Optimal throwing angle

Percentage Accurate: 67.5% → 99.7%

Time: 10.7s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot v - 19.6 \cdot H\\ t_1 := \frac{v}{\sqrt{t\_0}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{t\_0}}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (- (* v v) (* 19.6 H))) (t_1 (/ v (sqrt t_0))))
   (if (<= t_1 -5e-322)
     (atan (* v (sqrt (/ 1.0 t_0))))
     (if (<= t_1 0.0)
       (atan (/ v (fabs (- v (* 9.8 (/ H v))))))
       (atan (/ v (sqrt (fma v v (* H -19.6)))))))))

C

double code(double v, double H) {
	double t_0 = (v * v) - (19.6 * H);
	double t_1 = v / sqrt(t_0);
	double tmp;
	if (t_1 <= -5e-322) {
		tmp = atan((v * sqrt((1.0 / t_0))));
	} else if (t_1 <= 0.0) {
		tmp = atan((v / fabs((v - (9.8 * (H / v))))));
	} else {
		tmp = atan((v / sqrt(fma(v, v, (H * -19.6)))));
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = Float64(Float64(v * v) - Float64(19.6 * H))
	t_1 = Float64(v / sqrt(t_0))
	tmp = 0.0
	if (t_1 <= -5e-322)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / t_0))));
	elseif (t_1 <= 0.0)
		tmp = atan(Float64(v / abs(Float64(v - Float64(9.8 * Float64(H / v))))));
	else
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(H * -19.6)))));
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(v / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-322], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[ArcTan[N[(v / N[Abs[N[(v - N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < -4.99006e-322

   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. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   6. Step-by-step derivation

      1. pow2 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -4.99006e-322 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < 0.0

   1. Initial program 10.7%

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

      1. sqr-neg 10.7%

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

      2. sqr-neg 10.7%

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

      3. fma-neg 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 10.7%

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

      7. metadata-eval 10.7%

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

   3. Simplified 10.7%

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

   5. Taylor expanded in H around 0 65.7%

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

      1. add-sqr-sqrt 60.4%

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

      2. sqrt-unprod 10.7%

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

      3. pow2 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{{\left(v + -9.8 \cdot \frac{H}{v}\right)}^{2}}}}\right) \]

      4. +-commutative 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(-9.8 \cdot \frac{H}{v} + v\right)}}^{2}}}\right) \]

      5. fma-define 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}}^{2}}}\right) \]

   7. Applied egg-rr 10.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 10.7%

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

      2. rem-sqrt-square 99.4%

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

      3. fma-undefine 99.4%

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

      4. associate-*r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-/l* 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-*r/ 99.4%

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

      9. rem-cube-cbrt 99.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\left|H \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{19.6}\right)}^{3}}}{v}\right) + v\right|}\right) \]

      10. fma-define 99.4%

          \[\leadsto \tan^{-1} \left(\frac{v}{\left|\color{blue}{\mathsf{fma}\left(H, -0.5 \cdot \frac{{\left(\sqrt[3]{19.6}\right)}^{3}}{v}, v\right)}\right|}\right) \]

      11. rem-cube-cbrt 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in H around inf 99.4%

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

   if 0.0 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))

   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. sqr-neg 99.7%

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

      2. sqr-neg 99.7%

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

      3. fma-neg 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 99.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{elif}\;\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot v - 19.6 \cdot H\\ t_1 := \frac{v}{\sqrt{t\_0}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{t\_0}}\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (- (* v v) (* 19.6 H))) (t_1 (/ v (sqrt t_0))))
   (if (<= t_1 -5e-322)
     (atan (* v (sqrt (/ 1.0 t_0))))
     (if (<= t_1 0.0) (atan (/ v (fabs (- v (* 9.8 (/ H v)))))) (atan t_1)))))

