Optimal throwing angle

Percentage Accurate: 67.0% → 99.4%

Time: 8.5s

Alternatives: 10

Speedup: 1.3×

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: 67.0% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{v}{\mathsf{fma}\left(-19.6, \frac{H}{v}, v\right)}}\\ \mathbf{if}\;v \leq -3.5 \cdot 10^{+41}:\\ \;\;\;\;\tan^{-1} \left(-t\_0\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(H, -19.6, v \cdot v\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (sqrt (/ v (fma -19.6 (/ H v) v)))))
   (if (<= v -3.5e+41)
     (atan (- t_0))
     (if (<= v 8e-68) (atan (/ v (sqrt (fma H -19.6 (* v v))))) (atan t_0)))))

C

double code(double v, double H) {
	double t_0 = sqrt((v / fma(-19.6, (H / v), v)));
	double tmp;
	if (v <= -3.5e+41) {
		tmp = atan(-t_0);
	} else if (v <= 8e-68) {
		tmp = atan((v / sqrt(fma(H, -19.6, (v * v)))));
	} else {
		tmp = atan(t_0);
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = sqrt(Float64(v / fma(-19.6, Float64(H / v), v)))
	tmp = 0.0
	if (v <= -3.5e+41)
		tmp = atan(Float64(-t_0));
	elseif (v <= 8e-68)
		tmp = atan(Float64(v / sqrt(fma(H, -19.6, Float64(v * v)))));
	else
		tmp = atan(t_0);
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[Sqrt[N[(v / N[(-19.6 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[v, -3.5e+41], N[ArcTan[(-t$95$0)], $MachinePrecision], If[LessEqual[v, 8e-68], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6 + N[(v * v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.4999999999999999e41

   1. Initial program 40.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 40.9

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in v around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in v around -inf

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

      1. lower-atan.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 N/A

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

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

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -3.4999999999999999e41 < v < 8.00000000000000053e-68

   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
   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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   if 8.00000000000000053e-68 < v

   1. Initial program 57.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 57.1

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in v around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-atan.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

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

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

      15. metadata-eval N/A

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

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\tan^{-1} \left(\sqrt{\frac{v}{\mathsf{fma}\left(-19.6, \frac{H}{v}, v\right)}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-68}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(H, -19.6, v \cdot v\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{v}{\mathsf{fma}\left(-19.6, \frac{H}{v}, v\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+155)
   (atan -1.0)
   (if (<= v 8e-68)
     (atan (/ v (sqrt (fma H -19.6 (* v v)))))
     (atan (sqrt (/ v (fma -19.6 (/ H v) v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+155) {
		tmp = atan(-1.0);
	} else if (v <= 8e-68) {
		tmp = atan((v / sqrt(fma(H, -19.6, (v * v)))));
	} else {
		tmp = atan(sqrt((v / fma(-19.6, (H / v), v))));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+155)
		tmp = atan(-1.0);
	elseif (v <= 8e-68)
		tmp = atan(Float64(v / sqrt(fma(H, -19.6, Float64(v * v)))));
	else
		tmp = atan(sqrt(Float64(v / fma(-19.6, Float64(H / v), v))));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 8e-68], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6 + N[(v * v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Sqrt[N[(v / N[(-19.6 * N[(H / v), $MachinePrecision] + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000001e155

   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 -2.00000000000000001e155 < v < 8.00000000000000053e-68

   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
   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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   if 8.00000000000000053e-68 < v

   1. Initial program 57.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 57.1

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in v around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-atan.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

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

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

      15. metadata-eval N/A

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

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\tan^{-1} \left(\sqrt{\frac{v}{\mathsf{fma}\left(-19.6, \frac{H}{v}, v\right)}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+155}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+137}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(H, -19.6, v \cdot v\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+155)
   (atan -1.0)
   (if (<= v 1e+137) (atan (/ v (sqrt (fma H -19.6 (* v v))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+155) {
		tmp = atan(-1.0);
	} else if (v <= 1e+137) {
		tmp = atan((v / sqrt(fma(H, -19.6, (v * v)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+155)
		tmp = atan(-1.0);
	elseif (v <= 1e+137)
		tmp = atan(Float64(v / sqrt(fma(H, -19.6, Float64(v * v)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+155], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+137], N[ArcTan[N[(v / N[Sqrt[N[(H * -19.6 + N[(v * v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000001e155

   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 -2.00000000000000001e155 < v < 1e137

   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. 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. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   if 1e137 < v

   1. Initial program 15.9%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   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 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -0.000135:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(H, \frac{-9.8}{v}, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -0.000135)
   (atan -1.0)
   (if (<= v 2.3e-62)
     (atan (/ v (sqrt (* -19.6 H))))
     (atan (/ v (fma H (/ -9.8 v) v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -0.000135) {
		tmp = atan(-1.0);
	} else if (v <= 2.3e-62) {
		tmp = atan((v / sqrt((-19.6 * H))));
	} else {
		tmp = atan((v / fma(H, (-9.8 / v), v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -0.000135)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-62)
		tmp = atan(Float64(v / sqrt(Float64(-19.6 * H))));
	else
		tmp = atan(Float64(v / fma(H, Float64(-9.8 / v), v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -0.000135], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.3e-62], N[ArcTan[N[(v / N[Sqrt[N[(-19.6 * H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(H * N[(-9.8 / v), $MachinePrecision] + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.35000000000000002e-4

