Optimal throwing angle

Percentage Accurate: 67.3% → 99.2%

Time: 9.2s

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: 67.3% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 2e+86) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+86) {
		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 <= (-5d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+86) 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 <= -5e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+86) {
		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 <= -5e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+86:
		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 <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+86)
		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 <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+86)
		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, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+86], 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 < -5.00000000000000004e154

   1. Initial program 3.1%

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

      1. sqr-neg 3.1%

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

      2. sqr-neg 3.1%

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

      3. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -5.00000000000000004e154 < v < 2e86

   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 2e86 < v

   1. Initial program 32.6%

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

      1. sqr-neg 32.6%

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

      2. sqr-neg 32.6%

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

      3. metadata-eval 32.6%

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

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)} \]
   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: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.9e-15)
   (atan -1.0)
   (if (<= v 2.1e-55)
     (atan (/ v (sqrt (* H -19.6))))
     (atan (* v (/ 1.0 (+ v (* -9.8 (/ H v)))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.9e-15) {
		tmp = atan(-1.0);
	} else if (v <= 2.1e-55) {
		tmp = atan((v / sqrt((H * -19.6))));
	} else {
		tmp = atan((v * (1.0 / (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 <= (-3.9d-15)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.1d-55) then
        tmp = atan((v / sqrt((h * (-19.6d0)))))
    else
        tmp = atan((v * (1.0d0 / (v + ((-9.8d0) * (h / v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -3.9e-15) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.1e-55) {
		tmp = Math.atan((v / Math.sqrt((H * -19.6))));
	} else {
		tmp = Math.atan((v * (1.0 / (v + (-9.8 * (H / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.9e-15:
		tmp = math.atan(-1.0)
	elif v <= 2.1e-55:
		tmp = math.atan((v / math.sqrt((H * -19.6))))
	else:
		tmp = math.atan((v * (1.0 / (v + (-9.8 * (H / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.9e-15)
		tmp = atan(-1.0);
	elseif (v <= 2.1e-55)
		tmp = atan(Float64(v / sqrt(Float64(H * -19.6))));
	else
		tmp = atan(Float64(v * Float64(1.0 / Float64(v + Float64(-9.8 * Float64(H / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.9e-15)
		tmp = atan(-1.0);
	elseif (v <= 2.1e-55)
		tmp = atan((v / sqrt((H * -19.6))));
	else
		tmp = atan((v * (1.0 / (v + (-9.8 * (H / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.9e-15], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.1e-55], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v * N[(1.0 / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.90000000000000026e-15

   1. Initial program 54.6%

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

      1. sqr-neg 54.6%

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

      2. sqr-neg 54.6%

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

      3. metadata-eval 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in v around -inf 92.4%

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

   if -3.90000000000000026e-15 < v < 2.1000000000000002e-55

   1. Initial program 99.5%

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

      1. sqr-neg 99.5%

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

      2. sqr-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 98.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{\left(\sqrt[3]{19.6 \cdot H} \cdot \sqrt[3]{19.6 \cdot H}\right) \cdot \sqrt[3]{19.6 \cdot H}}}}\right) \]

      2. pow3 98.8%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   6. Applied egg-rr 98.8%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   7. Taylor expanded in H around -inf 0.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{-1 \cdot \left(\sqrt{H \cdot {\left(\sqrt[3]{-19.6}\right)}^{3}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{-\sqrt{H \cdot {\left(\sqrt[3]{-19.6}\right)}^{3}} \cdot {\left(\sqrt{-1}\right)}^{2}}}\right) \]

      2. *-commutative 0.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{-\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{H \cdot {\left(\sqrt[3]{-19.6}\right)}^{3}}}}\right) \]

      3. unpow2 0.0%

         \[\leadsto \tan^{-1} \left(\frac{v}{-\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{H \cdot {\left(\sqrt[3]{-19.6}\right)}^{3}}}\right) \]

      4. rem-square-sqrt 88.5%

         \[\leadsto \tan^{-1} \left(\frac{v}{-\color{blue}{-1} \cdot \sqrt{H \cdot {\left(\sqrt[3]{-19.6}\right)}^{3}}}\right) \]

      5. rem-cube-cbrt 90.0%

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

      6. distribute-lft-neg-in 90.0%

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

      7. metadata-eval 90.0%

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

      8. *-lft-identity 90.0%

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

      9. *-commutative 90.0%

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

   9. Simplified 90.0%

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

   if 2.1000000000000002e-55 < v

   1. Initial program 57.3%

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

      1. sqr-neg 57.3%

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

      2. sqr-neg 57.3%

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

      3. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

      1. add-cube-cbrt 57.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{\left(\sqrt[3]{19.6 \cdot H} \cdot \sqrt[3]{19.6 \cdot H}\right) \cdot \sqrt[3]{19.6 \cdot H}}}}\right) \]