C

double code(double v, double H) {
	double t_0 = (v * v) - (19.6 * H);
	double t_1 = v / sqrt(t_0);
	double tmp;
	if (t_1 <= -5e-322) {
		tmp = atan((v * sqrt((1.0 / t_0))));
	} else if (t_1 <= 0.0) {
		tmp = atan((v / fabs((v - (9.8 * (H / v))))));
	} else {
		tmp = atan(t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (v * v) - (19.6d0 * h)
    t_1 = v / sqrt(t_0)
    if (t_1 <= (-5d-322)) then
        tmp = atan((v * sqrt((1.0d0 / t_0))))
    else if (t_1 <= 0.0d0) then
        tmp = atan((v / abs((v - (9.8d0 * (h / v))))))
    else
        tmp = atan(t_1)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = (v * v) - (19.6 * H);
	double t_1 = v / Math.sqrt(t_0);
	double tmp;
	if (t_1 <= -5e-322) {
		tmp = Math.atan((v * Math.sqrt((1.0 / t_0))));
	} else if (t_1 <= 0.0) {
		tmp = Math.atan((v / Math.abs((v - (9.8 * (H / v))))));
	} else {
		tmp = Math.atan(t_1);
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = (v * v) - (19.6 * H)
	t_1 = v / math.sqrt(t_0)
	tmp = 0
	if t_1 <= -5e-322:
		tmp = math.atan((v * math.sqrt((1.0 / t_0))))
	elif t_1 <= 0.0:
		tmp = math.atan((v / math.fabs((v - (9.8 * (H / v))))))
	else:
		tmp = math.atan(t_1)
	return tmp

Julia

function code(v, H)
	t_0 = Float64(Float64(v * v) - Float64(19.6 * H))
	t_1 = Float64(v / sqrt(t_0))
	tmp = 0.0
	if (t_1 <= -5e-322)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / t_0))));
	elseif (t_1 <= 0.0)
		tmp = atan(Float64(v / abs(Float64(v - Float64(9.8 * Float64(H / v))))));
	else
		tmp = atan(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = (v * v) - (19.6 * H);
	t_1 = v / sqrt(t_0);
	tmp = 0.0;
	if (t_1 <= -5e-322)
		tmp = atan((v * sqrt((1.0 / t_0))));
	elseif (t_1 <= 0.0)
		tmp = atan((v / abs((v - (9.8 * (H / v))))));
	else
		tmp = atan(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(v / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-322], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[ArcTan[N[(v / N[Abs[N[(v - N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < -4.99006e-322

   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. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   6. Step-by-step derivation

      1. pow2 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -4.99006e-322 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))) < 0.0

   1. Initial program 10.7%

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

      1. sqr-neg 10.7%

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

      2. sqr-neg 10.7%

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

      3. fma-neg 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 10.7%

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

      7. metadata-eval 10.7%

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

   3. Simplified 10.7%

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

   5. Taylor expanded in H around 0 65.7%

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

      1. add-sqr-sqrt 60.4%

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

      2. sqrt-unprod 10.7%

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

      3. pow2 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{{\left(v + -9.8 \cdot \frac{H}{v}\right)}^{2}}}}\right) \]

      4. +-commutative 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(-9.8 \cdot \frac{H}{v} + v\right)}}^{2}}}\right) \]

      5. fma-define 10.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}}^{2}}}\right) \]

   7. Applied egg-rr 10.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 10.7%

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

      2. rem-sqrt-square 99.4%

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

      3. fma-undefine 99.4%

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

      4. associate-*r/ 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-/l* 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-*r/ 99.4%

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

      9. rem-cube-cbrt 99.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\left|H \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{19.6}\right)}^{3}}}{v}\right) + v\right|}\right) \]

      10. fma-define 99.4%

          \[\leadsto \tan^{-1} \left(\frac{v}{\left|\color{blue}{\mathsf{fma}\left(H, -0.5 \cdot \frac{{\left(\sqrt[3]{19.6}\right)}^{3}}{v}, v\right)}\right|}\right) \]

      11. rem-cube-cbrt 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in H around inf 99.4%

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

   if 0.0 < (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))

   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. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{elif}\;\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}} \leq 0:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= (* 19.6 H) -1e-301)
   (atan (* v (/ 1.0 (hypot v (sqrt (* H -19.6))))))
   (atan (/ v (fabs (- v (* 9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if ((19.6 * H) <= -1e-301) {
		tmp = atan((v * (1.0 / hypot(v, sqrt((H * -19.6))))));
	} else {
		tmp = atan((v / fabs((v - (9.8 * (H / v))))));
	}
	return tmp;
}

