Optimal throwing angle

Percentage Accurate: 67.1% → 99.8%

Time: 9.2s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, H \cdot -19.6\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan (/ v (fabs (fma -9.8 (/ H v) v))))
   (if (<= v 2e+145) (atan (/ v (sqrt (fma v v (* H -19.6))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan((v / fabs(fma(-9.8, (H / v), v))));
	} else if (v <= 2e+145) {
		tmp = atan((v / sqrt(fma(v, v, (H * -19.6)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+154)
		tmp = atan(Float64(v / abs(fma(-9.8, Float64(H / v), v))));
	elseif (v <= 2e+145)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(H * -19.6)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+154], N[ArcTan[N[(v / N[Abs[N[(-9.8 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 2e+145], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(H * -19.6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.00000000000000004e154

   1. Initial program 3.0%

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

      1. sqr-neg 3.0%

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

      2. sqr-neg 3.0%

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

      3. fmm-def 3.0%

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

      4. *-commutative 3.0%

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

      5. distribute-rgt-neg-in 3.0%

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

      6. metadata-eval 3.0%

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

      7. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in H around 0 1.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.1%

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

      3. pow2 3.1%

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

      4. +-commutative 3.1%

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

      5. fma-define 3.1%

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

   7. Applied egg-rr 3.1%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 3.1%

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

      2. rem-sqrt-square 99.1%

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

   9. Simplified 99.1%

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

   if -1.00000000000000004e154 < v < 2e145

   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. fmm-def 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 2e145 < v

   1. Initial program 14.1%

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

      1. sqr-neg 14.1%

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

      2. sqr-neg 14.1%

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

      3. metadata-eval 14.1%

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

   3. Simplified 14.1%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.2e+154)
   (atan (/ v (fabs (fma -9.8 (/ H v) v))))
   (if (<= v 5e+141)
     (atan (/ v (sqrt (- (* v v) (* H 19.6)))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.2e+154) {
		tmp = atan((v / fabs(fma(-9.8, (H / v), v))));
	} else if (v <= 5e+141) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.2e+154)
		tmp = atan(Float64(v / abs(fma(-9.8, Float64(H / v), v))));
	elseif (v <= 5e+141)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.2e+154], N[ArcTan[N[(v / N[Abs[N[(-9.8 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5e+141], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $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 < -1.20000000000000007e154

   1. Initial program 3.0%

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

      1. sqr-neg 3.0%

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

      2. sqr-neg 3.0%

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

      3. fmm-def 3.0%

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

      4. *-commutative 3.0%

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

      5. distribute-rgt-neg-in 3.0%

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

      6. metadata-eval 3.0%

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

      7. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in H around 0 1.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.1%

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

      3. pow2 3.1%

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

      4. +-commutative 3.1%

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

      5. fma-define 3.1%

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

   7. Applied egg-rr 3.1%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right)}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. unpow2 3.1%

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

      2. rem-sqrt-square 99.1%

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

   9. Simplified 99.1%

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

   if -1.20000000000000007e154 < v < 5.00000000000000025e141

   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 5.00000000000000025e141 < v

   1. Initial program 16.0%

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

      1. sqr-neg 16.0%

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

      2. sqr-neg 16.0%

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

      3. fmm-def 16.0%

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

      4. *-commutative 16.0%

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

      5. distribute-rgt-neg-in 16.0%

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

      6. metadata-eval 16.0%

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

      7. metadata-eval 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in H around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\left|\mathsf{fma}\left(-9.8, \frac{H}{v}, v\right)\right|}\right)\\ \mathbf{elif}\;v \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1e+154)
   (atan (+ (* -9.8 (/ (/ H v) v)) -1.0))
   (if (<= v 8e+142)
     (atan (/ v (sqrt (- (* v v) (* H 19.6)))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1e+154) {
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 8e+142) {
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	} 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 <= (-1d+154)) then
        tmp = atan((((-9.8d0) * ((h / v) / v)) + (-1.0d0)))
    else if (v <= 8d+142) then
        tmp = atan((v / sqrt(((v * v) - (h * 19.6d0)))))
    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 <= -1e+154) {
		tmp = Math.atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 8e+142) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (H * 19.6)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1e+154:
		tmp = math.atan(((-9.8 * ((H / v) / v)) + -1.0))
	elif v <= 8e+142:
		tmp = math.atan((v / math.sqrt(((v * v) - (H * 19.6)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1e+154)
		tmp = atan(Float64(Float64(-9.8 * Float64(Float64(H / v) / v)) + -1.0));
	elseif (v <= 8e+142)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(H * 19.6)))));
	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 <= -1e+154)
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	elseif (v <= 8e+142)
		tmp = atan((v / sqrt(((v * v) - (H * 19.6)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1e+154], N[ArcTan[N[(N[(-9.8 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 8e+142], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(H * 19.6), $MachinePrecision]), $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 < -1.00000000000000004e154

