Optimal throwing angle

Percentage Accurate: 67.6% → 98.8%

Time: 8.8s

Alternatives: 7

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: 67.6% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 6.8 \cdot 10^{+55}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4e+154)
   (atan -1.0)
   (if (<= v 6.8e+55) (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 <= 6.8e+55) {
		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 <= 6.8d+55) 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 <= 6.8e+55) {
		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 <= 6.8e+55:
		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 <= 6.8e+55)
		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 <= 6.8e+55)
		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, 6.8e+55], 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 < 6.7999999999999996e55

   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 6.7999999999999996e55 < v

   1. Initial program 51.6%

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

      1. sqr-neg 51.6%

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

      2. sqr-neg 51.6%

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

      3. metadata-eval 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 2: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.45 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.4 \cdot 10^{-74}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.45e-10)
   (atan -1.0)
   (if (<= v 4.4e-74)
     (atan (* v (pow (* H -19.6) -0.5)))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.45e-10) {
		tmp = atan(-1.0);
	} else if (v <= 4.4e-74) {
		tmp = atan((v * pow((H * -19.6), -0.5)));
	} 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 <= (-1.45d-10)) then
        tmp = atan((-1.0d0))
    else if (v <= 4.4d-74) then
        tmp = atan((v * ((h * (-19.6d0)) ** (-0.5d0))))
    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 <= -1.45e-10) {
		tmp = Math.atan(-1.0);
	} else if (v <= 4.4e-74) {
		tmp = Math.atan((v * Math.pow((H * -19.6), -0.5)));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.45e-10:
		tmp = math.atan(-1.0)
	elif v <= 4.4e-74:
		tmp = math.atan((v * math.pow((H * -19.6), -0.5)))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.45e-10)
		tmp = atan(-1.0);
	elseif (v <= 4.4e-74)
		tmp = atan(Float64(v * (Float64(H * -19.6) ^ -0.5)));
	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 <= -1.45e-10)
		tmp = atan(-1.0);
	elseif (v <= 4.4e-74)
		tmp = atan((v * ((H * -19.6) ^ -0.5)));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.45e-10], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.4e-74], N[ArcTan[N[(v * N[Power[N[(H * -19.6), $MachinePrecision], -0.5], $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 < -1.4499999999999999e-10

   1. Initial program 43.4%

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

      1. sqr-neg 43.4%

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

      2. sqr-neg 43.4%

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

      3. metadata-eval 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in v around -inf 98.8%

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

   if -1.4499999999999999e-10 < v < 4.40000000000000021e-74

   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. Step-by-step derivation

      1. clear-num 97.9%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. *-commutative 99.6%

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

      8. distribute-rgt-neg-in 99.6%

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

      9. fma-define 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow2 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in H around inf 92.7%

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

      1. *-commutative 92.7%

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

   9. Simplified 92.7%

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

   if 4.40000000000000021e-74 < v

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in H around 0 87.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.45 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.4 \cdot 10^{-74}:\\ \;\;\;\;\tan^{-1} \left(v \cdot {\left(H \cdot -19.6\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.35 \cdot 10^{-74}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-11)
   (atan -1.0)
   (if (<= v 3.35e-74)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-11) {
		tmp = atan(-1.0);
	} else if (v <= 3.35e-74) {
		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 <= (-3.6d-11)) then
        tmp = atan((-1.0d0))
    else if (v <= 3.35d-74) 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 <= -3.6e-11) {
		tmp = Math.atan(-1.0);
	} else if (v <= 3.35e-74) {
		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 <= -3.6e-11:
		tmp = math.atan(-1.0)
	elif v <= 3.35e-74:
		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 <= -3.6e-11)
		tmp = atan(-1.0);
	elseif (v <= 3.35e-74)
		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 <= -3.6e-11)
		tmp = atan(-1.0);
	elseif (v <= 3.35e-74)
		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, -3.6e-11], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3.35e-74], 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 < -3.59999999999999985e-11