Java

public static double code(double v, double H) {
	double tmp;
	if ((19.6 * H) <= -1e-301) {
		tmp = Math.atan((v * (1.0 / Math.hypot(v, Math.sqrt((H * -19.6))))));
	} else {
		tmp = Math.atan((v / Math.abs((v - (9.8 * (H / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if (19.6 * H) <= -1e-301:
		tmp = math.atan((v * (1.0 / math.hypot(v, math.sqrt((H * -19.6))))))
	else:
		tmp = math.atan((v / math.fabs((v - (9.8 * (H / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (Float64(19.6 * H) <= -1e-301)
		tmp = atan(Float64(v * Float64(1.0 / hypot(v, sqrt(Float64(H * -19.6))))));
	else
		tmp = atan(Float64(v / abs(Float64(v - Float64(9.8 * Float64(H / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if ((19.6 * H) <= -1e-301)
		tmp = atan((v * (1.0 / hypot(v, sqrt((H * -19.6))))));
	else
		tmp = atan((v / abs((v - (9.8 * (H / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[N[(19.6 * H), $MachinePrecision], -1e-301], N[ArcTan[N[(v * N[(1.0 / N[Sqrt[v ^ 2 + N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[Abs[N[(v - N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H) < -1.00000000000000007e-301

   1. Initial program 72.7%

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

      1. metadata-eval 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in v around 0 72.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   6. Step-by-step derivation

      1. sqrt-div 72.6%

         \[\leadsto \tan^{-1} \left(v \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{{v}^{2} - 19.6 \cdot H}}}\right) \]

      2. metadata-eval 72.6%

         \[\leadsto \tan^{-1} \left(v \cdot \frac{\color{blue}{1}}{\sqrt{{v}^{2} - 19.6 \cdot H}}\right) \]

      3. add-cube-cbrt 72.3%

         \[\leadsto \tan^{-1} \left(v \cdot \frac{1}{\sqrt{{v}^{2} - \color{blue}{\left(\sqrt[3]{19.6 \cdot H} \cdot \sqrt[3]{19.6 \cdot H}\right) \cdot \sqrt[3]{19.6 \cdot H}}}}\right) \]

      4. unpow3 72.3%

         \[\leadsto \tan^{-1} \left(v \cdot \frac{1}{\sqrt{{v}^{2} - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

      5. pow2 72.3%

         \[\leadsto \tan^{-1} \left(v \cdot \frac{1}{\sqrt{\color{blue}{v \cdot v} - {\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}\right) \]

      6. div-inv 72.4%

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{v}{\sqrt{v \cdot v - {\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}\right)} \]

      7. clear-num 71.7%

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{\sqrt{v \cdot v - {\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}{v}}\right)} \]

   7. Applied egg-rr 72.0%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{1}{\frac{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}{v}}\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ 72.6%

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

      2. fma-undefine 72.6%

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

      3. add-sqr-sqrt 72.6%

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

      4. hypot-define 99.2%

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

   9. Applied egg-rr 99.2%

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

   if -1.00000000000000007e-301 < (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)

   1. Initial program 59.7%

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

      1. sqr-neg 59.7%

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

      2. sqr-neg 59.7%

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

      3. fma-neg 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 59.7%

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

      7. metadata-eval 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in H around 0 52.7%

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

      1. add-sqr-sqrt 51.8%

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

      2. sqrt-unprod 59.7%

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

      3. pow2 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{{\left(v + -9.8 \cdot \frac{H}{v}\right)}^{2}}}}\right) \]

      4. +-commutative 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(-9.8 \cdot \frac{H}{v} + v\right)}}^{2}}}\right) \]

      5. fma-define 59.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}}^{2}}}\right) \]

   7. Applied egg-rr 59.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 59.7%

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

      2. rem-sqrt-square 100.0%

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

      3. fma-undefine 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-*r/ 100.0%

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

      9. rem-cube-cbrt 100.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\left|H \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{19.6}\right)}^{3}}}{v}\right) + v\right|}\right) \]

      10. fma-define 100.0%

          \[\leadsto \tan^{-1} \left(\frac{v}{\left|\color{blue}{\mathsf{fma}\left(H, -0.5 \cdot \frac{{\left(\sqrt[3]{19.6}\right)}^{3}}{v}, v\right)}\right|}\right) \]

