Optimal throwing angle

Percentage Accurate: 67.1% → 98.9%

Time: 9.3s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{+58}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+156)
   (atan -1.0)
   (if (<= v 5.2e+58) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+156) {
		tmp = atan(-1.0);
	} else if (v <= 5.2e+58) {
		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 <= (-1d+156)) then
        tmp = atan((-1.0d0))
    else if (v <= 5.2d+58) 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 <= -1e+156) {
		tmp = Math.atan(-1.0);
	} else if (v <= 5.2e+58) {
		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 <= -1e+156:
		tmp = math.atan(-1.0)
	elif v <= 5.2e+58:
		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 <= -1e+156)
		tmp = atan(-1.0);
	elseif (v <= 5.2e+58)
		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 <= -1e+156)
		tmp = atan(-1.0);
	elseif (v <= 5.2e+58)
		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, -1e+156], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5.2e+58], 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 < -9.9999999999999998e155

   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 -9.9999999999999998e155 < v < 5.19999999999999976e58

   1. Initial program 99.8%

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

      1. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   if 5.19999999999999976e58 < v

   1. Initial program 41.1%

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

      1. sqr-neg 41.1%

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

      2. sqr-neg 41.1%

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

      3. metadata-eval 41.1%

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

   3. Simplified 41.1%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 2: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9.8 \cdot \frac{H}{v}\\ \mathbf{if}\;v \leq -2.2 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{t\_0}{v}\right)\\ \mathbf{elif}\;v \leq 1.86 \cdot 10^{-112}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (* -9.8 (/ H v))))
   (if (<= v -2.2e-53)
     (atan (+ -1.0 (/ t_0 v)))
     (if (<= v 1.86e-112)
       (atan (/ v (sqrt (* H -19.6))))
       (atan (/ v (+ v t_0)))))))

C

double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -2.2e-53) {
		tmp = atan((-1.0 + (t_0 / v)));
	} else if (v <= 1.86e-112) {
		tmp = atan((v / sqrt((H * -19.6))));
	} else {
		tmp = atan((v / (v + t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-9.8d0) * (h / v)
    if (v <= (-2.2d-53)) then
        tmp = atan(((-1.0d0) + (t_0 / v)))
    else if (v <= 1.86d-112) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    else
        tmp = atan((v / (v + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -2.2e-53) {
		tmp = Math.atan((-1.0 + (t_0 / v)));
	} else if (v <= 1.86e-112) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v / (v + t_0)));
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = -9.8 * (H / v)
	tmp = 0
	if v <= -2.2e-53:
		tmp = math.atan((-1.0 + (t_0 / v)))
	elif v <= 1.86e-112:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((v / (v + t_0)))
	return tmp

Julia

function code(v, H)
	t_0 = Float64(-9.8 * Float64(H / v))
	tmp = 0.0
	if (v <= -2.2e-53)
		tmp = atan(Float64(-1.0 + Float64(t_0 / v)));
	elseif (v <= 1.86e-112)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	else
		tmp = atan(Float64(v / Float64(v + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = -9.8 * (H / v);
	tmp = 0.0;
	if (v <= -2.2e-53)
		tmp = atan((-1.0 + (t_0 / v)));
	elseif (v <= 1.86e-112)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v / (v + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.2e-53], N[ArcTan[N[(-1.0 + N[(t$95$0 / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.86e-112], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.20000000000000018e-53

   1. Initial program 46.1%

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

      1. sqr-neg 46.1%

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

      2. sqr-neg 46.1%

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

      3. metadata-eval 46.1%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in v around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. unpow2 90.6%

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

      3. associate-/r* 90.6%

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

      4. associate-*r/ 90.6%

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

   7. Applied egg-rr 90.6%

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

   if -2.20000000000000018e-53 < v < 1.86e-112

   1. Initial program 99.8%

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

      1. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

   if 1.86e-112 < v

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in H around 0 91.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.2 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{-9.8 \cdot \frac{H}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.86 \cdot 10^{-112}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9.8 \cdot \frac{H}{v}\\ \mathbf{if}\;v \leq -1.32 \cdot 10^{-51}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{t\_0}{v}\right)\\ \mathbf{elif}\;v \leq 1.86 \cdot 10^{-112}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (* -9.8 (/ H v))))
   (if (<= v -1.32e-51)
     (atan (+ -1.0 (/ t_0 v)))
     (if (<= v 1.86e-112)
       (atan (* v (sqrt (/ -0.05102040816326531 H))))
       (atan (/ v (+ v t_0)))))))