   1. Initial program 3.0%

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

      1. sqr-neg 3.0%

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

      2. sqr-neg 3.0%

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

      3. metadata-eval 3.0%

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

   3. Simplified 3.0%

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

   5. Taylor expanded in v around -inf 98.4%

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

      1. add-sqr-sqrt 57.9%

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

      2. unpow2 57.9%

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

      3. times-frac 58.5%

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

   7. Applied egg-rr 58.5%

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

      1. associate-*l/ 58.5%

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

      2. associate-*r/ 58.5%

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

      3. rem-square-sqrt 99.0%

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

   9. Simplified 99.0%

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

   if -1.00000000000000004e154 < v < 8.00000000000000041e142

   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 8.00000000000000041e142 < v

   1. Initial program 16.0%

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

      1. sqr-neg 16.0%

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

      2. sqr-neg 16.0%

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

      3. fmm-def 16.0%

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

      4. *-commutative 16.0%

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

      5. distribute-rgt-neg-in 16.0%

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

      6. metadata-eval 16.0%

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

      7. metadata-eval 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in H around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - H \cdot 19.6}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.06 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{H} \cdot -0.05102040816326531}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.06e-16)
   (atan (+ (* -9.8 (/ (/ H v) v)) -1.0))
   (if (<= v 1.4e-70)
     (atan (* v (sqrt (* (/ 1.0 H) -0.05102040816326531))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.06e-16) {
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 1.4e-70) {
		tmp = atan((v * sqrt(((1.0 / H) * -0.05102040816326531))));
	} 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 <= (-2.06d-16)) then
        tmp = atan((((-9.8d0) * ((h / v) / v)) + (-1.0d0)))
    else if (v <= 1.4d-70) then
        tmp = atan((v * sqrt(((1.0d0 / h) * (-0.05102040816326531d0)))))
    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 <= -2.06e-16) {
		tmp = Math.atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 1.4e-70) {
		tmp = Math.atan((v * Math.sqrt(((1.0 / H) * -0.05102040816326531))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.06e-16:
		tmp = math.atan(((-9.8 * ((H / v) / v)) + -1.0))
	elif v <= 1.4e-70:
		tmp = math.atan((v * math.sqrt(((1.0 / H) * -0.05102040816326531))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.06e-16)
		tmp = atan(Float64(Float64(-9.8 * Float64(Float64(H / v) / v)) + -1.0));
	elseif (v <= 1.4e-70)
		tmp = atan(Float64(v * sqrt(Float64(Float64(1.0 / H) * -0.05102040816326531))));
	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 <= -2.06e-16)
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	elseif (v <= 1.4e-70)
		tmp = atan((v * sqrt(((1.0 / H) * -0.05102040816326531))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.06e-16], N[ArcTan[N[(N[(-9.8 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.4e-70], N[ArcTan[N[(v * N[Sqrt[N[(N[(1.0 / H), $MachinePrecision] * -0.05102040816326531), $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 < -2.0599999999999999e-16

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. sqr-neg 53.9%

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

      3. metadata-eval 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in v around -inf 94.6%

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

      1. add-sqr-sqrt 53.1%

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

      2. unpow2 53.1%

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

      3. times-frac 53.4%

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

   7. Applied egg-rr 53.4%

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

      1. associate-*l/ 53.4%

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

      2. associate-*r/ 53.4%

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

      3. rem-square-sqrt 94.9%

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

   9. Simplified 94.9%

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

   if -2.0599999999999999e-16 < v < 1.4e-70

   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. Taylor expanded in v around 0 99.4%

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

   6. Taylor expanded in v around 0 88.2%

      \[\leadsto \tan^{-1} \left(v \cdot \sqrt{\color{blue}{\frac{-0.05102040816326531}{H}}}\right) \]
   7. Step-by-step derivation

      1. clear-num 88.1%

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

      2. associate-/r/ 88.2%

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

   8. Applied egg-rr 88.2%

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

   if 1.4e-70 < v

   1. Initial program 62.9%

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

      1. sqr-neg 62.9%

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

      2. sqr-neg 62.9%

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

      3. fmm-def 62.9%

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

      4. *-commutative 62.9%

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

      5. distribute-rgt-neg-in 62.9%

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

      6. metadata-eval 62.9%

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

      7. metadata-eval 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in H around 0 90.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.06 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{H} \cdot -0.05102040816326531}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.78 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.78e-16)
   (atan (+ (* -9.8 (/ (/ H v) v)) -1.0))
   (if (<= v 1.4e-70)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.78e-16) {
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 1.4e-70) {
		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 <= (-1.78d-16)) then
        tmp = atan((((-9.8d0) * ((h / v) / v)) + (-1.0d0)))
    else if (v <= 1.4d-70) 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 <= -1.78e-16) {
		tmp = Math.atan(((-9.8 * ((H / v) / v)) + -1.0));
	} else if (v <= 1.4e-70) {
		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 <= -1.78e-16:
		tmp = math.atan(((-9.8 * ((H / v) / v)) + -1.0))
	elif v <= 1.4e-70:
		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 <= -1.78e-16)
		tmp = atan(Float64(Float64(-9.8 * Float64(Float64(H / v) / v)) + -1.0));
	elseif (v <= 1.4e-70)
		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 <= -1.78e-16)
		tmp = atan(((-9.8 * ((H / v) / v)) + -1.0));
	elseif (v <= 1.4e-70)
		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, -1.78e-16], N[ArcTan[N[(N[(-9.8 * N[(N[(H / v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.4e-70], 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 < -1.78000000000000008e-16

