Optimal throwing angle

Percentage Accurate: 67.0% → 98.6%

Time: 16.4s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+157}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.22 \cdot 10^{+57}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+157)
   (atan -1.0)
   (if (<= v 1.22e+57)
     (atan (/ v (sqrt (- (* v v) (* 19.6 H)))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+157) {
		tmp = atan(-1.0);
	} else if (v <= 1.22e+57) {
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d+157)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.22d+57) then
        tmp = atan((v / sqrt(((v * v) - (19.6d0 * h)))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e+157) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.22e+57) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+157:
		tmp = math.atan(-1.0)
	elif v <= 1.22e+57:
		tmp = math.atan((v / math.sqrt(((v * v) - (19.6 * H)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+157)
		tmp = atan(-1.0);
	elseif (v <= 1.22e+57)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+157)
		tmp = atan(-1.0);
	elseif (v <= 1.22e+57)
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+157], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.22e+57], 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[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.99999999999999997e157

   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 -1.99999999999999997e157 < v < 1.22e57

   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 1.22e57 < v

   1. Initial program 31.8%

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

      1. sqr-neg 31.8%

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

      2. sqr-neg 31.8%

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

      3. metadata-eval 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in H around 0 98.6%

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

Alternative 2: 88.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.7 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-79}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{elif}\;v \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v} \cdot \frac{-9.8}{v}\right)}\right)\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{+31}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.7e-19)
   (atan -1.0)
   (if (<= v 2.3e-79)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (if (<= v 8.5e+29)
       (atan (/ v (* v (+ 1.0 (* (/ H v) (/ -9.8 v))))))
       (if (<= v 6.8e+31)
         (atan (* v (/ 1.0 (sqrt (* H -19.6)))))
         (atan (/ v (+ v (* -9.8 (/ H v))))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.7e-19) {
		tmp = atan(-1.0);
	} else if (v <= 2.3e-79) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else if (v <= 8.5e+29) {
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	} else if (v <= 6.8e+31) {
		tmp = atan((v * (1.0 / sqrt((H * -19.6)))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.7d-19)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.3d-79) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else if (v <= 8.5d+29) then
        tmp = atan((v / (v * (1.0d0 + ((h / v) * ((-9.8d0) / v))))))
    else if (v <= 6.8d+31) then
        tmp = atan((v * (1.0d0 / sqrt((h * (-19.6d0))))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.7e-19) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.3e-79) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else if (v <= 8.5e+29) {
		tmp = Math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	} else if (v <= 6.8e+31) {
		tmp = Math.atan((v * (1.0 / Math.sqrt((H * -19.6)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.7e-19:
		tmp = math.atan(-1.0)
	elif v <= 2.3e-79:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	elif v <= 8.5e+29:
		tmp = math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))))
	elif v <= 6.8e+31:
		tmp = math.atan((v * (1.0 / math.sqrt((H * -19.6)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.7e-19)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-79)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	elseif (v <= 8.5e+29)
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / v) * Float64(-9.8 / v))))));
	elseif (v <= 6.8e+31)
		tmp = atan(Float64(v * Float64(1.0 / sqrt(Float64(H * -19.6)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.7e-19)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-79)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	elseif (v <= 8.5e+29)
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	elseif (v <= 6.8e+31)
		tmp = atan((v * (1.0 / sqrt((H * -19.6)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.7e-19], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.3e-79], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 8.5e+29], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / v), $MachinePrecision] * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 6.8e+31], N[ArcTan[N[(v * N[(1.0 / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if v < -1.7000000000000001e-19

   1. Initial program 53.5%

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

      1. sqr-neg 53.5%

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

      2. sqr-neg 53.5%

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

      3. metadata-eval 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in v around -inf 91.4%

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

   if -1.7000000000000001e-19 < v < 2.30000000000000012e-79

   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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in v around 0 89.0%

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

   if 2.30000000000000012e-79 < v < 8.5000000000000006e29

   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

   5. Taylor expanded in v around inf 71.4%

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

      1. associate-*r/ 71.4%

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

   7. Simplified 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. times-frac 71.4%

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

   9. Applied egg-rr 71.4%

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

   if 8.5000000000000006e29 < v < 6.7999999999999996e31

   1. Initial program 99.2%

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

      1. sqr-neg 99.2%

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

      2. sqr-neg 99.2%

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

      3. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in v around 0 99.2%

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

   6. Taylor expanded in v around 0 99.2%

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

      1. clear-num 99.2%

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

      2. sqrt-div 99.2%

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

      3. metadata-eval 99.2%

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

      4. div-inv 99.2%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 6.7999999999999996e31 < v

