Optimal throwing angle

Percentage Accurate: 67.6% → 99.2%

Time: 9.4s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.6% 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: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7.6 \cdot 10^{-75}:\\ \;\;\;\;\tan^{-1} \left(-\sqrt{\frac{1}{1 + -19.6 \cdot \frac{H}{{v}^{2}}}}\right)\\ \mathbf{elif}\;v \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{1 - 19.6 \cdot \frac{\frac{H}{v}}{v}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -7.6e-75)
   (atan (- (sqrt (/ 1.0 (+ 1.0 (* -19.6 (/ H (pow v 2.0))))))))
   (if (<= v 6.5e-41)
     (atan (/ v (sqrt (- (* v v) (* H 19.6)))))
     (atan (sqrt (/ 1.0 (- 1.0 (* 19.6 (/ (/ H v) v)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -7.6e-75) {
		tmp = atan(-sqrt((1.0 / (1.0 + (-19.6 * (H / pow(v, 2.0)))))));
	} else if (v <= 6.5e-41) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = atan(sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-7.6d-75)) then
        tmp = atan(-sqrt((1.0d0 / (1.0d0 + ((-19.6d0) * (h / (v ** 2.0d0)))))))
    else if (v <= 6.5d-41) then
        tmp = atan((v / sqrt(((v * v) - (h * 19.6d0)))))
    else
        tmp = atan(sqrt((1.0d0 / (1.0d0 - (19.6d0 * ((h / v) / v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -7.6e-75) {
		tmp = Math.atan(-Math.sqrt((1.0 / (1.0 + (-19.6 * (H / Math.pow(v, 2.0)))))));
	} else if (v <= 6.5e-41) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = Math.atan(Math.sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -7.6e-75:
		tmp = math.atan(-math.sqrt((1.0 / (1.0 + (-19.6 * (H / math.pow(v, 2.0)))))))
	elif v <= 6.5e-41:
		tmp = math.atan((v / math.sqrt(((v * v) - (H * 19.6)))))
	else:
		tmp = math.atan(math.sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -7.6e-75)
		tmp = atan(Float64(-sqrt(Float64(1.0 / Float64(1.0 + Float64(-19.6 * Float64(H / (v ^ 2.0))))))));
	elseif (v <= 6.5e-41)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	else
		tmp = atan(sqrt(Float64(1.0 / Float64(1.0 - Float64(19.6 * Float64(Float64(H / v) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -7.6e-75)
		tmp = atan(-sqrt((1.0 / (1.0 + (-19.6 * (H / (v ^ 2.0)))))));
	elseif (v <= 6.5e-41)
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	else
		tmp = atan(sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -7.6e-75], N[ArcTan[(-N[Sqrt[N[(1.0 / N[(1.0 + N[(-19.6 * N[(H / N[Power[v, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision], If[LessEqual[v, 6.5e-41], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Sqrt[N[(1.0 / N[(1.0 - N[(19.6 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.59999999999999987e-75

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. sqr-neg 53.1%

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

      3. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

      1. clear-num 53.1%

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

      2. associate-/r/ 53.2%

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

      3. pow1/2 53.2%

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

      4. pow-flip 53.2%

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

      5. sub-neg 53.2%

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

      6. +-commutative 53.2%

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

      7. *-commutative 53.2%

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

      8. distribute-rgt-neg-in 53.2%

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

      9. fma-define 53.2%

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

      10. metadata-eval 53.2%

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

      11. pow2 53.2%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      12. metadata-eval 53.2%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{-0.5}} \cdot v\right) \]

   6. Applied egg-rr 53.2%

      \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.5} \cdot v\right)} \]

   7. Taylor expanded in v around inf 53.2%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left({v}^{2} \cdot \left(1 + -19.6 \cdot \frac{H}{{v}^{2}}\right)\right)}}^{-0.5} \cdot v\right) \]

