(\frac{H \cdot -96.04}{v \cdot \left(v \cdot \left(v \cdot v\right)\right)} + \frac{-9.8}{v \cdot v}\right)\right)\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-121}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.9e-63)
   (atan
    (+ -1.0 (* H (+ (/ (* H -96.04) (* v (* v (* v v)))) (/ -9.8 (* v v))))))
   (if (<= v 4e-121)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.9e-63) {
		tmp = atan((-1.0 + (H * (((H * -96.04) / (v * (v * (v * v)))) + (-9.8 / (v * v))))));
	} else if (v <= 4e-121) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} 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 <= (-2.9d-63)) then
        tmp = atan(((-1.0d0) + (h * (((h * (-96.04d0)) / (v * (v * (v * v)))) + ((-9.8d0) / (v * v))))))
    else if (v <= 4d-121) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    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 <= -2.9e-63) {
		tmp = Math.atan((-1.0 + (H * (((H * -96.04) / (v * (v * (v * v)))) + (-9.8 / (v * v))))));
	} else if (v <= 4e-121) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.9e-63:
		tmp = math.atan((-1.0 + (H * (((H * -96.04) / (v * (v * (v * v)))) + (-9.8 / (v * v))))))
	elif v <= 4e-121:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.9e-63)
		tmp = atan(Float64(-1.0 + Float64(H * Float64(Float64(Float64(H * -96.04) / Float64(v * Float64(v * Float64(v * v)))) + Float64(-9.8 / Float64(v * v))))));
	elseif (v <= 4e-121)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	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 <= -2.9e-63)
		tmp = atan((-1.0 + (H * (((H * -96.04) / (v * (v * (v * v)))) + (-9.8 / (v * v))))));
	elseif (v <= 4e-121)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.9e-63], N[ArcTan[N[(-1.0 + N[(H * N[(N[(N[(H * -96.04), $MachinePrecision] / N[(v * N[(v * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-9.8 / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 4e-121], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $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 < -2.89999999999999975e-63

   1. Initial program 59.1%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 59.1%

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

   3. Simplified 59.1%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

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

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

      12. associate-*r/ N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 91.2%

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

   9. Simplified 91.2%

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

   10. Taylor expanded in H around 0

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

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

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

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

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

       6. sub-neg N/A

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

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

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

   12. Simplified 91.4%

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

   if -2.89999999999999975e-63 < v < 3.9999999999999999e-121

   1. Initial program 99.7%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in H around inf

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

      1. /-lowering-/.f64 89.9%

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

   10. Simplified 89.9%

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

   if 3.9999999999999999e-121 < v

   1. Initial program 68.3%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in H around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 79.8%

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

   7. Simplified 79.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 79.8%

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

   9. Applied egg-rr 79.8%

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

Alternative 4: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.6 \cdot 10^{-139}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.16 \cdot 10^{-172}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{\frac{H \cdot 9.8}{v}}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.6e-139)
   (atan -1.0)
   (if (<= v 1.16e-172) (atan (/ 1.0 (/ (/ (* H 9.8) v) v))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-139) {
		tmp = atan(-1.0);
	} else if (v <= 1.16e-172) {
		tmp = atan((1.0 / (((H * 9.8) / v) / 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 <= (-9.6d-139)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.16d-172) then
        tmp = atan((1.0d0 / (((h * 9.8d0) / v) / 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 <= -9.6e-139) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.16e-172) {
		tmp = Math.atan((1.0 / (((H * 9.8) / v) / v)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.6e-139:
		tmp = math.atan(-1.0)
	elif v <= 1.16e-172:
		tmp = math.atan((1.0 / (((H * 9.8) / v) / v)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.6e-139)
		tmp = atan(-1.0);
	elseif (v <= 1.16e-172)
		tmp = atan(Float64(1.0 / Float64(Float64(Float64(H * 9.8) / v) / v)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.6e-139)
		tmp = atan(-1.0);
	elseif (v <= 1.16e-172)
		tmp = atan((1.0 / (((H * 9.8) / v) / v)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.6e-139], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.16e-172], N[ArcTan[N[(1.0 / N[(N[(N[(H * 9.8), $MachinePrecision] / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -9.60000000000000059e-139