      2. pow3 57.2%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   6. Applied egg-rr 57.2%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   7. Taylor expanded in H around 0 89.5%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + -0.5 \cdot \frac{H \cdot {\left(\sqrt[3]{19.6}\right)}^{3}}{v}}}\right) \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 89.5%

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

      2. associate-/l* 90.4%

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

   9. Simplified 90.4%

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

   10. Taylor expanded in H around inf 90.4%

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

       1. div-inv 90.4%

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

       2. cancel-sign-sub-inv 90.4%

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

       3. metadata-eval 90.4%

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

   12. Applied egg-rr 90.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.1 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{H \cdot -19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \frac{1}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e-178) (atan -1.0) (atan (* v (/ 1.0 (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e-178) {
		tmp = atan(-1.0);
	} else {
		tmp = atan((v * (1.0 / (v + (-9.8 * (H / v))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1d-178)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan((v * (1.0d0 / (v + ((-9.8d0) * (h / v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1e-178) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan((v * (1.0 / (v + (-9.8 * (H / v))))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e-178:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan((v * (1.0 / (v + (-9.8 * (H / v))))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e-178)
		tmp = atan(-1.0);
	else
		tmp = atan(Float64(v * Float64(1.0 / Float64(v + Float64(-9.8 * Float64(H / v))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1e-178)
		tmp = atan(-1.0);
	else
		tmp = atan((v * (1.0 / (v + (-9.8 * (H / v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e-178], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[N[(v * N[(1.0 / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -9.9999999999999995e-179

   1. Initial program 66.4%

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

      1. sqr-neg 66.4%

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

      2. sqr-neg 66.4%

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

      3. metadata-eval 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in v around -inf 74.4%

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

   if -9.9999999999999995e-179 < 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. Step-by-step derivation

      1. add-cube-cbrt 68.3%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{\left(\sqrt[3]{19.6 \cdot H} \cdot \sqrt[3]{19.6 \cdot H}\right) \cdot \sqrt[3]{19.6 \cdot H}}}}\right) \]

      2. pow3 68.4%

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   6. Applied egg-rr 68.4%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{{\left(\sqrt[3]{19.6 \cdot H}\right)}^{3}}}}\right) \]

   7. Taylor expanded in H around 0 70.6%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + -0.5 \cdot \frac{H \cdot {\left(\sqrt[3]{19.6}\right)}^{3}}{v}}}\right) \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 70.6%

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

      2. associate-/l* 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in H around inf 71.3%

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

       1. div-inv 71.3%

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

       2. cancel-sign-sub-inv 71.3%

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

       3. metadata-eval 71.3%

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

   12. Applied egg-rr 71.3%

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

Alternative 4: 71.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{-177}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.2e-177) (atan -1.0) (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.2e-177) {
		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 <= (-1.2d-177)) 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 <= -1.2e-177) {
		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 <= -1.2e-177:
		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 <= -1.2e-177)
		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 <= -1.2e-177)
		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, -1.2e-177], 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 < -1.1999999999999999e-177

   1. Initial program 66.4%

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

      1. sqr-neg 66.4%

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

      2. sqr-neg 66.4%

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

      3. metadata-eval 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in v around -inf 74.4%

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

   if -1.1999999999999999e-177 < 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 H around 0 71.3%

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

Alternative 5: 66.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -5e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e-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 <= (-5d-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 <= -5e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -5e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e-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 <= -5e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -4.999999999999985e-310

   1. Initial program 70.2%

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

      1. sqr-neg 70.2%

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

      2. sqr-neg 70.2%

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

      3. metadata-eval 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in v around -inf 66.2%

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

   if -4.999999999999985e-310 < v

   1. Initial program 65.4%

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

      1. sqr-neg 65.4%

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

      2. sqr-neg 65.4%

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

      3. metadata-eval 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in v around inf 74.4%

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

Alternative 6: 34.9% 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 67.7%

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

   1. sqr-neg 67.7%

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

   2. sqr-neg 67.7%

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

   3. metadata-eval 67.7%

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

3. Simplified 67.7%

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

5. Taylor expanded in v around -inf 32.2%

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

Reproduce

herbie shell --seed 2024186 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))