   8. Taylor expanded in v around -inf 99.9%

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

   if -7.59999999999999987e-75 < v < 6.5000000000000004e-41

   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. 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

   if 6.5000000000000004e-41 < v

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. sqr-neg 54.9%

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

      3. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

      1. clear-num 54.9%

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

      2. associate-/r/ 54.9%

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

      3. pow1/2 54.9%

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

      4. pow-flip 54.9%

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

      5. sub-neg 54.9%

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

      6. +-commutative 54.9%

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

      7. *-commutative 54.9%

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

      8. distribute-rgt-neg-in 54.9%

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

      9. fma-define 54.9%

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

      10. metadata-eval 54.9%

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

      11. pow2 54.9%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      12. metadata-eval 54.9%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{-0.5}} \cdot v\right) \]

   6. Applied egg-rr 54.9%

      \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.5} \cdot v\right)} \]

   7. Taylor expanded in v around inf 55.0%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left({v}^{2} \cdot \left(1 + -19.6 \cdot \frac{H}{{v}^{2}}\right)\right)}}^{-0.5} \cdot v\right) \]

   8. Taylor expanded in H around inf 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7.6 \cdot 10^{-75}:\\ \;\;\;\;\tan^{-1} \left(-\sqrt{\frac{1}{1 + -19.6 \cdot \frac{H}{{v}^{2}}}}\right)\\ \mathbf{elif}\;v \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{1 - 19.6 \cdot \frac{\frac{H}{v}}{v}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{1 - 19.6 \cdot \frac{\frac{H}{v}}{v}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 4e-41)
     (atan (/ v (sqrt (- (* v v) (* H 19.6)))))
     (atan (sqrt (/ 1.0 (- 1.0 (* 19.6 (/ (/ H v) v)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 4e-41) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = atan(sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1d+155)) then
        tmp = atan((-1.0d0))
    else if (v <= 4d-41) then
        tmp = atan((v / sqrt(((v * v) - (h * 19.6d0)))))
    else
        tmp = atan(sqrt((1.0d0 / (1.0d0 - (19.6d0 * ((h / v) / v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4e-41) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = Math.atan(Math.sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e+155:
		tmp = math.atan(-1.0)
	elif v <= 4e-41:
		tmp = math.atan((v / math.sqrt(((v * v) - (H * 19.6)))))
	else:
		tmp = math.atan(math.sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 4e-41)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	else
		tmp = atan(sqrt(Float64(1.0 / Float64(1.0 - Float64(19.6 * Float64(Float64(H / v) / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 4e-41)
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	else
		tmp = atan(sqrt((1.0 / (1.0 - (19.6 * ((H / v) / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4e-41], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Sqrt[N[(1.0 / N[(1.0 - N[(19.6 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -1.00000000000000001e155 < v < 4.00000000000000002e-41

   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. 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 4.00000000000000002e-41 < v

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. sqr-neg 54.9%

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

      3. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

      1. clear-num 54.9%

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

      2. associate-/r/ 54.9%

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

      3. pow1/2 54.9%

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

      4. pow-flip 54.9%

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

      5. sub-neg 54.9%

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

      6. +-commutative 54.9%

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

      7. *-commutative 54.9%

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

      8. distribute-rgt-neg-in 54.9%

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

      9. fma-define 54.9%

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

      10. metadata-eval 54.9%

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

      11. pow2 54.9%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \color{blue}{{v}^{2}}\right)\right)}^{\left(-0.5\right)} \cdot v\right) \]

      12. metadata-eval 54.9%

          \[\leadsto \tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{\color{blue}{-0.5}} \cdot v\right) \]

   6. Applied egg-rr 54.9%

      \[\leadsto \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(H, -19.6, {v}^{2}\right)\right)}^{-0.5} \cdot v\right)} \]

   7. Taylor expanded in v around inf 55.0%

      \[\leadsto \tan^{-1} \left({\color{blue}{\left({v}^{2} \cdot \left(1 + -19.6 \cdot \frac{H}{{v}^{2}}\right)\right)}}^{-0.5} \cdot v\right) \]