   1. Initial program 66.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 81.6%

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

   if -9.60000000000000059e-139 < v < 1.1599999999999999e-172

   1. Initial program 99.7%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

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

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

      12. associate-*r/ N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 31.6%

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

   9. Simplified 31.6%

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

   10. Taylor expanded in v around 0

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

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

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

       6. associate-*r/ N/A

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

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

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 31.6%

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

   12. Simplified 31.6%

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

   if 1.1599999999999999e-172 < v

   1. Initial program 70.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 74.9%

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

Alternative 5: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.4 \cdot 10^{-139}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-147}:\\ \;\;\;\;\tan^{-1} \left(\left(v \cdot -0.10204081632653061\right) \cdot \frac{v}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.4e-139)
   (atan -1.0)
   (if (<= v 1.7e-147)
     (atan (* (* v -0.10204081632653061) (/ v H)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.4e-139) {
		tmp = atan(-1.0);
	} else if (v <= 1.7e-147) {
		tmp = atan(((v * -0.10204081632653061) * (v / H)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5.4d-139)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.7d-147) then
        tmp = atan(((v * (-0.10204081632653061d0)) * (v / h)))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5.4e-139) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.7e-147) {
		tmp = Math.atan(((v * -0.10204081632653061) * (v / H)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.4e-139:
		tmp = math.atan(-1.0)
	elif v <= 1.7e-147:
		tmp = math.atan(((v * -0.10204081632653061) * (v / H)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.4e-139)
		tmp = atan(-1.0);
	elseif (v <= 1.7e-147)
		tmp = atan(Float64(Float64(v * -0.10204081632653061) * Float64(v / H)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5.4e-139)
		tmp = atan(-1.0);
	elseif (v <= 1.7e-147)
		tmp = atan(((v * -0.10204081632653061) * (v / H)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.4e-139], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.7e-147], N[ArcTan[N[(N[(v * -0.10204081632653061), $MachinePrecision] * N[(v / H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.3999999999999997e-139

   1. Initial program 66.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 81.6%

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

   if -5.3999999999999997e-139 < v < 1.69999999999999998e-147

   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 29.4%

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

   7. Simplified 29.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. *-lowering-*.f64 29.4%

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

   9. Applied egg-rr 29.4%

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

   10. Taylor expanded in v around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 29.4%

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

   12. Simplified 29.4%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 29.4%

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

   14. Applied egg-rr 29.4%

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

   if 1.69999999999999998e-147 < v

   1. Initial program 69.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 69.5%

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

   3. Simplified 69.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 77.5%

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

Alternative 6: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 1.4e-292)
   (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 <= 1.4e-292) {
		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 <= 1.4d-292) 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 <= 1.4e-292) {
		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 <= 1.4e-292:
		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 <= 1.4e-292)
		tmp = atan(Float64(v / Float64(Float64(Float64(H * 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 <= 1.4e-292)
		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, 1.4e-292], N[ArcTan[N[(v / N[(N[(N[(H * 9.8), $MachinePrecision] / v), $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 < 1.4000000000000001e-292

   1. Initial program 73.2%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in v around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

   7. Simplified 50.2%

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

   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. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 72.3%

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

   10. Simplified 72.3%

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

   if 1.4000000000000001e-292 < v

   1. Initial program 75.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in H around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 68.2%

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

   7. Simplified 68.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 68.2%

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

   9. Applied egg-rr 68.2%

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

Alternative 7: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7 \cdot 10^{-139}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -7e-139) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -7e-139) {
		tmp = atan(-1.0);
	} 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 <= (-7d-139)) then
        tmp = atan((-1.0d0))
    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 <= -7e-139) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -7e-139:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -7e-139)
		tmp = atan(-1.0);
	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 <= -7e-139)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -7e-139], N[ArcTan[-1.0], $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 < -7.00000000000000002e-139

   1. Initial program 66.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 66.7%

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

   3. Simplified 66.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 81.6%

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

   if -7.00000000000000002e-139 < v

   1. Initial program 79.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in H around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. 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 62.4%

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

   7. Simplified 62.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 62.4%

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

   9. Applied egg-rr 62.4%

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

Alternative 8: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.7 \cdot 10^{-271}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -2.7e-271) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.7e-271) {
		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 <= (-2.7d-271)) 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 <= -2.7e-271) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.7e-271:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.7e-271)
		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 <= -2.7e-271)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.7e-271], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.6999999999999999e-271

   1. Initial program 71.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 71.9%

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

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
   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 69.4%

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

   if -2.6999999999999999e-271 < v

   1. Initial program 76.2%

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

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

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

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

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

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

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

      4. sub-neg N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in v around inf

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

   7. Simplified 61.2%

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

Alternative 9: 34.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} -1 \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (atan -1.0))

C

double code(double v, double H) {
	return atan(-1.0);
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((-1.0d0))
end function

Java

public static double code(double v, double H) {
	return Math.atan(-1.0);
}

Python

def code(v, H):
	return math.atan(-1.0)

Julia

function code(v, H)
	return atan(-1.0)
end

MATLAB

function tmp = code(v, H)
	tmp = atan(-1.0);
end

Wolfram

code[v_, H_] := N[ArcTan[-1.0], $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
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 74.1%

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

3. Simplified 74.1%

   \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v + H \cdot -19.6}}\right)} \]
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 33.8%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))