Optimal throwing angle

Percentage Accurate: 67.8% → 98.7%

Time: 11.1s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.8% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.9 \cdot 10^{+58}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 8.2 \cdot 10^{+120}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-1}{19.6 \cdot H - v \cdot v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.9e+58)
   (atan -1.0)
   (if (<= v 8.2e+120)
     (atan (* v (sqrt (/ -1.0 (- (* 19.6 H) (* v v))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.9e+58) {
		tmp = atan(-1.0);
	} else if (v <= 8.2e+120) {
		tmp = atan((v * sqrt((-1.0 / ((19.6 * H) - (v * v))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-4.9d+58)) then
        tmp = atan((-1.0d0))
    else if (v <= 8.2d+120) then
        tmp = atan((v * sqrt(((-1.0d0) / ((19.6d0 * h) - (v * v))))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -4.9e+58) {
		tmp = Math.atan(-1.0);
	} else if (v <= 8.2e+120) {
		tmp = Math.atan((v * Math.sqrt((-1.0 / ((19.6 * H) - (v * v))))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4.9e+58:
		tmp = math.atan(-1.0)
	elif v <= 8.2e+120:
		tmp = math.atan((v * math.sqrt((-1.0 / ((19.6 * H) - (v * v))))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.9e+58)
		tmp = atan(-1.0);
	elseif (v <= 8.2e+120)
		tmp = atan(Float64(v * sqrt(Float64(-1.0 / Float64(Float64(19.6 * H) - Float64(v * v))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4.9e+58)
		tmp = atan(-1.0);
	elseif (v <= 8.2e+120)
		tmp = atan((v * sqrt((-1.0 / ((19.6 * H) - (v * v))))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.9e+58], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 8.2e+120], N[ArcTan[N[(v * N[Sqrt[N[(-1.0 / N[(N[(19.6 * H), $MachinePrecision] - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.90000000000000018e58

   1. Initial program 39.5%

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

      1. sqr-neg 39.5%

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

      2. sqr-neg 39.5%

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

      3. metadata-eval 39.5%

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

   3. Simplified 39.5%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -4.90000000000000018e58 < v < 8.2e120

   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 99.7%

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

      1. unpow2 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 8.2e120 < v

   1. Initial program 25.9%

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

      1. sqr-neg 25.9%

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

      2. sqr-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in v around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.9 \cdot 10^{+58}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 8.2 \cdot 10^{+120}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-1}{19.6 \cdot H - v \cdot v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 8.2 \cdot 10^{+120}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+159)
   (atan -1.0)
   (if (<= v 8.2e+120) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+159) {
		tmp = atan(-1.0);
	} else if (v <= 8.2e+120) {
		tmp = atan((v / sqrt(((v * v) - (19.6 * 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 <= (-2d+159)) then
        tmp = atan((-1.0d0))
    else if (v <= 8.2d+120) then
        tmp = atan((v / sqrt(((v * v) - (19.6d0 * 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 <= -2e+159) {
		tmp = Math.atan(-1.0);
	} else if (v <= 8.2e+120) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+159:
		tmp = math.atan(-1.0)
	elif v <= 8.2e+120:
		tmp = math.atan((v / math.sqrt(((v * v) - (19.6 * H)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+159)
		tmp = atan(-1.0);
	elseif (v <= 8.2e+120)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+159)
		tmp = atan(-1.0);
	elseif (v <= 8.2e+120)
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+159], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 8.2e+120], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.9999999999999999e159

   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 -1.9999999999999999e159 < v < 8.2e120

   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 8.2e120 < v

   1. Initial program 25.9%

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

      1. sqr-neg 25.9%

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

      2. sqr-neg 25.9%

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

      3. metadata-eval 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.6 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 3.05 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.6e-13)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (if (<= v 3.05e-71)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* (/ H v) -9.8)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.6e-13) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 3.05e-71) {
		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 <= (-4.6d-13)) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else if (v <= 3.05d-71) 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 <= -4.6e-13) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else if (v <= 3.05e-71) {
		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 <= -4.6e-13:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	elif v <= 3.05e-71:
		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 <= -4.6e-13)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	elseif (v <= 3.05e-71)
		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 <= -4.6e-13)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	elseif (v <= 3.05e-71)
		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, -4.6e-13], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 3.05e-71], 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 < -4.59999999999999958e-13