C

double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -1.32e-51) {
		tmp = atan((-1.0 + (t_0 / v)));
	} else if (v <= 1.86e-112) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / (v + t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-9.8d0) * (h / v)
    if (v <= (-1.32d-51)) then
        tmp = atan(((-1.0d0) + (t_0 / v)))
    else if (v <= 1.86d-112) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((v / (v + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -1.32e-51) {
		tmp = Math.atan((-1.0 + (t_0 / v)));
	} else if (v <= 1.86e-112) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + t_0)));
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = -9.8 * (H / v)
	tmp = 0
	if v <= -1.32e-51:
		tmp = math.atan((-1.0 + (t_0 / v)))
	elif v <= 1.86e-112:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + t_0)))
	return tmp

Julia

function code(v, H)
	t_0 = Float64(-9.8 * Float64(H / v))
	tmp = 0.0
	if (v <= -1.32e-51)
		tmp = atan(Float64(-1.0 + Float64(t_0 / v)));
	elseif (v <= 1.86e-112)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = -9.8 * (H / v);
	tmp = 0.0;
	if (v <= -1.32e-51)
		tmp = atan((-1.0 + (t_0 / v)));
	elseif (v <= 1.86e-112)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.32e-51], N[ArcTan[N[(-1.0 + N[(t$95$0 / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.86e-112], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.31999999999999998e-51

   1. Initial program 46.1%

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

      1. sqr-neg 46.1%

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

      2. sqr-neg 46.1%

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

      3. metadata-eval 46.1%

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

   3. Simplified 46.1%

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

   5. Taylor expanded in v around -inf 90.6%

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

      1. associate-*r/ 90.6%

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

      2. unpow2 90.6%

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

      3. associate-/r* 90.6%

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

      4. associate-*r/ 90.6%

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

   7. Applied egg-rr 90.6%

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

   if -1.31999999999999998e-51 < v < 1.86e-112

   1. Initial program 99.8%

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

      1. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 94.9%

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

      2. associate-/r/ 99.8%

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

      3. pow1/2 99.8%

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

      4. pow-flip 99.8%

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

      5. sub-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. distribute-rgt-neg-in 99.8%

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

      9. fma-define 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow2 99.8%

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

      12. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in H around inf 90.3%

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

      1. *-commutative 90.3%

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

   9. Simplified 90.3%

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

       1. add-sqr-sqrt 90.0%

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

       2. sqrt-unprod 90.3%

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

       3. pow-prod-up 90.4%

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

       4. metadata-eval 90.4%

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

   11. Applied egg-rr 90.4%

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

       1. unpow-1 90.4%

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

       2. *-commutative 90.4%

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

       3. rem-square-sqrt 0.0%

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

       4. unpow2 0.0%

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

       5. associate-/r* 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 90.2%

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

       8. metadata-eval 90.2%

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

   13. Simplified 90.2%

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

   if 1.86e-112 < v

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in H around 0 91.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.32 \cdot 10^{-51}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{-9.8 \cdot \frac{H}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 1.86 \cdot 10^{-112}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.8 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{-9.8 \cdot \frac{H}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.8e-143)
   (atan (+ -1.0 (/ (* -9.8 (/ H v)) v)))
   (if (<= v 4.2e-123)
     (atan (* v (* -0.10204081632653061 (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.8e-143) {
		tmp = atan((-1.0 + ((-9.8 * (H / v)) / v)));
	} else if (v <= 4.2e-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 <= (-1.8d-143)) then
        tmp = atan(((-1.0d0) + (((-9.8d0) * (h / v)) / v)))
    else if (v <= 4.2d-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 <= -1.8e-143) {
		tmp = Math.atan((-1.0 + ((-9.8 * (H / v)) / v)));
	} else if (v <= 4.2e-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 <= -1.8e-143:
		tmp = math.atan((-1.0 + ((-9.8 * (H / v)) / v)))
	elif v <= 4.2e-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 <= -1.8e-143)
		tmp = atan(Float64(-1.0 + Float64(Float64(-9.8 * Float64(H / v)) / v)));
	elseif (v <= 4.2e-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 <= -1.8e-143)
		tmp = atan((-1.0 + ((-9.8 * (H / v)) / v)));
	elseif (v <= 4.2e-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, -1.8e-143], N[ArcTan[N[(-1.0 + N[(N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 4.2e-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 < -1.7999999999999999e-143

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. sqr-neg 55.9%

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

      3. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in v around -inf 80.9%

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

      1. associate-*r/ 80.9%

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

      2. unpow2 80.9%

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

      3. associate-/r* 80.9%

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

      4. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

   if -1.7999999999999999e-143 < v < 4.1999999999999998e-123

   1. Initial program 99.8%

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

      1. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in H around 0 27.6%

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

   6. Taylor expanded in v around 0 27.6%

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

      1. associate-*r/ 27.6%

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

      2. *-commutative 27.6%

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

   8. Simplified 27.6%

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

      1. associate-/r/ 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. *-commutative 27.6%