   8. Taylor expanded in H around inf 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-lft-identity 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{1}{1 - 19.6 \cdot \frac{\frac{H}{v}}{v}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+150}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+155)
   (atan -1.0)
   (if (<= v 1e+150) (atan (/ v (sqrt (- (* v v) (* H 19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = atan(-1.0);
	} else if (v <= 1e+150) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1d+155)) then
        tmp = atan((-1.0d0))
    else if (v <= 1d+150) then
        tmp = atan((v / sqrt(((v * v) - (h * 19.6d0)))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1e+155) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1e+150) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e+155:
		tmp = math.atan(-1.0)
	elif v <= 1e+150:
		tmp = math.atan((v / math.sqrt(((v * v) - (H * 19.6)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 1e+150)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1e+155)
		tmp = atan(-1.0);
	elseif (v <= 1e+150)
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+150], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000001e155

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -1.00000000000000001e155 < v < 9.99999999999999981e149

   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. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. 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

   if 9.99999999999999981e149 < v

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+150}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{\frac{H \cdot -9.8}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.52 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.5e-62)
   (atan (+ -1.0 (/ (/ (* H -9.8) v) v)))
   (if (<= v 1.52e-43)
     (atan (/ v (sqrt (* -19.6 H))))
     (atan (/ v (+ v (* (/ H v) -9.8)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.5e-62) {
		tmp = atan((-1.0 + (((H * -9.8) / v) / v)));
	} else if (v <= 1.52e-43) {
		tmp = atan((v / sqrt((-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 <= (-3.5d-62)) then
        tmp = atan(((-1.0d0) + (((h * (-9.8d0)) / v) / v)))
    else if (v <= 1.52d-43) then
        tmp = atan((v / sqrt(((-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 <= -3.5e-62) {
		tmp = Math.atan((-1.0 + (((H * -9.8) / v) / v)));
	} else if (v <= 1.52e-43) {
		tmp = Math.atan((v / Math.sqrt((-19.6 * H))));
	} else {
		tmp = Math.atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.5e-62:
		tmp = math.atan((-1.0 + (((H * -9.8) / v) / v)))
	elif v <= 1.52e-43:
		tmp = math.atan((v / math.sqrt((-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 <= -3.5e-62)
		tmp = atan(Float64(-1.0 + Float64(Float64(Float64(H * -9.8) / v) / v)));
	elseif (v <= 1.52e-43)
		tmp = atan(Float64(v / sqrt(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 <= -3.5e-62)
		tmp = atan((-1.0 + (((H * -9.8) / v) / v)));
	elseif (v <= 1.52e-43)
		tmp = atan((v / sqrt((-19.6 * H))));
	else
		tmp = atan((v / (v + ((H / v) * -9.8))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.5e-62], N[ArcTan[N[(-1.0 + N[(N[(N[(H * -9.8), $MachinePrecision] / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.52e-43], N[ArcTan[N[(v / N[Sqrt[N[(-19.6 * H), $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 < -3.5000000000000001e-62

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. sqr-neg 52.1%

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

      3. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in v around -inf 92.7%

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

      1. associate-*r/ 92.7%

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

      2. unpow2 92.7%

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

      3. associate-/r* 92.7%

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

      4. *-commutative 92.7%

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

   7. Applied egg-rr 92.7%

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

   if -3.5000000000000001e-62 < v < 1.52e-43

   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. 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. Step-by-step derivation

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in H around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 88.6%