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. sqr-neg 53.9%

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

      3. metadata-eval 53.9%

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

   3. Simplified 53.9%

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

   5. Taylor expanded in v around -inf 94.6%

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

      1. add-sqr-sqrt 53.1%

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

      2. unpow2 53.1%

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

      3. times-frac 53.4%

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

   7. Applied egg-rr 53.4%

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

      1. associate-*l/ 53.4%

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

      2. associate-*r/ 53.4%

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

      3. rem-square-sqrt 94.9%

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

   9. Simplified 94.9%

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

   if -1.78000000000000008e-16 < v < 1.4e-70

   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. Taylor expanded in v around 0 99.4%

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

   6. Taylor expanded in v around 0 88.2%

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

   if 1.4e-70 < v

   1. Initial program 62.9%

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

      1. sqr-neg 62.9%

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

      2. sqr-neg 62.9%

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

      3. fmm-def 62.9%

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

      4. *-commutative 62.9%

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

      5. distribute-rgt-neg-in 62.9%

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

      6. metadata-eval 62.9%

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

      7. metadata-eval 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in H around 0 90.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.78 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1} \left(-9.8 \cdot \frac{\frac{H}{v}}{v} + -1\right)\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 3.4 \cdot 10^{-139}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-9.8 \cdot \frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.05e-185)
   (atan -1.0)
   (if (<= v 3.4e-139) (atan (/ v (* -9.8 (/ H v)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.05e-185) {
		tmp = atan(-1.0);
	} else if (v <= 3.4e-139) {
		tmp = atan((v / (-9.8 * (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 <= (-1.05d-185)) then
        tmp = atan((-1.0d0))
    else if (v <= 3.4d-139) then
        tmp = atan((v / ((-9.8d0) * (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 <= -1.05e-185) {
		tmp = Math.atan(-1.0);
	} else if (v <= 3.4e-139) {
		tmp = Math.atan((v / (-9.8 * (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.05e-185:
		tmp = math.atan(-1.0)
	elif v <= 3.4e-139:
		tmp = math.atan((v / (-9.8 * (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.05e-185)
		tmp = atan(-1.0);
	elseif (v <= 3.4e-139)
		tmp = atan(Float64(v / Float64(-9.8 * Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.05e-185)
		tmp = atan(-1.0);
	elseif (v <= 3.4e-139)
		tmp = atan((v / (-9.8 * (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.05e-185], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 3.4e-139], N[ArcTan[N[(v / N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.05e-185

   1. Initial program 65.3%

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

      1. sqr-neg 65.3%

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

      2. sqr-neg 65.3%

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

      3. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in v around -inf 76.8%

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

   if -1.05e-185 < v < 3.39999999999999999e-139

   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. fmm-def 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in H around 0 36.0%

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

   6. Taylor expanded in v around 0 36.0%

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

   if 3.39999999999999999e-139 < v

   1. Initial program 67.3%

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

      1. sqr-neg 67.3%

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

      2. sqr-neg 67.3%

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

      3. metadata-eval 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in v around inf 83.2%

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

Alternative 7: 71.9% 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 65.3%

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

      1. sqr-neg 65.3%

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

      2. sqr-neg 65.3%

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

      3. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in v around -inf 76.8%

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

   if -1.05e-185 < v

   1. Initial program 75.0%

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

      1. sqr-neg 75.0%

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

      2. sqr-neg 75.0%

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

      3. fmm-def 75.0%

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

      4. *-commutative 75.0%

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

      5. distribute-rgt-neg-in 75.0%

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

      6. metadata-eval 75.0%

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

      7. metadata-eval 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in H around 0 72.6%

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

Alternative 8: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 6 \cdot 10^{-307}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 6e-307) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 6e-307) {
		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 <= 6d-307) 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 <= 6e-307) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 6e-307:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 6e-307)
		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 <= 6e-307)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 6e-307], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 5.9999999999999999e-307

   1. Initial program 69.9%

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

      1. sqr-neg 69.9%

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

      2. sqr-neg 69.9%

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

      3. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in v around -inf 67.0%

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

   if 5.9999999999999999e-307 < v

   1. Initial program 72.1%

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

      1. sqr-neg 72.1%

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

      2. sqr-neg 72.1%

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

      3. metadata-eval 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in v around inf 71.5%

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

Alternative 9: 36.1% 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 71.0%

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

   1. sqr-neg 71.0%

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

   2. sqr-neg 71.0%

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

   3. metadata-eval 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in v around -inf 32.4%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))