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

      4. times-frac 27.6%

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

      5. metadata-eval 27.6%

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

   10. Applied egg-rr 27.6%

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

   if 4.1999999999999998e-123 < v

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in v around inf 90.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.8 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{-9.8 \cdot \frac{H}{v}}{v}\right)\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -8.6 \cdot 10^{-153}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -8.6e-153)
   (atan -1.0)
   (if (<= v 4.2e-123)
     (atan (* v (* -0.10204081632653061 (/ v H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -8.6e-153) {
		tmp = atan(-1.0);
	} else if (v <= 4.2e-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 <= (-8.6d-153)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.2d-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 <= -8.6e-153) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.2e-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 <= -8.6e-153:
		tmp = math.atan(-1.0)
	elif v <= 4.2e-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 <= -8.6e-153)
		tmp = atan(-1.0);
	elseif (v <= 4.2e-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 <= -8.6e-153)
		tmp = atan(-1.0);
	elseif (v <= 4.2e-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, -8.6e-153], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.2e-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 < -8.6000000000000001e-153

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. sqr-neg 55.9%

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

      3. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in v around -inf 80.8%

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

   if -8.6000000000000001e-153 < v < 4.1999999999999998e-123

   1. Initial program 99.8%

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

      1. sqr-neg 99.8%

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

      2. sqr-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in H around 0 27.6%

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

   6. Taylor expanded in v around 0 27.6%

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

      1. associate-*r/ 27.6%

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

      2. *-commutative 27.6%

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

   8. Simplified 27.6%

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

      1. associate-/r/ 27.6%

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

      2. *-un-lft-identity 27.6%

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

      3. *-commutative 27.6%

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

      4. times-frac 27.6%

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

      5. metadata-eval 27.6%

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

   10. Applied egg-rr 27.6%

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

   if 4.1999999999999998e-123 < v

   1. Initial program 59.9%

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

      1. sqr-neg 59.9%

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

      2. sqr-neg 59.9%

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

      3. metadata-eval 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in v around inf 90.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8.6 \cdot 10^{-153}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(-0.10204081632653061 \cdot \frac{v}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -9.8 \cdot \frac{H}{v}\\ \mathbf{if}\;v \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{t\_0}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (* -9.8 (/ H v))))
   (if (<= v -3.3e-142) (atan (+ -1.0 (/ t_0 v))) (atan (/ v (+ v t_0))))))

C

double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -3.3e-142) {
		tmp = atan((-1.0 + (t_0 / v)));
	} else {
		tmp = atan((v / (v + t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-9.8d0) * (h / v)
    if (v <= (-3.3d-142)) then
        tmp = atan(((-1.0d0) + (t_0 / v)))
    else
        tmp = atan((v / (v + t_0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double t_0 = -9.8 * (H / v);
	double tmp;
	if (v <= -3.3e-142) {
		tmp = Math.atan((-1.0 + (t_0 / v)));
	} else {
		tmp = Math.atan((v / (v + t_0)));
	}
	return tmp;
}

Python

def code(v, H):
	t_0 = -9.8 * (H / v)
	tmp = 0
	if v <= -3.3e-142:
		tmp = math.atan((-1.0 + (t_0 / v)))
	else:
		tmp = math.atan((v / (v + t_0)))
	return tmp

Julia

function code(v, H)
	t_0 = Float64(-9.8 * Float64(H / v))
	tmp = 0.0
	if (v <= -3.3e-142)
		tmp = atan(Float64(-1.0 + Float64(t_0 / v)));
	else
		tmp = atan(Float64(v / Float64(v + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = -9.8 * (H / v);
	tmp = 0.0;
	if (v <= -3.3e-142)
		tmp = atan((-1.0 + (t_0 / v)));
	else
		tmp = atan((v / (v + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.3e-142], N[ArcTan[N[(-1.0 + N[(t$95$0 / v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.2999999999999997e-142

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. sqr-neg 55.9%

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

      3. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in v around -inf 80.9%

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

      1. associate-*r/ 80.9%

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

      2. unpow2 80.9%

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

      3. associate-/r* 80.9%

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

      4. associate-*r/ 80.9%

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

   7. Applied egg-rr 80.9%

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

   if -3.2999999999999997e-142 < v

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in H around 0 72.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.3 \cdot 10^{-142}:\\ \;\;\;\;\tan^{-1} \left(-1 + \frac{-9.8 \cdot \frac{H}{v}}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (v H) :precision binary64 (if (<= v -1e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e-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 <= (-1d-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 <= -1e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e-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 <= -1e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.999999999999969e-311

   1. Initial program 65.6%

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

      1. sqr-neg 65.6%

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

      2. sqr-neg 65.6%

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

      3. metadata-eval 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in v around -inf 63.8%

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

   if -9.999999999999969e-311 < v

   1. Initial program 66.1%

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

      1. sqr-neg 66.1%

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

      2. sqr-neg 66.1%

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

      3. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in v around inf 76.9%

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

Alternative 8: 35.4% 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 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. Taylor expanded in v around -inf 32.5%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))