Optimal throwing angle

Percentage Accurate: 67.5% → 99.7%

Time: 9.7s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan -1.0)
   (if (<= v 2e+134)
     (atan (* v (sqrt (/ 1.0 (- (* v v) (* 19.6 H))))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+134) {
		tmp = atan((v * sqrt((1.0 / ((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 <= (-1d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+134) then
        tmp = atan((v * sqrt((1.0d0 / ((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 <= -1e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+134) {
		tmp = Math.atan((v * Math.sqrt((1.0 / ((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 <= -1e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+134:
		tmp = math.atan((v * math.sqrt((1.0 / ((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 <= -1e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+134)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / 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 <= -1e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+134)
		tmp = atan((v * sqrt((1.0 / ((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, -1e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+134], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $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 < -1.00000000000000004e154

   1. Initial program 3.1%

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

      1. 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 -1.00000000000000004e154 < v < 1.99999999999999984e134

   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 99.7%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   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 1.99999999999999984e134 < v

   1. Initial program 15.9%

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

      1. sqr-neg 15.9%

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

      2. sqr-neg 15.9%

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

      3. metadata-eval 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in H around 0 100.0%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.3e+154)
   (atan -1.0)
   (if (<= v 2e+139) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.3e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+139) {
		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 <= (-1.3d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+139) 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 <= -1.3e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+139) {
		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 <= -1.3e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+139:
		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 <= -1.3e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+139)
		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 <= -1.3e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+139)
		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, -1.3e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+139], 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 < -1.29999999999999994e154

   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 -1.29999999999999994e154 < v < 2.00000000000000007e139

   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 2.00000000000000007e139 < v

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. sqr-neg 10.9%

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

      3. metadata-eval 10.9%

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

   3. Simplified 10.9%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)} \]
   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: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v} - v}\right)\\ \mathbf{elif}\;v \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.9e-61)
   (atan (/ v (- (/ (* H 9.8) v) v)))
   (if (<= v 2.15e-97)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-61) {
		tmp = atan((v / (((H * 9.8) / v) - v)));
	} else if (v <= 2.15e-97) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.9d-61)) then
        tmp = atan((v / (((h * 9.8d0) / v) - v)))
    else if (v <= 2.15d-97) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-61) {
		tmp = Math.atan((v / (((H * 9.8) / v) - v)));
	} else if (v <= 2.15e-97) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.9e-61:
		tmp = math.atan((v / (((H * 9.8) / v) - v)))
	elif v <= 2.15e-97:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.9e-61)
		tmp = atan(Float64(v / Float64(Float64(Float64(H * 9.8) / v) - v)));
	elseif (v <= 2.15e-97)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.9e-61)
		tmp = atan((v / (((H * 9.8) / v) - v)));
	elseif (v <= 2.15e-97)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.9e-61], N[ArcTan[N[(v / N[(N[(N[(H * 9.8), $MachinePrecision] / v), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2.15e-97], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.8999999999999999e-61

   1. Initial program 54.7%

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

      1. sqr-neg 54.7%

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

      2. sqr-neg 54.7%

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

      3. metadata-eval 54.7%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in v around -inf 90.4%

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

      1. associate-*r* 90.4%

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

      2. neg-mul-1 90.4%

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

      3. *-commutative 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in H around 0 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. associate-*r/ 90.4%

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

      5. *-commutative 90.4%

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

   10. Simplified 90.4%

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

   if -1.8999999999999999e-61 < v < 2.15e-97

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. sqr-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 99.4%

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

   6. Taylor expanded in v around 0 89.1%

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

   if 2.15e-97 < v

   1. Initial program 60.5%

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

      1. sqr-neg 60.5%

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

      2. sqr-neg 60.5%

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

      3. metadata-eval 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in H around 0 86.2%

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

Alternative 4: 72.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{-219}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - -9.8 \cdot \frac{\frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.9e-219)
   (atan (/ v (* v (- -1.0 (* -9.8 (/ (/ H v) v))))))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.9e-219) {
		tmp = atan((v / (v * (-1.0 - (-9.8 * ((H / v) / v))))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2.9d-219)) then
        tmp = atan((v / (v * ((-1.0d0) - ((-9.8d0) * ((h / v) / v))))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.9e-219) {
		tmp = Math.atan((v / (v * (-1.0 - (-9.8 * ((H / v) / v))))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.9e-219:
		tmp = math.atan((v / (v * (-1.0 - (-9.8 * ((H / v) / v))))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.9e-219)
		tmp = atan(Float64(v / Float64(v * Float64(-1.0 - Float64(-9.8 * Float64(Float64(H / v) / v))))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.9e-219)
		tmp = atan((v / (v * (-1.0 - (-9.8 * ((H / v) / v))))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.9e-219], N[ArcTan[N[(v / N[(v * N[(-1.0 - N[(-9.8 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $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 2 regimes