      11. rem-cube-cbrt 100.0%

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

      12. associate-*r/ 100.0%

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

      13. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in H around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;19.6 \cdot H \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{\mathsf{hypot}\left(v, \sqrt{H \cdot -19.6}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.1 \cdot 10^{+69}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{elif}\;v \leq 1.2 \cdot 10^{+60}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.1e+69)
   (atan (/ v (fabs (- v (* 9.8 (/ H v))))))
   (if (<= v 1.2e+60)
     (atan (/ v (sqrt (- (* v v) (* 19.6 H)))))
     (atan (/ v (+ v (* (/ H v) -9.8)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.1e+69) {
		tmp = atan((v / fabs((v - (9.8 * (H / v))))));
	} else if (v <= 1.2e+60) {
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-3.1d+69)) then
        tmp = atan((v / abs((v - (9.8d0 * (h / v))))))
    else if (v <= 1.2d+60) then
        tmp = atan((v / sqrt(((v * v) - (19.6d0 * h)))))
    else
        tmp = atan((v / (v + ((h / v) * (-9.8d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.1e+69) {
		tmp = Math.atan((v / Math.abs((v - (9.8 * (H / v))))));
	} else if (v <= 1.2e+60) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = Math.atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.1e+69:
		tmp = math.atan((v / math.fabs((v - (9.8 * (H / v))))))
	elif v <= 1.2e+60:
		tmp = math.atan((v / math.sqrt(((v * v) - (19.6 * H)))))
	else:
		tmp = math.atan((v / (v + ((H / v) * -9.8))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.1e+69)
		tmp = atan(Float64(v / abs(Float64(v - Float64(9.8 * Float64(H / v))))));
	elseif (v <= 1.2e+60)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(Float64(H / v) * -9.8))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.1e+69)
		tmp = atan((v / abs((v - (9.8 * (H / v))))));
	elseif (v <= 1.2e+60)
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	else
		tmp = atan((v / (v + ((H / v) * -9.8))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.1e+69], N[ArcTan[N[(v / N[Abs[N[(v - N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.2e+60], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.0999999999999998e69

   1. Initial program 31.7%

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

      1. sqr-neg 31.7%

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

      2. sqr-neg 31.7%

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

      3. fma-neg 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 31.7%

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

      7. metadata-eval 31.7%

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

   3. Simplified 31.7%

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

   5. Taylor expanded in H around 0 1.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 31.7%

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

      3. pow2 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{{\left(v + -9.8 \cdot \frac{H}{v}\right)}^{2}}}}\right) \]

      4. +-commutative 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(-9.8 \cdot \frac{H}{v} + v\right)}}^{2}}}\right) \]

      5. fma-define 31.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}}^{2}}}\right) \]

   7. Applied egg-rr 31.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 31.7%

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

      2. rem-sqrt-square 98.7%

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

      3. fma-undefine 98.7%

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

      4. associate-*r/ 97.8%

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

      5. *-commutative 97.8%

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

      6. associate-/l* 98.7%

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

      7. metadata-eval 98.7%

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

      8. associate-*r/ 98.7%

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

      9. rem-cube-cbrt 98.7%

         \[\leadsto \tan^{-1} \left(\frac{v}{\left|H \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{19.6}\right)}^{3}}}{v}\right) + v\right|}\right) \]

      10. fma-define 98.7%

          \[\leadsto \tan^{-1} \left(\frac{v}{\left|\color{blue}{\mathsf{fma}\left(H, -0.5 \cdot \frac{{\left(\sqrt[3]{19.6}\right)}^{3}}{v}, v\right)}\right|}\right) \]

      11. rem-cube-cbrt 98.7%

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

      12. associate-*r/ 98.7%

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

      13. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in H around inf 98.7%

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

   if -3.0999999999999998e69 < v < 1.2e60

   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. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 1.2e60 < v

   1. Initial program 34.1%

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

      1. sqr-neg 34.1%

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

      2. sqr-neg 34.1%

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

      3. fma-neg 34.1%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 34.1%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 34.1%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 34.1%

