Optimal throwing angle

Percentage Accurate: 67.4% → 99.4%

Time: 7.3s

Alternatives: 6

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: 67.4% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(H, 19.6, v \cdot v\right)\\ \mathbf{if}\;v \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v \cdot v, \frac{v \cdot v}{t\_0}, \frac{-H}{t\_0} \cdot \left(384.16 \cdot H\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (fma H 19.6 (* v v))))
   (if (<= v -2e+156)
     (atan -1.0)
     (if (<= v 5e+150)
       (atan
        (/
         v
         (sqrt (fma (* v v) (/ (* v v) t_0) (* (/ (- H) t_0) (* 384.16 H))))))
       (atan 1.0)))))

C

double code(double v, double H) {
	double t_0 = fma(H, 19.6, (v * v));
	double tmp;
	if (v <= -2e+156) {
		tmp = atan(-1.0);
	} else if (v <= 5e+150) {
		tmp = atan((v / sqrt(fma((v * v), ((v * v) / t_0), ((-H / t_0) * (384.16 * H))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = fma(H, 19.6, Float64(v * v))
	tmp = 0.0
	if (v <= -2e+156)
		tmp = atan(-1.0);
	elseif (v <= 5e+150)
		tmp = atan(Float64(v / sqrt(fma(Float64(v * v), Float64(Float64(v * v) / t_0), Float64(Float64(Float64(-H) / t_0) * Float64(384.16 * H))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[(H * 19.6 + N[(v * v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e+156], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5e+150], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[((-H) / t$95$0), $MachinePrecision] * N[(384.16 * H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2e156

   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 -2e156 < v < 5.00000000000000009e150

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v \cdot v, \frac{v \cdot v}{\mathsf{fma}\left(H, \color{blue}{\frac{98}{5}}, v \cdot v\right)}, \mathsf{neg}\left(\frac{\left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right) \cdot \left(\left(2 \cdot \frac{49}{5}\right) \cdot H\right)}{v \cdot v + \left(2 \cdot \frac{49}{5}\right) \cdot H}\right)\right)}}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v \cdot v, \frac{v \cdot v}{\mathsf{fma}\left(H, 19.6, v \cdot v\right)}, -\left(384.16 \cdot H\right) \cdot \frac{H}{\mathsf{fma}\left(H, 19.6, v \cdot v\right)}\right)}}}\right) \]

   if 5.00000000000000009e150 < v

   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%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v \cdot v, \frac{v \cdot v}{\mathsf{fma}\left(H, 19.6, v \cdot v\right)}, \frac{-H}{\mathsf{fma}\left(H, 19.6, v \cdot v\right)} \cdot \left(384.16 \cdot H\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+156)
   (atan -1.0)
   (if (<= v 5e+150) (atan (/ v (sqrt (fma v v (* -19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+156) {
		tmp = atan(-1.0);
	} else if (v <= 5e+150) {
		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 <= -2e+156)
		tmp = atan(-1.0);
	elseif (v <= 5e+150)
		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, -2e+156], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 5e+150], 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 < -2e156

   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 -2e156 < v < 5.00000000000000009e150

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-in 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. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   if 5.00000000000000009e150 < v

   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%

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

Alternative 3: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.2 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.4 \cdot 10^{-49}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-9.8}{v}, H, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.2e-62)
   (atan -1.0)
   (if (<= v 4.4e-49)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan (/ v (fma (/ -9.8 v) H v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.2e-62) {
		tmp = atan(-1.0);
	} else if (v <= 4.4e-49) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan((v / fma((-9.8 / v), H, v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.2e-62)
		tmp = atan(-1.0);
	elseif (v <= 4.4e-49)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(Float64(v / fma(Float64(-9.8 / v), H, v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.2e-62], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.4e-49], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(N[(-9.8 / v), $MachinePrecision] * H + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.1999999999999998e-62

   1. Initial program 47.7%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 91.9%

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

   if -4.1999999999999998e-62 < v < 4.3999999999999998e-49

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 92.6%

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

   if 4.3999999999999998e-49 < v

   1. Initial program 53.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 4: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.2 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(9.8, \frac{H}{v \cdot v}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.2e-62)
   (atan -1.0)
   (if (<= v 4.6e-49)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan (fma 9.8 (/ H (* v v)) 1.0)))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.2e-62) {
		tmp = atan(-1.0);
	} else if (v <= 4.6e-49) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan(fma(9.8, (H / (v * v)), 1.0));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.2e-62)
		tmp = atan(-1.0);
	elseif (v <= 4.6e-49)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(fma(9.8, Float64(H / Float64(v * v)), 1.0));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.2e-62], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4.6e-49], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.1999999999999998e-62

   1. Initial program 47.7%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 91.9%

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

   if -4.1999999999999998e-62 < v < 4.5999999999999998e-49

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in v around 0

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

   8. Applied rewrites 92.6%

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

   if 4.5999999999999998e-49 < v

   1. Initial program 53.1%

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

   3. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 5: 67.0% 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 63.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 66.8%

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

   if -3.999999999999988e-310 < v

   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 70.0%

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

Alternative 6: 34.1% 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 64.6%

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

3. Taylor expanded in v around -inf

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

5. Applied rewrites 34.5%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))