2. if v < -2.89999999999999984e-219

   1. Initial program 62.2%

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

      1. sqr-neg 62.2%

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

      2. sqr-neg 62.2%

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

      3. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in v around -inf 80.7%

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

      1. associate-*r* 80.7%

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

      2. neg-mul-1 80.7%

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

      3. *-commutative 80.7%

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

   7. Simplified 80.7%

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

      1. *-un-lft-identity 80.7%

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

      2. unpow2 80.7%

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

      3. times-frac 80.8%

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

   9. Applied egg-rr 80.8%

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

       1. associate-*l/ 80.8%

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

       2. *-lft-identity 80.8%

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

   11. Simplified 80.8%

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

   if -2.89999999999999984e-219 < v

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. metadata-eval 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in H around 0 73.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{-219}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v \cdot \left(-1 - -9.8 \cdot \frac{\frac{H}{v}}{v}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.8 \cdot 10^{-144}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \left(v \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.1e-169)
   (atan -1.0)
   (if (<= v 4.8e-144)
     (atan (* -0.10204081632653061 (* v (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.1e-169) {
		tmp = atan(-1.0);
	} else if (v <= 4.8e-144) {
		tmp = atan((-0.10204081632653061 * (v * (v / H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-5.1d-169)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.8d-144) then
        tmp = atan(((-0.10204081632653061d0) * (v * (v / h))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -5.1e-169) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.8e-144) {
		tmp = Math.atan((-0.10204081632653061 * (v * (v / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5.1e-169:
		tmp = math.atan(-1.0)
	elif v <= 4.8e-144:
		tmp = math.atan((-0.10204081632653061 * (v * (v / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5.1e-169)
		tmp = atan(-1.0);
	elseif (v <= 4.8e-144)
		tmp = atan(Float64(-0.10204081632653061 * Float64(v * Float64(v / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -5.1e-169)
		tmp = atan(-1.0);
	elseif (v <= 4.8e-144)
		tmp = atan((-0.10204081632653061 * (v * (v / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5.1e-169], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.8e-144], N[ArcTan[N[(-0.10204081632653061 * N[(v * N[(v / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.09999999999999997e-169

   1. Initial program 60.4%

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

      1. sqr-neg 60.4%

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

      2. sqr-neg 60.4%

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

      3. metadata-eval 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in v around -inf 83.8%

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

   if -5.09999999999999997e-169 < v < 4.79999999999999988e-144

   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 H around 0 35.4%

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

   6. Taylor expanded in v around 0 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. *-commutative 35.4%

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

      3. times-frac 35.4%

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

      4. clear-num 35.4%

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

      5. div-inv 35.4%

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

      6. metadata-eval 35.4%

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

   8. Applied egg-rr 35.4%

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

      1. associate-*r* 35.4%

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

      2. associate-/r/ 35.4%

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

      3. *-commutative 35.4%

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

      4. associate-/r/ 35.4%

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

   10. Simplified 35.4%

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

   if 4.79999999999999988e-144 < v

   1. Initial program 63.9%

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

      1. sqr-neg 63.9%

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

      2. sqr-neg 63.9%

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

      3. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in v around inf 79.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.8 \cdot 10^{-144}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \left(v \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{-219}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\frac{H \cdot 9.8}{v} - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.9e-219)
   (atan (/ v (- (/ (* H 9.8) v) v)))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.9e-219) {
		tmp = atan((v / (((H * 9.8) / v) - v)));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2.9d-219)) then
        tmp = atan((v / (((h * 9.8d0) / v) - v)))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.9e-219) {
		tmp = Math.atan((v / (((H * 9.8) / v) - v)));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.9e-219:
		tmp = math.atan((v / (((H * 9.8) / v) - v)))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.9e-219)
		tmp = atan(Float64(v / Float64(Float64(Float64(H * 9.8) / v) - v)));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.9e-219)
		tmp = atan((v / (((H * 9.8) / v) - v)));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.9e-219], N[ArcTan[N[(v / N[(N[(N[(H * 9.8), $MachinePrecision] / v), $MachinePrecision] - v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.89999999999999984e-219

   1. Initial program 62.2%

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

      1. sqr-neg 62.2%

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

      2. sqr-neg 62.2%

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

      3. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in v around -inf 80.7%

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

      1. associate-*r* 80.7%

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

      2. neg-mul-1 80.7%

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

      3. *-commutative 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in H around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 80.7%

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

      5. *-commutative 80.7%

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

   10. Simplified 80.7%

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

   if -2.89999999999999984e-219 < v

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. metadata-eval 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in H around 0 73.7%

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

Alternative 7: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5.1 \cdot 10^{-169}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5.1e-169) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5.1e-169) {
		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 <= (-5.1d-169)) 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 <= -5.1e-169) {
		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 <= -5.1e-169:
		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 <= -5.1e-169)
		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 <= -5.1e-169)
		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, -5.1e-169], 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 < -5.09999999999999997e-169

   1. Initial program 60.4%

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

      1. sqr-neg 60.4%

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

      2. sqr-neg 60.4%

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

      3. metadata-eval 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in v around -inf 83.8%

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

   if -5.09999999999999997e-169 < v

   1. Initial program 70.6%

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

      1. sqr-neg 70.6%

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

      2. sqr-neg 70.6%

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

      3. metadata-eval 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in H around 0 71.5%

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

Alternative 8: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -2e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e-310)
		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 <= -2e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.999999999999994e-310

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. sqr-neg 65.0%

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

      3. metadata-eval 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in v around -inf 74.2%

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

   if -1.999999999999994e-310 < v

   1. Initial program 67.8%

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

      1. sqr-neg 67.8%

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

      2. sqr-neg 67.8%

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

      3. metadata-eval 67.8%

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

   3. Simplified 67.8%

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

   5. Taylor expanded in v around inf 71.2%

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

Alternative 9: 34.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. sqr-neg 66.5%

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

   2. sqr-neg 66.5%

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

   3. metadata-eval 66.5%

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

3. Simplified 66.5%

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

5. Taylor expanded in v around -inf 34.9%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))