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

      7. metadata-eval 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in H around 0 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.1 \cdot 10^{+69}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{elif}\;v \leq 1.2 \cdot 10^{+60}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.22 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|v - 9.8 \cdot \frac{H}{v}\right|}\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-27}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{1}{\frac{v}{H \cdot -9.8}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.22e-76)
   (atan (/ v (fabs (- v (* 9.8 (/ H v))))))
   (if (<= v 8e-27)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (/ 1.0 (/ v (* H -9.8)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.22e-76) {
		tmp = atan((v / fabs((v - (9.8 * (H / v))))));
	} else if (v <= 8e-27) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.22d-76)) then
        tmp = atan((v / abs((v - (9.8d0 * (h / v))))))
    else if (v <= 8d-27) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((v / (v + (1.0d0 / (v / (h * (-9.8d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.22e-76) {
		tmp = Math.atan((v / Math.abs((v - (9.8 * (H / v))))));
	} else if (v <= 8e-27) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.22e-76:
		tmp = math.atan((v / math.fabs((v - (9.8 * (H / v))))))
	elif v <= 8e-27:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (1.0 / (v / (H * -9.8))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.22e-76)
		tmp = atan(Float64(v / abs(Float64(v - Float64(9.8 * Float64(H / v))))));
	elseif (v <= 8e-27)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(1.0 / Float64(v / Float64(H * -9.8))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.22e-76)
		tmp = atan((v / abs((v - (9.8 * (H / v))))));
	elseif (v <= 8e-27)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.22e-76], N[ArcTan[N[(v / N[Abs[N[(v - N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 8e-27], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(1.0 / N[(v / N[(H * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.22e-76

   1. Initial program 60.9%

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

      1. sqr-neg 60.9%

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

      2. sqr-neg 60.9%

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

      3. fma-neg 60.9%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 60.9%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 60.9%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 60.9%

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

      7. metadata-eval 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in H around 0 1.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 51.2%

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

      3. pow2 51.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{{\left(v + -9.8 \cdot \frac{H}{v}\right)}^{2}}}}\right) \]

      4. +-commutative 51.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(-9.8 \cdot \frac{H}{v} + v\right)}}^{2}}}\right) \]

      5. fma-define 51.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}}^{2}}}\right) \]

   7. Applied egg-rr 51.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 51.2%

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

      2. rem-sqrt-square 89.6%

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

      3. fma-undefine 89.6%

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

      4. associate-*r/ 89.1%

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

      5. *-commutative 89.1%

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

      6. associate-/l* 89.6%

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

      7. metadata-eval 89.6%

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

      8. associate-*r/ 89.6%

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

      9. rem-cube-cbrt 89.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\left|H \cdot \left(-0.5 \cdot \frac{\color{blue}{{\left(\sqrt[3]{19.6}\right)}^{3}}}{v}\right) + v\right|}\right) \]

      10. fma-define 89.6%

          \[\leadsto \tan^{-1} \left(\frac{v}{\left|\color{blue}{\mathsf{fma}\left(H, -0.5 \cdot \frac{{\left(\sqrt[3]{19.6}\right)}^{3}}{v}, v\right)}\right|}\right) \]

      11. rem-cube-cbrt 89.6%

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

      12. associate-*r/ 89.6%

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

      13. metadata-eval 89.6%

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

   9. Simplified 89.6%

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

   10. Taylor expanded in H around inf 89.6%

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

   if -1.22e-76 < v < 8.0000000000000003e-27

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]

   6. Taylor expanded in v around 0 90.9%

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

   if 8.0000000000000003e-27 < v

   1. Initial program 47.4%

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

      1. sqr-neg 47.4%

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

      2. sqr-neg 47.4%

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

      3. fma-neg 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 47.4%

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

      7. metadata-eval 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in H around 0 92.4%

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

      1. associate-*r/ 92.4%

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

      2. clear-num 92.4%

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

      3. *-commutative 92.4%

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

   7. Applied egg-rr 92.4%

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

Alternative 6: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.22 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.26 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{1}{\frac{v}{H \cdot -9.8}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.22e-76)
   (atan (+ (/ (* (/ H v) -9.8) v) -1.0))
   (if (<= v 1.26e-26)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (/ 1.0 (/ v (* H -9.8)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.22e-76) {
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	} else if (v <= 1.26e-26) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.22d-76)) then
        tmp = atan(((((h / v) * (-9.8d0)) / v) + (-1.0d0)))
    else if (v <= 1.26d-26) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((v / (v + (1.0d0 / (v / (h * (-9.8d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.22e-76) {
		tmp = Math.atan(((((H / v) * -9.8) / v) + -1.0));
	} else if (v <= 1.26e-26) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.22e-76:
		tmp = math.atan(((((H / v) * -9.8) / v) + -1.0))
	elif v <= 1.26e-26:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (1.0 / (v / (H * -9.8))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.22e-76)
		tmp = atan(Float64(Float64(Float64(Float64(H / v) * -9.8) / v) + -1.0));
	elseif (v <= 1.26e-26)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(1.0 / Float64(v / Float64(H * -9.8))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.22e-76)
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	elseif (v <= 1.26e-26)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.22e-76], N[ArcTan[N[(N[(N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.26e-26], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(1.0 / N[(v / N[(H * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.22e-76

