Optimal throwing angle

Percentage Accurate: 67.9% → 99.6%

Time: 5.6s

Alternatives: 5

Speedup: 1.0×

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: 67.9% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+154)
   (atan -1.0)
   (if (<= v 2e+132) (atan (/ v (sqrt (fma 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+132) {
		tmp = atan((v / sqrt(fma(v, v, (-19.6 * H)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+132)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(-19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+132], N[ArcTan[N[(v / N[Sqrt[N[(v * v + 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. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

   if -5.00000000000000004e154 < v < 1.99999999999999998e132

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{\left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v + \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} + \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H\right)}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{49}{5}}\right)\right) \cdot H\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(\color{blue}{\frac{98}{5}}\right)\right) \cdot H\right)}}\right) \]

      9. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   if 1.99999999999999998e132 < v

   1. Initial program 9.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

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

Alternative 2: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-73}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{H}{v \cdot v} - 1\right)\\ \mathbf{elif}\;v \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-73)
   (atan (- (* -9.8 (/ H (* v v))) 1.0))
   (if (<= v 3e-65)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan (/ v (fma (/ H v) -9.8 v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-73) {
		tmp = atan(((-9.8 * (H / (v * v))) - 1.0));
	} else if (v <= 3e-65) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan((v / fma((H / v), -9.8, v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.6e-73)
		tmp = atan(Float64(Float64(-9.8 * Float64(H / Float64(v * v))) - 1.0));
	elseif (v <= 3e-65)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(Float64(v / fma(Float64(H / v), -9.8, v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.6e-73], N[ArcTan[N[(N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 3e-65], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.5999999999999999e-73

   1. Initial program 51.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-49}{5} \cdot \frac{H}{{v}^{2}}} - 1\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \color{blue}{\frac{H}{{v}^{2}}} - 1\right) \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{\color{blue}{v \cdot v}} - 1\right) \]

      5. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if -3.5999999999999999e-73 < v < 2.99999999999999998e-65

   1. Initial program 99.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{H \cdot {\left(\sqrt{\frac{-98}{5}}\right)}^{2} + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} + H \cdot {\left(\sqrt{\frac{-98}{5}}\right)}^{2}}}}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\left(\sqrt{\frac{-98}{5}} \cdot \sqrt{\frac{-98}{5}}\right)}}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\frac{-98}{5}}}}\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{98}{5}\right)\right)}}}\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\left(\mathsf{neg}\left(H \cdot \frac{98}{5}\right)\right)}}}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\left(\mathsf{neg}\left(H\right)\right) \cdot \frac{98}{5}}}}\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} - H \cdot \frac{98}{5}}}}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - \color{blue}{\frac{98}{5} \cdot H}}}\right) \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} + \left(\mathsf{neg}\left(\frac{98}{5}\right)\right) \cdot H}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\frac{-98}{5}} \cdot H}}\right) \]

      11. +-commutative N/A

          \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{\frac{-98}{5} \cdot H + {v}^{2}}}}\right) \]

      12. lower-atan.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \left(\sqrt{\frac{\frac{-5}{98}}{H}} \cdot v\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

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

   if 2.99999999999999998e-65 < v

   1. Initial program 42.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in H around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{-49}{5} \cdot \frac{H}{v}}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{-49}{5} \cdot \frac{H}{v} + v}}\right) \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{H}{v} \cdot \frac{-49}{5}} + v}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\mathsf{fma}\left(\frac{H}{v}, \frac{-49}{5}, v\right)}}\right) \]

      4. lower-/.f64 92.8

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

   5. Applied rewrites 92.8%

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

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.6 \cdot 10^{-73}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{H}{v \cdot v} - 1\right)\\ \mathbf{elif}\;v \leq 3 \cdot 10^{-65}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.6e-73)
   (atan (- (* -9.8 (/ H (* v v))) 1.0))
   (if (<= v 3e-65)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.6e-73) {
		tmp = atan(((-9.8 * (H / (v * v))) - 1.0));
	} else if (v <= 3e-65) {
		tmp = atan((sqrt((-0.05102040816326531 / 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 <= (-3.6d-73)) then
        tmp = atan((((-9.8d0) * (h / (v * v))) - 1.0d0))
    else if (v <= 3d-65) then
        tmp = atan((sqrt(((-0.05102040816326531d0) / 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 <= -3.6e-73) {
		tmp = Math.atan(((-9.8 * (H / (v * v))) - 1.0));
	} else if (v <= 3e-65) {
		tmp = Math.atan((Math.sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -3.6e-73:
		tmp = math.atan(((-9.8 * (H / (v * v))) - 1.0))
	elif v <= 3e-65:
		tmp = math.atan((math.sqrt((-0.05102040816326531 / H)) * v))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.6e-73)
		tmp = atan(Float64(Float64(-9.8 * Float64(H / Float64(v * v))) - 1.0));
	elseif (v <= 3e-65)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -3.6e-73)
		tmp = atan(((-9.8 * (H / (v * v))) - 1.0));
	elseif (v <= 3e-65)
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.6e-73], N[ArcTan[N[(N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 3e-65], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.5999999999999999e-73

   1. Initial program 51.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\frac{-49}{5} \cdot \frac{H}{{v}^{2}}} - 1\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \color{blue}{\frac{H}{{v}^{2}}} - 1\right) \]

      4. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{\color{blue}{v \cdot v}} - 1\right) \]

      5. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if -3.5999999999999999e-73 < v < 2.99999999999999998e-65

   1. Initial program 99.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{H \cdot {\left(\sqrt{\frac{-98}{5}}\right)}^{2} + {v}^{2}}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} + H \cdot {\left(\sqrt{\frac{-98}{5}}\right)}^{2}}}}\right) \]

      2. unpow2 N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\left(\sqrt{\frac{-98}{5}} \cdot \sqrt{\frac{-98}{5}}\right)}}}\right) \]

      3. rem-square-sqrt N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\frac{-98}{5}}}}\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + H \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{98}{5}\right)\right)}}}\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\left(\mathsf{neg}\left(H \cdot \frac{98}{5}\right)\right)}}}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\left(\mathsf{neg}\left(H\right)\right) \cdot \frac{98}{5}}}}\right) \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} - H \cdot \frac{98}{5}}}}\right) \]

      8. *-commutative N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - \color{blue}{\frac{98}{5} \cdot H}}}\right) \]

      9. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{{v}^{2} + \left(\mathsf{neg}\left(\frac{98}{5}\right)\right) \cdot H}}}\right) \]

      10. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} + \color{blue}{\frac{-98}{5}} \cdot H}}\right) \]

      11. +-commutative N/A

          \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\frac{1}{\color{blue}{\frac{-98}{5} \cdot H + {v}^{2}}}}\right) \]

      12. lower-atan.f64 N/A

          \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \left(\sqrt{\frac{\frac{-5}{98}}{H}} \cdot v\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

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

   if 2.99999999999999998e-65 < v

   1. Initial program 42.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 92.7%

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

Alternative 4: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v -4e-310) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4e-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 <= (-4d-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 <= -4e-310) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4e-310:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4e-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 <= -4e-310)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4e-310], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.999999999999988e-310

   1. Initial program 65.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.0%

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

   if -3.999999999999988e-310 < v

   1. Initial program 60.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 66.4%

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

Alternative 5: 34.4% accurate, 1.4× 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 63.2%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around -inf

   \[\leadsto \tan^{-1} \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 39.6%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))