   1. Initial program 37.9%

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

      1. sqr-neg 37.9%

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

      2. sqr-neg 37.9%

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

      3. metadata-eval 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in H around 0 96.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + -9.8 \cdot \frac{H}{v}}}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{if}\;v \leq -4.5 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.2 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;v \leq 8.6 \cdot 10^{+29}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v} \cdot \frac{-9.8}{v}\right)}\right)\\ \mathbf{elif}\;v \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (atan (* v (sqrt (/ -0.05102040816326531 H))))))
   (if (<= v -4.5e-19)
     (atan -1.0)
     (if (<= v 2.2e-79)
       t_0
       (if (<= v 8.6e+29)
         (atan (/ v (* v (+ 1.0 (* (/ H v) (/ -9.8 v))))))
         (if (<= v 2.1e+31) t_0 (atan (/ v (+ v (* -9.8 (/ H v)))))))))))

C

double code(double v, double H) {
	double t_0 = atan((v * sqrt((-0.05102040816326531 / H))));
	double tmp;
	if (v <= -4.5e-19) {
		tmp = atan(-1.0);
	} else if (v <= 2.2e-79) {
		tmp = t_0;
	} else if (v <= 8.6e+29) {
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	} else if (v <= 2.1e+31) {
		tmp = t_0;
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    if (v <= (-4.5d-19)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.2d-79) then
        tmp = t_0
    else if (v <= 8.6d+29) then
        tmp = atan((v / (v * (1.0d0 + ((h / v) * ((-9.8d0) / v))))))
    else if (v <= 2.1d+31) then
        tmp = t_0
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	double tmp;
	if (v <= -4.5e-19) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.2e-79) {
		tmp = t_0;
	} else if (v <= 8.6e+29) {
		tmp = Math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	} else if (v <= 2.1e+31) {
		tmp = t_0;
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	tmp = 0
	if v <= -4.5e-19:
		tmp = math.atan(-1.0)
	elif v <= 2.2e-79:
		tmp = t_0
	elif v <= 8.6e+29:
		tmp = math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))))
	elif v <= 2.1e+31:
		tmp = t_0
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	t_0 = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))))
	tmp = 0.0
	if (v <= -4.5e-19)
		tmp = atan(-1.0);
	elseif (v <= 2.2e-79)
		tmp = t_0;
	elseif (v <= 8.6e+29)
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / v) * Float64(-9.8 / v))))));
	elseif (v <= 2.1e+31)
		tmp = t_0;
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = atan((v * sqrt((-0.05102040816326531 / H))));
	tmp = 0.0;
	if (v <= -4.5e-19)
		tmp = atan(-1.0);
	elseif (v <= 2.2e-79)
		tmp = t_0;
	elseif (v <= 8.6e+29)
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	elseif (v <= 2.1e+31)
		tmp = t_0;
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[v, -4.5e-19], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.2e-79], t$95$0, If[LessEqual[v, 8.6e+29], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / v), $MachinePrecision] * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2.1e+31], t$95$0, N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if v < -4.50000000000000013e-19

   1. Initial program 53.5%

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

      1. sqr-neg 53.5%

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

      2. sqr-neg 53.5%

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

      3. metadata-eval 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in v around -inf 91.4%

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

   if -4.50000000000000013e-19 < v < 2.1999999999999999e-79 or 8.6000000000000006e29 < v < 2.09999999999999979e31

   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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in v around 0 89.2%

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

   if 2.1999999999999999e-79 < v < 8.6000000000000006e29

   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

   5. Taylor expanded in v around inf 71.4%

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

      1. associate-*r/ 71.4%

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

   7. Simplified 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. times-frac 71.4%

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

   9. Applied egg-rr 71.4%

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

   if 2.09999999999999979e31 < v

   1. Initial program 37.9%

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

      1. sqr-neg 37.9%

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

      2. sqr-neg 37.9%

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

      3. metadata-eval 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in H around 0 96.3%

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

Alternative 4: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.6 \cdot 10^{-95}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(1 + \frac{H}{v} \cdot \frac{-9.8}{v}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.6e-95)
   (atan -1.0)
   (atan (/ v (* v (+ 1.0 (* (/ H v) (/ -9.8 v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-9.6d-95)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v / (v * (1.0d0 + ((h / v) * ((-9.8d0) / v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.6e-95:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v * Float64(1.0 + Float64(Float64(H / v) * Float64(-9.8 / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v * (1.0 + ((H / v) * (-9.8 / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.6e-95], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v * N[(1.0 + N[(N[(H / v), $MachinePrecision] * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.6e-95