   1. Initial program 60.9%

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

      1. metadata-eval 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in v around -inf 88.9%

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

      1. associate-*r/ 88.5%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-9.8 \cdot H}{{v}^{2}}} - 1\right) \]

      2. pow2 88.5%

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

      3. associate-/r* 88.5%

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

      4. associate-*r/ 89.0%

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

   7. Applied egg-rr 89.0%

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

   if -1.22e-76 < v < 1.26000000000000002e-26

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 99.5%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]

   6. Taylor expanded in v around 0 90.9%

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

   if 1.26000000000000002e-26 < v

   1. Initial program 47.4%

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

      1. sqr-neg 47.4%

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

      2. sqr-neg 47.4%

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

      3. fma-neg 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 47.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 47.4%

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

      7. metadata-eval 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in H around 0 92.4%

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

      1. associate-*r/ 92.4%

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

      2. clear-num 92.4%

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

      3. *-commutative 92.4%

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

   7. Applied egg-rr 92.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.22 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.26 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{1}{\frac{v}{H \cdot -9.8}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.65 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{elif}\;v \leq 9.2 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(v \cdot \frac{-0.10204081632653061}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.65e-137)
   (atan (+ (/ (* (/ H v) -9.8) v) -1.0))
   (if (<= v 9.2e-76)
     (atan (* v (* v (/ -0.10204081632653061 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.65e-137) {
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	} else if (v <= 9.2e-76) {
		tmp = atan((v * (v * (-0.10204081632653061 / 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 <= (-1.65d-137)) then
        tmp = atan(((((h / v) * (-9.8d0)) / v) + (-1.0d0)))
    else if (v <= 9.2d-76) then
        tmp = atan((v * (v * ((-0.10204081632653061d0) / 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 <= -1.65e-137) {
		tmp = Math.atan(((((H / v) * -9.8) / v) + -1.0));
	} else if (v <= 9.2e-76) {
		tmp = Math.atan((v * (v * (-0.10204081632653061 / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.65e-137:
		tmp = math.atan(((((H / v) * -9.8) / v) + -1.0))
	elif v <= 9.2e-76:
		tmp = math.atan((v * (v * (-0.10204081632653061 / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.65e-137)
		tmp = atan(Float64(Float64(Float64(Float64(H / v) * -9.8) / v) + -1.0));
	elseif (v <= 9.2e-76)
		tmp = atan(Float64(v * Float64(v * Float64(-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.65e-137)
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	elseif (v <= 9.2e-76)
		tmp = atan((v * (v * (-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.65e-137], N[ArcTan[N[(N[(N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 9.2e-76], N[ArcTan[N[(v * N[(v * N[(-0.10204081632653061 / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.6500000000000001e-137

   1. Initial program 65.4%

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

      1. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in v around -inf 84.5%

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

      1. associate-*r/ 84.2%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-9.8 \cdot H}{{v}^{2}}} - 1\right) \]

      2. pow2 84.2%

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

      3. associate-/r* 84.2%

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

      4. associate-*r/ 84.7%

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

   7. Applied egg-rr 84.7%

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

   if -1.6500000000000001e-137 < v < 9.20000000000000025e-76

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. fma-neg 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in H around 0 18.6%