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

      5. distribute-lft-neg-in 88.6%

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

      6. metadata-eval 88.6%

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

      7. rem-cube-cbrt 90.1%

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

      8. *-commutative 90.1%

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

   9. Simplified 90.1%

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

       1. *-un-lft-identity 90.1%

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

       2. *-un-lft-identity 90.1%

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

       3. *-commutative 90.1%

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

   11. Applied egg-rr 90.1%

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

       1. *-lft-identity 90.1%

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

   13. Simplified 90.1%

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

   if 1.52e-43 < v

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. sqr-neg 54.9%

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

      3. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in H around 0 91.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.5 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{\frac{H \cdot -9.8}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.52 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.22 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{\frac{H \cdot -9.8}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.22e-62)
   (atan (+ -1.0 (/ (/ (* H -9.8) v) v)))
   (if (<= v 1.6e-43)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* (/ H v) -9.8)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.22e-62) {
		tmp = atan((-1.0 + (((H * -9.8) / v) / v)));
	} else if (v <= 1.6e-43) {
		tmp = atan((v * sqrt((-0.05102040816326531 / 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 <= (-1.22d-62)) then
        tmp = atan(((-1.0d0) + (((h * (-9.8d0)) / v) / v)))
    else if (v <= 1.6d-43) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / 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 <= -1.22e-62) {
		tmp = Math.atan((-1.0 + (((H * -9.8) / v) / v)));
	} else if (v <= 1.6e-43) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.22e-62:
		tmp = math.atan((-1.0 + (((H * -9.8) / v) / v)))
	elif v <= 1.6e-43:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + ((H / v) * -9.8))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.22e-62)
		tmp = atan(Float64(-1.0 + Float64(Float64(Float64(H * -9.8) / v) / v)));
	elseif (v <= 1.6e-43)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / 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 <= -1.22e-62)
		tmp = atan((-1.0 + (((H * -9.8) / v) / v)));
	elseif (v <= 1.6e-43)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + ((H / v) * -9.8))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.22e-62], N[ArcTan[N[(-1.0 + N[(N[(N[(H * -9.8), $MachinePrecision] / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.6e-43], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $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 < -1.2199999999999999e-62

   1. Initial program 52.1%

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

      1. sqr-neg 52.1%

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

      2. sqr-neg 52.1%

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

      3. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in v around -inf 92.7%

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

      1. associate-*r/ 92.7%

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

      2. unpow2 92.7%

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

      3. associate-/r* 92.7%

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

      4. *-commutative 92.7%

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

   7. Applied egg-rr 92.7%

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

   if -1.2199999999999999e-62 < v < 1.59999999999999992e-43

   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. 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 99.4%

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

   6. Taylor expanded in v around 0 90.0%

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

   if 1.59999999999999992e-43 < v

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. sqr-neg 54.9%

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

      3. metadata-eval 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in H around 0 91.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.22 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{\frac{H \cdot -9.8}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.2 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.2e-137) (atan -1.0) (atan (/ v (+ v (* (/ H v) -9.8))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.2e-137) {
		tmp = atan(-1.0);
	} 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 <= (-2.2d-137)) then
        tmp = atan((-1.0d0))
    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 <= -2.2e-137) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.2e-137:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + ((H / v) * -9.8))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.2e-137)
		tmp = atan(-1.0);
	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 <= -2.2e-137)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + ((H / v) * -9.8))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.2e-137], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(N[(H / v), $MachinePrecision] * -9.8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.2000000000000001e-137

   1. Initial program 59.2%

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

      1. sqr-neg 59.2%

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

      2. sqr-neg 59.2%

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

      3. metadata-eval 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in v around -inf 81.5%

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

   if -2.2000000000000001e-137 < v

   1. Initial program 71.3%

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

      1. sqr-neg 71.3%

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

      2. sqr-neg 71.3%

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

      3. metadata-eval 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in H around 0 66.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.2 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.78 \cdot 10^{-274}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 1.78e-274) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 1.78e-274) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= 1.78d-274) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 1.78e-274) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 1.78e-274:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 1.78e-274)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= 1.78e-274)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 1.78e-274], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.77999999999999991e-274

   1. Initial program 67.8%

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

      1. sqr-neg 67.8%

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

      2. sqr-neg 67.8%

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

      3. metadata-eval 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in v around -inf 65.0%

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

   if 1.77999999999999991e-274 < v

   1. Initial program 65.3%

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

      1. sqr-neg 65.3%

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

      2. sqr-neg 65.3%

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

      3. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in v around inf 74.7%

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

Alternative 8: 34.4% 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 66.5%

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

   1. sqr-neg 66.5%

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

   2. sqr-neg 66.5%

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

   3. metadata-eval 66.5%

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

3. Simplified 66.5%

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

5. Taylor expanded in v around -inf 33.1%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))