   1. Initial program 61.4%

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

      1. sqr-neg 61.4%

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

      2. sqr-neg 61.4%

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

      3. metadata-eval 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in v around -inf 84.3%

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

   if -9.6e-95 < v

   1. Initial program 71.0%

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

      1. sqr-neg 71.0%

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

      2. sqr-neg 71.0%

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

      3. metadata-eval 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in v around inf 63.4%

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

      1. associate-*r/ 63.4%

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

   7. Simplified 63.4%

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

      1. *-commutative 63.4%

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

      2. unpow2 63.4%

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

      3. times-frac 63.4%

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

   9. Applied egg-rr 63.4%

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

Alternative 5: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.08 \cdot 10^{-158}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.05 \cdot 10^{-253}:\\ \;\;\;\;\tan^{-1} \left(\frac{v \cdot \left(v \cdot -0.10204081632653061\right)}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.08e-158)
   (atan -1.0)
   (if (<= v 2.05e-253)
     (atan (/ (* v (* v -0.10204081632653061)) H))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.08e-158) {
		tmp = atan(-1.0);
	} else if (v <= 2.05e-253) {
		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 <= (-1.08d-158)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.05d-253) 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 <= -1.08e-158) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.05e-253) {
		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 <= -1.08e-158:
		tmp = math.atan(-1.0)
	elif v <= 2.05e-253:
		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 <= -1.08e-158)
		tmp = atan(-1.0);
	elseif (v <= 2.05e-253)
		tmp = atan(Float64(Float64(v * Float64(v * -0.10204081632653061)) / H));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.08e-158)
		tmp = atan(-1.0);
	elseif (v <= 2.05e-253)
		tmp = atan(((v * (v * -0.10204081632653061)) / H));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.08e-158], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.05e-253], N[ArcTan[N[(N[(v * N[(v * -0.10204081632653061), $MachinePrecision]), $MachinePrecision] / H), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.08e-158

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. metadata-eval 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in v around -inf 78.9%

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

   if -1.08e-158 < v < 2.05000000000000001e-253

   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

   5. Taylor expanded in H around 0 29.1%

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

      1. associate-*r/ 29.1%

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

   7. Simplified 29.1%

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

      1. clear-num 29.1%

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

      2. inv-pow 29.1%

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

      3. *-un-lft-identity 29.1%

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

      4. times-frac 29.1%

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

      5. metadata-eval 29.1%

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

   9. Applied egg-rr 29.1%

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

       1. unpow-1 29.1%

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

   11. Simplified 29.1%

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

       1. clear-num 29.1%

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

       2. associate-/r/ 29.1%

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

       3. associate-/r* 29.1%

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

       4. metadata-eval 29.1%

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

   13. Applied egg-rr 29.1%

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

   14. Taylor expanded in v around 0 29.1%

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

       1. *-commutative 29.1%

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

       2. associate-*r/ 29.1%

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

       3. *-commutative 29.1%

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

       4. associate-*r/ 29.2%

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

   16. Applied egg-rr 29.2%

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

   if 2.05000000000000001e-253 < v

   1. Initial program 62.0%

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

      1. sqr-neg 62.0%

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

      2. sqr-neg 62.0%

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

      3. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in v around inf 72.4%

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

Alternative 6: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6 \cdot 10^{-162}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.05 \cdot 10^{-253}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6e-162)
   (atan -1.0)
   (if (<= v 2.05e-253)
     (atan (* v (* -0.10204081632653061 (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6e-162) {
		tmp = atan(-1.0);
	} else if (v <= 2.05e-253) {
		tmp = atan((v * (-0.10204081632653061 * (v / H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-6d-162)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.05d-253) then
        tmp = atan((v * ((-0.10204081632653061d0) * (v / h))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -6e-162) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.05e-253) {
		tmp = Math.atan((v * (-0.10204081632653061 * (v / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -6e-162:
		tmp = math.atan(-1.0)
	elif v <= 2.05e-253:
		tmp = math.atan((v * (-0.10204081632653061 * (v / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6e-162)
		tmp = atan(-1.0);
	elseif (v <= 2.05e-253)
		tmp = atan(Float64(v * Float64(-0.10204081632653061 * Float64(v / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -6e-162)
		tmp = atan(-1.0);
	elseif (v <= 2.05e-253)
		tmp = atan((v * (-0.10204081632653061 * (v / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6e-162], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.05e-253], N[ArcTan[N[(v * N[(-0.10204081632653061 * N[(v / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.99999999999999997e-162

   1. Initial program 64.6%

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

      1. sqr-neg 64.6%

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

      2. sqr-neg 64.6%

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

      3. metadata-eval 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in v around -inf 78.9%