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

   6. Taylor expanded in v around 0 18.6%

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

      1. *-un-lft-identity 18.6%

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

      2. *-un-lft-identity 18.6%

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

      3. *-commutative 18.6%

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

      4. times-frac 18.6%

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

      5. clear-num 18.6%

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

      6. div-inv 18.6%

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

      7. metadata-eval 18.6%

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

   8. Applied egg-rr 18.6%

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

      1. *-lft-identity 18.6%

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

      2. *-commutative 18.6%

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

      3. associate-*r/ 18.6%

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

      4. associate-*l/ 18.6%

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

      5. *-commutative 18.6%

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

      6. associate-/l* 18.6%

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

   10. Simplified 18.6%

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

   if 9.20000000000000025e-76 < v

   1. Initial program 52.4%

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

      1. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in v around inf 87.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.65 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{elif}\;v \leq 9.2 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(v \cdot \frac{-0.10204081632653061}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.55 \cdot 10^{-173}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 9.2 \cdot 10^{-76}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(v \cdot \frac{-0.10204081632653061}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.55e-173)
   (atan -1.0)
   (if (<= v 9.2e-76)
     (atan (* v (* v (/ -0.10204081632653061 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.55e-173) {
		tmp = atan(-1.0);
	} else if (v <= 9.2e-76) {
		tmp = atan((v * (v * (-0.10204081632653061 / 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 <= (-1.55d-173)) then
        tmp = atan((-1.0d0))
    else if (v <= 9.2d-76) then
        tmp = atan((v * (v * ((-0.10204081632653061d0) / 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 <= -1.55e-173) {
		tmp = Math.atan(-1.0);
	} else if (v <= 9.2e-76) {
		tmp = Math.atan((v * (v * (-0.10204081632653061 / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.55e-173:
		tmp = math.atan(-1.0)
	elif v <= 9.2e-76:
		tmp = math.atan((v * (v * (-0.10204081632653061 / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.55e-173)
		tmp = atan(-1.0);
	elseif (v <= 9.2e-76)
		tmp = atan(Float64(v * Float64(v * Float64(-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.55e-173)
		tmp = atan(-1.0);
	elseif (v <= 9.2e-76)
		tmp = atan((v * (v * (-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.55e-173], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 9.2e-76], N[ArcTan[N[(v * N[(v * N[(-0.10204081632653061 / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.55000000000000003e-173

   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. metadata-eval 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in v around -inf 77.7%

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

   if -1.55000000000000003e-173 < v < 9.20000000000000025e-76

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. fma-neg 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 99.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in H around 0 20.7%

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

   6. Taylor expanded in v around 0 20.7%

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

      1. *-un-lft-identity 20.7%

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

      2. *-un-lft-identity 20.7%

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

      3. *-commutative 20.7%

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

      4. times-frac 20.7%

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

      5. clear-num 20.7%

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

      6. div-inv 20.7%

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

      7. metadata-eval 20.7%

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

   8. Applied egg-rr 20.7%

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

      1. *-lft-identity 20.7%

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

      2. *-commutative 20.7%

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

      3. associate-*r/ 20.6%

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

      4. associate-*l/ 20.7%

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

      5. *-commutative 20.7%

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

      6. associate-/l* 20.7%

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

   10. Simplified 20.7%

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

   if 9.20000000000000025e-76 < v

   1. Initial program 52.4%

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

      1. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in v around inf 87.0%

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

Alternative 9: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.4 \cdot 10^{-138}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{1}{\frac{v}{H \cdot -9.8}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.4e-138)
   (atan (+ (/ (* (/ H v) -9.8) v) -1.0))
   (atan (/ v (+ v (/ 1.0 (/ v (* H -9.8))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.4e-138) {
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	} else {
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2.4d-138)) then
        tmp = atan(((((h / v) * (-9.8d0)) / v) + (-1.0d0)))
    else
        tmp = atan((v / (v + (1.0d0 / (v / (h * (-9.8d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.4e-138) {
		tmp = Math.atan(((((H / v) * -9.8) / v) + -1.0));
	} else {
		tmp = Math.atan((v / (v + (1.0 / (v / (H * -9.8))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.4e-138:
		tmp = math.atan(((((H / v) * -9.8) / v) + -1.0))
	else:
		tmp = math.atan((v / (v + (1.0 / (v / (H * -9.8))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.4e-138)
		tmp = atan(Float64(Float64(Float64(Float64(H / v) * -9.8) / v) + -1.0));
	else
		tmp = atan(Float64(v / Float64(v + Float64(1.0 / Float64(v / Float64(H * -9.8))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.4e-138)
		tmp = atan(((((H / v) * -9.8) / v) + -1.0));
	else
		tmp = atan((v / (v + (1.0 / (v / (H * -9.8))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.4e-138], N[ArcTan[N[(N[(N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(1.0 / N[(v / N[(H * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.3999999999999999e-138

   1. Initial program 65.4%

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

      1. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in v around -inf 84.5%

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

      1. associate-*r/ 84.2%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-9.8 \cdot H}{{v}^{2}}} - 1\right) \]