   1. Initial program 47.5%

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

      1. sqr-neg 47.5%

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

      2. sqr-neg 47.5%

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

      3. metadata-eval 47.5%

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

   3. Simplified 47.5%

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

      1. frac-2neg 47.5%

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

      2. div-inv 47.5%

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

      3. sub-neg 47.5%

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

      4. add-sqr-sqrt 25.0%

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

      5. hypot-define 54.2%

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

      6. *-commutative 54.2%

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

      7. distribute-rgt-neg-in 54.2%

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

      8. metadata-eval 54.2%

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

   6. Applied egg-rr 54.2%

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

   7. Taylor expanded in H around 0 0.0%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \frac{H \cdot {\left(\sqrt{-19.6}\right)}^{2}}{v}\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 1.6%

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

   9. Simplified 1.6%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-neg-out 1.6%

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

       2. neg-sub0 1.6%

          \[\leadsto \tan^{-1} \color{blue}{\left(0 - v \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right)} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       4. sqrt-prod 52.7%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\sqrt{v \cdot v}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       5. sqr-neg 52.7%

          \[\leadsto \tan^{-1} \left(0 - \sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       6. sqrt-unprod 96.3%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       7. add-sqr-sqrt 96.7%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(-v\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       8. associate-*r/ 96.7%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{\left(-v\right) \cdot 1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]

       9. frac-2neg 96.7%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{-\left(-v\right) \cdot 1}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}}\right) \]

       10. *-rgt-identity 96.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\left(-v\right)}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       11. add-sqr-sqrt 96.2%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       12. sqrt-unprod 52.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       13. sqr-neg 52.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       14. sqrt-prod 0.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       15. add-sqr-sqrt 1.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       16. add-sqr-sqrt 1.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       17. sqrt-unprod 1.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       18. sqr-neg 1.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       19. sqrt-prod 0.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       20. add-sqr-sqrt 96.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       21. remove-double-neg 96.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)}}\right) \]

       22. +-commutative 96.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right) + v}}\right) \]

       23. fma-define 96.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{\mathsf{fma}\left(0.5, H \cdot \frac{-19.6}{v}, v\right)}}\right) \]

   11. Applied egg-rr 96.7%

       \[\leadsto \tan^{-1} \color{blue}{\left(0 - \frac{v}{\mathsf{fma}\left(0.5, -19.6 \cdot \frac{H}{v}, v\right)}\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 96.7%

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

       2. distribute-neg-frac2 96.7%

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

       3. fma-define 96.7%

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

       4. distribute-neg-in 96.7%

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

       5. *-rgt-identity 96.7%

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

       6. unsub-neg 96.7%

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

       7. *-rgt-identity 96.7%

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

       8. associate-*r* 96.7%

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

       9. metadata-eval 96.7%

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

       10. distribute-lft-neg-in 96.7%

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

       11. metadata-eval 96.7%

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

   13. Simplified 96.7%

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

   if -4.59999999999999958e-13 < v < 3.0499999999999999e-71

   1. Initial program 99.5%

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

      1. sqr-neg 99.5%

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

      2. sqr-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in v around 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 90.5%

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

   if 3.0499999999999999e-71 < v

   1. Initial program 57.0%

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

      1. sqr-neg 57.0%

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

      2. sqr-neg 57.0%

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

      3. metadata-eval 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in H around 0 89.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.6 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{elif}\;v \leq 3.05 \cdot 10^{-71}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 2e-296)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (atan (* v (/ 1.0 (+ v (* 0.5 (* H (/ -19.6 v)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= 2e-296) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = atan((v * (1.0 / (v + (0.5 * (H * (-19.6 / 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-296) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    else
        tmp = atan((v * (1.0d0 / (v + (0.5d0 * (h * ((-19.6d0) / v)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 2e-296) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = Math.atan((v * (1.0 / (v + (0.5 * (H * (-19.6 / v)))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 2e-296:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	else:
		tmp = math.atan((v * (1.0 / (v + (0.5 * (H * (-19.6 / v)))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 2e-296)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	else
		tmp = atan(Float64(v * Float64(1.0 / Float64(v + Float64(0.5 * Float64(H * Float64(-19.6 / v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= 2e-296)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	else
		tmp = atan((v * (1.0 / (v + (0.5 * (H * (-19.6 / v)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 2e-296], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v * N[(1.0 / N[(v + N[(0.5 * N[(H * N[(-19.6 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2e-296