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

   if -5.99999999999999997e-162 < v < 2.05000000000000001e-253

   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

   5. Taylor expanded in H around 0 29.1%

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

      1. associate-*r/ 29.1%

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

   7. Simplified 29.1%

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

      1. clear-num 29.1%

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

      2. inv-pow 29.1%

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

      3. *-un-lft-identity 29.1%

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

      4. times-frac 29.1%

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

      5. metadata-eval 29.1%

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

   9. Applied egg-rr 29.1%

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

       1. unpow-1 29.1%

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

   11. Simplified 29.1%

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

       1. clear-num 29.1%

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

       2. associate-/r/ 29.1%

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

       3. associate-/r* 29.1%

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

       4. metadata-eval 29.1%

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

   13. Applied egg-rr 29.1%

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

   14. Taylor expanded in v around 0 29.1%

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

   if 2.05000000000000001e-253 < v

   1. Initial program 62.0%

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

      1. sqr-neg 62.0%

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

      2. sqr-neg 62.0%

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

      3. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in v around inf 72.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6 \cdot 10^{-162}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.05 \cdot 10^{-253}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.6 \cdot 10^{-95}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{1}{\frac{v + H \cdot \frac{-9.8}{v}}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.6e-95) (atan -1.0) (atan (/ 1.0 (/ (+ v (* H (/ -9.8 v))) v)))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((1.0 / ((v + (H * (-9.8 / 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 <= (-9.6d-95)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((1.0d0 / ((v + (h * ((-9.8d0) / v))) / v)))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((1.0 / ((v + (H * (-9.8 / v))) / v)));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.6e-95:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((1.0 / ((v + (H * (-9.8 / v))) / v)))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(1.0 / Float64(Float64(v + Float64(H * Float64(-9.8 / v))) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan((1.0 / ((v + (H * (-9.8 / v))) / v)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.6e-95], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(1.0 / N[(N[(v + N[(H * N[(-9.8 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.6e-95

   1. Initial program 61.4%

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

      1. sqr-neg 61.4%

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

      2. sqr-neg 61.4%

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

      3. metadata-eval 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in v around -inf 84.3%

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

   if -9.6e-95 < v

   1. Initial program 71.0%

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

      1. sqr-neg 71.0%

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

      2. sqr-neg 71.0%

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

      3. metadata-eval 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in H around 0 63.4%

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

      1. associate-*r/ 63.4%

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

   7. Simplified 63.4%

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

      1. clear-num 63.4%

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

      2. inv-pow 63.4%

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

      3. *-un-lft-identity 63.4%

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

      4. times-frac 63.4%

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

      5. metadata-eval 63.4%

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

   9. Applied egg-rr 63.4%

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

       1. unpow-1 63.4%

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

   11. Simplified 63.4%

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

       1. associate-*r/ 63.4%

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

       2. associate-/r/ 63.4%

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

       3. *-commutative 63.4%

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

   13. Applied egg-rr 63.4%

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

       1. clear-num 63.4%

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

       2. inv-pow 63.4%

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

       3. associate-*l/ 63.4%

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

       4. *-commutative 63.4%

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

       5. times-frac 63.4%

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

       6. metadata-eval 63.4%

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

   15. Applied egg-rr 63.4%

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

       1. unpow-1 63.4%

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

       2. associate-*r/ 63.4%

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

       3. *-commutative 63.4%

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

       4. associate-/l* 63.4%

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

   17. Simplified 63.4%

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

Alternative 8: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.6 \cdot 10^{-95}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -9.6e-95) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-9.6d-95)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-95) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.6e-95:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -9.6e-95)
		tmp = atan(-1.0);
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.6e-95], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.6e-95

   1. Initial program 61.4%

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

      1. sqr-neg 61.4%

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

      2. sqr-neg 61.4%

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

      3. metadata-eval 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in v around -inf 84.3%

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

   if -9.6e-95 < v

   1. Initial program 71.0%

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

      1. sqr-neg 71.0%

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

      2. sqr-neg 71.0%

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

      3. metadata-eval 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in H around 0 63.4%

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

Alternative 9: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -9.6 \cdot 10^{-60}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -9.6e-60) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -9.6e-60) {
		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 <= (-9.6d-60)) 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 <= -9.6e-60) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -9.6e-60:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -9.6e-60)
		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 <= -9.6e-60)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -9.6e-60], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.60000000000000038e-60

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. metadata-eval 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in v around -inf 88.7%

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

   if -9.60000000000000038e-60 < v

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in v around inf 53.7%

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

Alternative 10: 35.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. sqr-neg 67.7%

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

   2. sqr-neg 67.7%

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

   3. metadata-eval 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in v around -inf 30.9%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))