      2. pow2 84.2%

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

      3. associate-/r* 84.2%

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

      4. associate-*r/ 84.7%

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

   7. Applied egg-rr 84.7%

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

   if -2.3999999999999999e-138 < 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. sqr-neg 70.6%

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

      2. sqr-neg 70.6%

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

      3. fma-neg 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 70.6%

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

      7. metadata-eval 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in H around 0 61.1%

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

      1. associate-*r/ 61.1%

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

      2. clear-num 61.1%

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

      3. *-commutative 61.1%

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

   7. Applied egg-rr 61.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.4 \cdot 10^{-138}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{1}{\frac{v}{H \cdot -9.8}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{H}{v} \cdot -9.8\\ \mathbf{if}\;v \leq -2.5 \cdot 10^{-141}:\\ \;\;\;\;\tan^{-1} \left(\frac{t\_0}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (* (/ H v) -9.8)))
   (if (<= v -2.5e-141) (atan (+ (/ t_0 v) -1.0)) (atan (/ v (+ v t_0))))))

C

double code(double v, double H) {
	double t_0 = (H / v) * -9.8;
	double tmp;
	if (v <= -2.5e-141) {
		tmp = atan(((t_0 / v) + -1.0));
	} else {
		tmp = atan((v / (v + t_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 = (h / v) * (-9.8d0)
    if (v <= (-2.5d-141)) then
        tmp = atan(((t_0 / v) + (-1.0d0)))
    else
        tmp = atan((v / (v + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = (H / v) * -9.8;
	double tmp;
	if (v <= -2.5e-141) {
		tmp = Math.atan(((t_0 / v) + -1.0));
	} else {
		tmp = Math.atan((v / (v + t_0)));
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = (H / v) * -9.8
	tmp = 0
	if v <= -2.5e-141:
		tmp = math.atan(((t_0 / v) + -1.0))
	else:
		tmp = math.atan((v / (v + t_0)))
	return tmp

Julia

function code(v, H)
	t_0 = Float64(Float64(H / v) * -9.8)
	tmp = 0.0
	if (v <= -2.5e-141)
		tmp = atan(Float64(Float64(t_0 / v) + -1.0));
	else
		tmp = atan(Float64(v / Float64(v + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = (H / v) * -9.8;
	tmp = 0.0;
	if (v <= -2.5e-141)
		tmp = atan(((t_0 / v) + -1.0));
	else
		tmp = atan((v / (v + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision]}, If[LessEqual[v, -2.5e-141], N[ArcTan[N[(N[(t$95$0 / v), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.5e-141

   1. Initial program 65.4%

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

      1. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in v around -inf 84.5%

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

      1. associate-*r/ 84.2%

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-9.8 \cdot H}{{v}^{2}}} - 1\right) \]

      2. pow2 84.2%

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

      3. associate-/r* 84.2%

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

      4. associate-*r/ 84.7%

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

   7. Applied egg-rr 84.7%

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

   if -2.5e-141 < 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. sqr-neg 70.6%

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

      2. sqr-neg 70.6%

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

      3. fma-neg 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -\left(2 \cdot 9.8\right) \cdot H\right)}}}\right) \]

      4. *-commutative 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -\color{blue}{H \cdot \left(2 \cdot 9.8\right)}\right)}}\right) \]

      5. distribute-rgt-neg-in 70.6%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{H \cdot \left(-2 \cdot 9.8\right)}\right)}}\right) \]

      6. metadata-eval 70.6%

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

      7. metadata-eval 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in H around 0 61.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.5 \cdot 10^{-141}:\\ \;\;\;\;\tan^{-1} \left(\frac{\frac{H}{v} \cdot -9.8}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.8 \cdot 10^{-304}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5.8e-304) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.8e-304) {
		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 <= (-5.8d-304)) 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 <= -5.8e-304) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.8e-304:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.8e-304)
		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 <= -5.8e-304)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.8e-304], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -5.8e-304

   1. Initial program 73.0%

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

      1. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in v around -inf 66.5%

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

   if -5.8e-304 < v

   1. Initial program 65.5%

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

      1. metadata-eval 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in v around inf 64.1%

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

Alternative 12: 34.9% 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 68.8%

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

   1. metadata-eval 68.8%

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

3. Simplified 68.8%

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

5. Taylor expanded in v around -inf 30.1%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))