   1. Initial program 65.8%

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

      1. sqr-neg 65.8%

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

      2. sqr-neg 65.8%

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

      3. metadata-eval 65.8%

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

   3. Simplified 65.8%

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

      1. frac-2neg 65.8%

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

      2. div-inv 65.8%

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

      3. sub-neg 65.8%

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

      4. add-sqr-sqrt 48.9%

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

      5. hypot-define 67.8%

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

      6. *-commutative 67.8%

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

      7. distribute-rgt-neg-in 67.8%

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

      8. metadata-eval 67.8%

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

   6. Applied egg-rr 67.8%

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

   7. Taylor expanded in H around 0 0.0%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \frac{H \cdot {\left(\sqrt{-19.6}\right)}^{2}}{v}\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 5.6%

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

   9. Simplified 5.6%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-neg-out 5.6%

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

       2. neg-sub0 5.6%

          \[\leadsto \tan^{-1} \color{blue}{\left(0 - v \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right)} \]

       3. add-sqr-sqrt 0.1%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       4. sqrt-prod 43.4%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\sqrt{v \cdot v}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       5. sqr-neg 43.4%

          \[\leadsto \tan^{-1} \left(0 - \sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       6. sqrt-unprod 71.6%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       7. add-sqr-sqrt 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(-v\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       8. associate-*r/ 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{\left(-v\right) \cdot 1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]

       9. frac-2neg 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{-\left(-v\right) \cdot 1}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}}\right) \]

       10. *-rgt-identity 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\left(-v\right)}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       11. add-sqr-sqrt 71.5%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       12. sqrt-unprod 43.4%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       13. sqr-neg 43.4%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       14. sqrt-prod 0.1%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       15. add-sqr-sqrt 5.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       16. add-sqr-sqrt 5.5%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       17. sqrt-unprod 5.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       18. sqr-neg 5.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       19. sqrt-prod 0.1%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       20. add-sqr-sqrt 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       21. remove-double-neg 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)}}\right) \]

       22. +-commutative 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right) + v}}\right) \]

       23. fma-define 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{\mathsf{fma}\left(0.5, H \cdot \frac{-19.6}{v}, v\right)}}\right) \]

   11. Applied egg-rr 72.0%

       \[\leadsto \tan^{-1} \color{blue}{\left(0 - \frac{v}{\mathsf{fma}\left(0.5, -19.6 \cdot \frac{H}{v}, v\right)}\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 72.0%

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

       2. distribute-neg-frac2 72.0%

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

       3. fma-define 72.0%

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

       4. distribute-neg-in 72.0%

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

       5. *-rgt-identity 72.0%

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

       6. unsub-neg 72.0%

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

       7. *-rgt-identity 72.0%

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

       8. associate-*r* 72.0%

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

       9. metadata-eval 72.0%

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

       10. distribute-lft-neg-in 72.0%

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

       11. metadata-eval 72.0%

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

   13. Simplified 72.0%

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

   if 2e-296 < v

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. sqr-neg 70.2%

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

      3. metadata-eval 70.2%

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

   3. Simplified 70.2%

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

      1. frac-2neg 70.2%

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

      2. div-inv 70.2%

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

      3. sub-neg 70.2%

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

      4. add-sqr-sqrt 52.6%

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

      5. hypot-define 68.5%

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

      6. *-commutative 68.5%

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

      7. distribute-rgt-neg-in 68.5%

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

      8. metadata-eval 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in H around 0 0.0%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \frac{H \cdot {\left(\sqrt{-19.6}\right)}^{2}}{v}\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 68.9%

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

   9. Simplified 68.9%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.4 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 9.8 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.4e-143)
   (atan -1.0)
   (if (<= v 9.8e-123)
     (atan (* v (* -0.10204081632653061 (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.4e-143) {
		tmp = atan(-1.0);
	} else if (v <= 9.8e-123) {
		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 <= (-6.4d-143)) then
        tmp = atan((-1.0d0))
    else if (v <= 9.8d-123) 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 <= -6.4e-143) {
		tmp = Math.atan(-1.0);
	} else if (v <= 9.8e-123) {
		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 <= -6.4e-143:
		tmp = math.atan(-1.0)
	elif v <= 9.8e-123:
		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 <= -6.4e-143)
		tmp = atan(-1.0);
	elseif (v <= 9.8e-123)
		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 <= -6.4e-143)
		tmp = atan(-1.0);
	elseif (v <= 9.8e-123)
		tmp = atan((v * (-0.10204081632653061 * (v / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.4e-143], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 9.8e-123], 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 < -6.3999999999999997e-143

