Optimal throwing angle

Percentage Accurate: 68.0% → 99.1%

Time: 8.3s

Alternatives: 6

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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4e+154)
   (atan -1.0)
   (if (<= v 5.4e+73) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4e+154) {
		tmp = atan(-1.0);
	} else if (v <= 5.4e+73) {
		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 <= (-4d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.4d+73) 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 <= -4e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.4e+73) {
		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 <= -4e+154:
		tmp = math.atan(-1.0)
	elif v <= 5.4e+73:
		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 <= -4e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.4e+73)
		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 <= -4e+154)
		tmp = atan(-1.0);
	elseif (v <= 5.4e+73)
		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, -4e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.4e+73], 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 < -4.00000000000000015e154

   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 -4.00000000000000015e154 < v < 5.3999999999999998e73

   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 5.3999999999999998e73 < v

   1. Initial program 46.4%

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

      1. sqr-neg 46.4%

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

      2. sqr-neg 46.4%

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

      3. metadata-eval 46.4%

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

   3. Simplified 46.4%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 2: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.75e-8)
   (atan -1.0)
   (if (<= v 5.5e-36)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.75e-8) {
		tmp = atan(-1.0);
	} else if (v <= 5.5e-36) {
		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 <= (-1.75d-8)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.5d-36) 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 <= -1.75e-8) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.5e-36) {
		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 <= -1.75e-8:
		tmp = math.atan(-1.0)
	elif v <= 5.5e-36:
		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 <= -1.75e-8)
		tmp = atan(-1.0);
	elseif (v <= 5.5e-36)
		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 <= -1.75e-8)
		tmp = atan(-1.0);
	elseif (v <= 5.5e-36)
		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, -1.75e-8], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.5e-36], 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 < -1.75000000000000012e-8

   1. Initial program 51.5%

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

      1. sqr-neg 51.5%

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

      2. sqr-neg 51.5%

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

      3. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in v around -inf 97.8%

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

   if -1.75000000000000012e-8 < v < 5.49999999999999984e-36

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 99.4%

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

   6. Taylor expanded in v around 0 91.3%

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

   if 5.49999999999999984e-36 < v

   1. Initial program 61.9%

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

      1. sqr-neg 61.9%

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

      2. sqr-neg 61.9%

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

      3. metadata-eval 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in H around 0 94.1%

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

Alternative 3: 70.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.9 \cdot 10^{-189}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 6.4 \cdot 10^{-83}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.9e-189)
   (atan -1.0)
   (if (<= v 6.4e-83)
     (atan (* -0.10204081632653061 (/ v (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.9e-189) {
		tmp = atan(-1.0);
	} else if (v <= 6.4e-83) {
		tmp = atan((-0.10204081632653061 * (v / (H / 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-189)) then
        tmp = atan((-1.0d0))
    else if (v <= 6.4d-83) then
        tmp = atan(((-0.10204081632653061d0) * (v / (h / 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-189) {
		tmp = Math.atan(-1.0);
	} else if (v <= 6.4e-83) {
		tmp = Math.atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4.9e-189:
		tmp = math.atan(-1.0)
	elif v <= 6.4e-83:
		tmp = math.atan((-0.10204081632653061 * (v / (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.9e-189)
		tmp = atan(-1.0);
	elseif (v <= 6.4e-83)
		tmp = atan(Float64(-0.10204081632653061 * Float64(v / Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4.9e-189)
		tmp = atan(-1.0);
	elseif (v <= 6.4e-83)
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.9e-189], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 6.4e-83], N[ArcTan[N[(-0.10204081632653061 * N[(v / N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.8999999999999997e-189

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. metadata-eval 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in v around -inf 74.9%

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

   if -4.8999999999999997e-189 < v < 6.4000000000000002e-83

   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 H around 0 26.6%

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

   6. Taylor expanded in v around 0 26.6%

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

      1. *-un-lft-identity 26.6%

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

      2. times-frac 26.6%

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

      3. metadata-eval 26.6%

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

   8. Applied egg-rr 26.6%

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

   if 6.4000000000000002e-83 < v

   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 90.0%

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

Alternative 4: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -8.2 \cdot 10^{-182}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -8.2e-182) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -8.2e-182) {
		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 <= (-8.2d-182)) 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 <= -8.2e-182) {
		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 <= -8.2e-182:
		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 <= -8.2e-182)
		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 <= -8.2e-182)
		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, -8.2e-182], 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 < -8.2000000000000003e-182

   1. Initial program 65.9%

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

      1. sqr-neg 65.9%

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

      2. sqr-neg 65.9%

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

      3. metadata-eval 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in v around -inf 74.9%

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

   if -8.2000000000000003e-182 < v

   1. Initial program 78.2%

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

      1. sqr-neg 78.2%

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

      2. sqr-neg 78.2%

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

      3. metadata-eval 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in H around 0 65.9%

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

Alternative 5: 67.7% 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 69.3%

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

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in v around -inf 67.8%

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

   if -1.999999999999994e-310 < v

   1. Initial program 75.9%

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

      1. sqr-neg 75.9%

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

      2. sqr-neg 75.9%

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

      3. metadata-eval 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in v around inf 62.9%

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

Alternative 6: 34.6% 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 72.6%

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

   1. sqr-neg 72.6%

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

   2. sqr-neg 72.6%

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

   3. metadata-eval 72.6%

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

3. Simplified 72.6%

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

5. Taylor expanded in v around -inf 35.4%

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

Reproduce

herbie shell --seed 2024166 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))