   1. Initial program 43.4%

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

      1. sqr-neg 43.4%

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

      2. sqr-neg 43.4%

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

      3. metadata-eval 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in v around -inf 98.8%

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

   if -3.59999999999999985e-11 < v < 3.3499999999999998e-74

   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. Step-by-step derivation

      1. clear-num 97.9%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.6%

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

      5. sub-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. *-commutative 99.6%

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

      8. distribute-rgt-neg-in 99.6%

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

      9. fma-define 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow2 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.6%

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

      3. pow-prod-up 99.5%

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

      4. metadata-eval 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in H around inf 92.7%

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

   if 3.3499999999999998e-74 < v

   1. Initial program 67.0%

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

      1. sqr-neg 67.0%

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

      2. sqr-neg 67.0%

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

      3. metadata-eval 67.0%

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

   3. Simplified 67.0%

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

   5. Taylor expanded in H around 0 87.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.35 \cdot 10^{-74}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.1 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-114}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.1e-185)
   (atan -1.0)
   (if (<= v 1.7e-114)
     (atan (* -0.10204081632653061 (/ v (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.1e-185) {
		tmp = atan(-1.0);
	} else if (v <= 1.7e-114) {
		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 <= (-1.1d-185)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.7d-114) 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 <= -1.1e-185) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.7e-114) {
		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 <= -1.1e-185:
		tmp = math.atan(-1.0)
	elif v <= 1.7e-114:
		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 <= -1.1e-185)
		tmp = atan(-1.0);
	elseif (v <= 1.7e-114)
		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 <= -1.1e-185)
		tmp = atan(-1.0);
	elseif (v <= 1.7e-114)
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.1e-185], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.7e-114], 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 < -1.1e-185

   1. Initial program 57.6%

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

      1. sqr-neg 57.6%

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

      2. sqr-neg 57.6%

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

      3. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in v around -inf 79.9%

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

   if -1.1e-185 < v < 1.69999999999999991e-114

   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 24.9%

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

      1. +-commutative 24.9%

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

      2. fma-define 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in H around inf 24.9%

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

      1. *-un-lft-identity 24.9%

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

      2. times-frac 24.9%

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

      3. metadata-eval 24.9%

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

   10. Applied egg-rr 24.9%

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

   if 1.69999999999999991e-114 < v

   1. Initial program 68.6%

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

      1. sqr-neg 68.6%

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

      2. sqr-neg 68.6%

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

      3. metadata-eval 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in v around inf 84.2%

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

Alternative 5: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.05e-185) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.05e-185) {
		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 <= (-1.05d-185)) 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 <= -1.05e-185) {
		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 <= -1.05e-185:
		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 <= -1.05e-185)
		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 <= -1.05e-185)
		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, -1.05e-185], 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 < -1.05e-185

   1. Initial program 57.6%

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

      1. sqr-neg 57.6%

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

      2. sqr-neg 57.6%

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

      3. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in v around -inf 79.9%

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

   if -1.05e-185 < v

   1. Initial program 78.3%

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

      1. sqr-neg 78.3%

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

      2. sqr-neg 78.3%

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

      3. metadata-eval 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in H around 0 65.7%

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

Alternative 6: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -1e-309) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e-309) {
		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-309)) 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-309) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e-309:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e-309)
		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-309)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e-309], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -1.000000000000002e-309

   1. Initial program 63.5%

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

      1. sqr-neg 63.5%

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

      2. sqr-neg 63.5%

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

      3. metadata-eval 63.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in v around -inf 69.1%

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

   if -1.000000000000002e-309 < v

   1. Initial program 75.6%

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

      1. sqr-neg 75.6%

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

      2. sqr-neg 75.6%

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

      3. metadata-eval 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in v around inf 66.0%

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

Alternative 7: 35.7% 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 70.0%

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

   1. sqr-neg 70.0%

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

   2. sqr-neg 70.0%

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

   3. metadata-eval 70.0%

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

3. Simplified 70.0%

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

5. Taylor expanded in v around -inf 33.4%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))