   1. Initial program 58.0%

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

      1. sqr-neg 58.0%

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

      2. sqr-neg 58.0%

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

      3. metadata-eval 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in v around -inf 83.2%

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

   if -6.3999999999999997e-143 < v < 9.7999999999999996e-123

   1. Initial program 99.5%

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

      1. sqr-neg 99.5%

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

      2. sqr-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

      3. sub-neg 99.5%

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

      4. add-sqr-sqrt 99.5%

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

      5. hypot-define 99.5%

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

      6. *-commutative 99.5%

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

      7. distribute-rgt-neg-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in H around 0 0.0%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \frac{H \cdot {\left(\sqrt{-19.6}\right)}^{2}}{v}\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 23.0%

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

   9. Simplified 23.0%

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

   10. Taylor expanded in v around 0 23.0%

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

       1. *-commutative 23.0%

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

   12. Simplified 23.0%

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

       1. distribute-lft-neg-out 23.0%

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

   14. Applied egg-rr 23.0%

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

       1. distribute-rgt-neg-in 23.0%

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

       2. *-commutative 23.0%

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

       3. distribute-lft-neg-in 23.0%

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

       4. metadata-eval 23.0%

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

   16. Simplified 23.0%

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

   if 9.7999999999999996e-123 < v

   1. Initial program 61.6%

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

      1. sqr-neg 61.6%

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

      2. sqr-neg 61.6%

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

      3. metadata-eval 61.6%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in v around inf 81.6%

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

Alternative 6: 71.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v 2e-296)
   (atan (/ v (- (* 9.8 (/ H v)) v)))
   (atan (/ v (+ v (* (/ H v) -9.8))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= 2e-296) {
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	} 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 <= 2d-296) then
        tmp = atan((v / ((9.8d0 * (h / v)) - v)))
    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 <= 2e-296) {
		tmp = Math.atan((v / ((9.8 * (H / v)) - v)));
	} else {
		tmp = Math.atan((v / (v + ((H / v) * -9.8))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 2e-296:
		tmp = math.atan((v / ((9.8 * (H / v)) - v)))
	else:
		tmp = math.atan((v / (v + ((H / v) * -9.8))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 2e-296)
		tmp = atan(Float64(v / Float64(Float64(9.8 * Float64(H / v)) - v)));
	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 <= 2e-296)
		tmp = atan((v / ((9.8 * (H / v)) - v)));
	else
		tmp = atan((v / (v + ((H / v) * -9.8))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 2e-296], N[ArcTan[N[(v / N[(N[(9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] - v), $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 2 regimes

2. if v < 2e-296

   1. Initial program 65.8%

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

      1. sqr-neg 65.8%

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

      2. sqr-neg 65.8%

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

      3. metadata-eval 65.8%

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

   3. Simplified 65.8%

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

      1. frac-2neg 65.8%

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

      2. div-inv 65.8%

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

      3. sub-neg 65.8%

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

      4. add-sqr-sqrt 48.9%

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

      5. hypot-define 67.8%

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

      6. *-commutative 67.8%

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

      7. distribute-rgt-neg-in 67.8%

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

      8. metadata-eval 67.8%

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

   6. Applied egg-rr 67.8%

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

   7. Taylor expanded in H around 0 0.0%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \frac{H \cdot {\left(\sqrt{-19.6}\right)}^{2}}{v}\right)}}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 5.6%

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

   9. Simplified 5.6%

      \[\leadsto \tan^{-1} \left(\left(-v\right) \cdot \frac{1}{-\color{blue}{\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. distribute-lft-neg-out 5.6%

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

       2. neg-sub0 5.6%

          \[\leadsto \tan^{-1} \color{blue}{\left(0 - v \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right)} \]

       3. add-sqr-sqrt 0.1%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       4. sqrt-prod 43.4%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\sqrt{v \cdot v}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       5. sqr-neg 43.4%

          \[\leadsto \tan^{-1} \left(0 - \sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)}} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       6. sqrt-unprod 71.6%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       7. add-sqr-sqrt 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\left(-v\right)} \cdot \frac{1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}\right) \]

       8. associate-*r/ 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{\left(-v\right) \cdot 1}{-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)}}\right) \]

       9. frac-2neg 72.0%

          \[\leadsto \tan^{-1} \left(0 - \color{blue}{\frac{-\left(-v\right) \cdot 1}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}}\right) \]

       10. *-rgt-identity 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\left(-v\right)}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       11. add-sqr-sqrt 71.5%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       12. sqrt-unprod 43.4%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       13. sqr-neg 43.4%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       14. sqrt-prod 0.1%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       15. add-sqr-sqrt 5.6%

           \[\leadsto \tan^{-1} \left(0 - \frac{-\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       16. add-sqr-sqrt 5.5%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       17. sqrt-unprod 5.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       18. sqr-neg 5.7%

           \[\leadsto \tan^{-1} \left(0 - \frac{\sqrt{\color{blue}{v \cdot v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       19. sqrt-prod 0.1%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       20. add-sqr-sqrt 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{\color{blue}{v}}{-\left(-\left(v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)\right)\right)}\right) \]

       21. remove-double-neg 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{v + 0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right)}}\right) \]

       22. +-commutative 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{0.5 \cdot \left(H \cdot \frac{-19.6}{v}\right) + v}}\right) \]

       23. fma-define 72.0%

           \[\leadsto \tan^{-1} \left(0 - \frac{v}{\color{blue}{\mathsf{fma}\left(0.5, H \cdot \frac{-19.6}{v}, v\right)}}\right) \]

   11. Applied egg-rr 72.0%

       \[\leadsto \tan^{-1} \color{blue}{\left(0 - \frac{v}{\mathsf{fma}\left(0.5, -19.6 \cdot \frac{H}{v}, v\right)}\right)} \]
   12. Step-by-step derivation

       1. neg-sub0 72.0%

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

       2. distribute-neg-frac2 72.0%

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

       3. fma-define 72.0%

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

       4. distribute-neg-in 72.0%

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

       5. *-rgt-identity 72.0%

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

       6. unsub-neg 72.0%

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

       7. *-rgt-identity 72.0%

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

       8. associate-*r* 72.0%

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

       9. metadata-eval 72.0%

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

       10. distribute-lft-neg-in 72.0%

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

       11. metadata-eval 72.0%

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

   13. Simplified 72.0%

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

   if 2e-296 < v

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. sqr-neg 70.2%

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

      3. metadata-eval 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in H around 0 68.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{9.8 \cdot \frac{H}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.8 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.8e-143) (atan -1.0) (atan (/ v (+ v (* (/ H v) -9.8))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.8e-143) {
		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 <= (-6.8d-143)) 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 <= -6.8e-143) {
		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 <= -6.8e-143:
		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 <= -6.8e-143)
		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 <= -6.8e-143)
		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, -6.8e-143], 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 < -6.79999999999999966e-143

   1. Initial program 58.0%

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

      1. sqr-neg 58.0%

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

      2. sqr-neg 58.0%

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

      3. metadata-eval 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in v around -inf 83.2%

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

   if -6.79999999999999966e-143 < v

   1. Initial program 74.9%

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

      1. sqr-neg 74.9%

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

      2. sqr-neg 74.9%

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

      3. metadata-eval 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in H around 0 61.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.8 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + \frac{H}{v} \cdot -9.8}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.35 \cdot 10^{-298}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.35e-298) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.35e-298) {
		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 <= (-1.35d-298)) 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 <= -1.35e-298) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.35e-298:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.35e-298)
		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 <= -1.35e-298)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.35e-298], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.3500000000000001e-298

   1. Initial program 64.7%

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

      1. sqr-neg 64.7%

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

      2. sqr-neg 64.7%

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

      3. metadata-eval 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in v around -inf 70.5%

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

   if -1.3500000000000001e-298 < v

   1. Initial program 71.1%

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

      1. sqr-neg 71.1%

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

      2. sqr-neg 71.1%

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

      3. metadata-eval 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in v around inf 62.3%

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

Alternative 9: 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 68.0%

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

   1. sqr-neg 68.0%

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

   2. sqr-neg 68.0%

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

   3. metadata-eval 68.0%

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

3. Simplified 68.0%

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

5. Taylor expanded in v around -inf 35.1%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))