   1. Initial program 45.9%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 89.6%

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

   if -1.35000000000000002e-4 < v < 2.3e-62

   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

   3. Taylor expanded in v around 0

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

      1. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 2.3e-62 < v

   1. Initial program 56.3%

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

Alternative 5: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -0.000135:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -0.000135)
   (atan -1.0)
   (if (<= v 2.3e-62) (atan (/ v (sqrt (* -19.6 H)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -0.000135) {
		tmp = atan(-1.0);
	} else if (v <= 2.3e-62) {
		tmp = atan((v / sqrt((-19.6 * H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-0.000135d0)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.3d-62) then
        tmp = atan((v / sqrt(((-19.6d0) * h))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -0.000135) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.3e-62) {
		tmp = Math.atan((v / Math.sqrt((-19.6 * H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -0.000135:
		tmp = math.atan(-1.0)
	elif v <= 2.3e-62:
		tmp = math.atan((v / math.sqrt((-19.6 * H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -0.000135)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-62)
		tmp = atan(Float64(v / sqrt(Float64(-19.6 * H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -0.000135)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-62)
		tmp = atan((v / sqrt((-19.6 * H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -0.000135], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.3e-62], N[ArcTan[N[(v / N[Sqrt[N[(-19.6 * H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.35000000000000002e-4

   1. Initial program 45.9%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 89.6%

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

   if -1.35000000000000002e-4 < v < 2.3e-62

   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

   3. Taylor expanded in v around 0

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

      1. lower-*.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 2.3e-62 < v

   1. Initial program 56.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 93.7%

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

Alternative 6: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -0.000135:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -0.000135)
   (atan -1.0)
   (if (<= v 2.3e-62)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -0.000135) {
		tmp = atan(-1.0);
	} else if (v <= 2.3e-62) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-0.000135d0)) then
        tmp = atan((-1.0d0))
    else if (v <= 2.3d-62) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -0.000135) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2.3e-62) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -0.000135:
		tmp = math.atan(-1.0)
	elif v <= 2.3e-62:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -0.000135)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-62)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -0.000135)
		tmp = atan(-1.0);
	elseif (v <= 2.3e-62)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -0.000135], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2.3e-62], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.35000000000000002e-4

   1. Initial program 45.9%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 89.6%

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

   if -1.35000000000000002e-4 < v < 2.3e-62

   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
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in v around 0

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt 90.7

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

   7. Applied rewrites 90.7%

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

   if 2.3e-62 < v

   1. Initial program 56.3%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 93.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -0.000135:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;\tan^{-1} \left(\left(v \cdot v\right) \cdot \frac{-0.10204081632653061}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.9e-136)
   (atan -1.0)
   (if (<= v 1.1e-128)
     (atan (* (* v v) (/ -0.10204081632653061 H)))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-136) {
		tmp = atan(-1.0);
	} else if (v <= 1.1e-128) {
		tmp = atan(((v * v) * (-0.10204081632653061 / H)));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.9d-136)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.1d-128) then
        tmp = atan(((v * v) * ((-0.10204081632653061d0) / h)))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-136) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.1e-128) {
		tmp = Math.atan(((v * v) * (-0.10204081632653061 / H)));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.9e-136:
		tmp = math.atan(-1.0)
	elif v <= 1.1e-128:
		tmp = math.atan(((v * v) * (-0.10204081632653061 / H)))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.9e-136)
		tmp = atan(-1.0);
	elseif (v <= 1.1e-128)
		tmp = atan(Float64(Float64(v * v) * Float64(-0.10204081632653061 / H)));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.9e-136)
		tmp = atan(-1.0);
	elseif (v <= 1.1e-128)
		tmp = atan(((v * v) * (-0.10204081632653061 / H)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.9e-136], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.1e-128], N[ArcTan[N[(N[(v * v), $MachinePrecision] * N[(-0.10204081632653061 / H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.9000000000000001e-136

   1. Initial program 57.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 76.2%

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

   if -1.9000000000000001e-136 < v < 1.10000000000000005e-128

   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

   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. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 31.5

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

   8. Applied rewrites 31.5%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(v \cdot v\right) \cdot -0.10204081632653061}{H}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if 1.10000000000000005e-128 < v

   1. Initial program 59.4%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 89.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;\tan^{-1} \left(\left(v \cdot v\right) \cdot \frac{-0.10204081632653061}{H}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(v \cdot \frac{-0.10204081632653061}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -1.9e-136)
   (atan -1.0)
   (if (<= v 1.1e-128)
     (atan (* v (* v (/ -0.10204081632653061 H))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-136) {
		tmp = atan(-1.0);
	} else if (v <= 1.1e-128) {
		tmp = atan((v * (v * (-0.10204081632653061 / H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-1.9d-136)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.1d-128) then
        tmp = atan((v * (v * ((-0.10204081632653061d0) / h))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -1.9e-136) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.1e-128) {
		tmp = Math.atan((v * (v * (-0.10204081632653061 / H))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -1.9e-136:
		tmp = math.atan(-1.0)
	elif v <= 1.1e-128:
		tmp = math.atan((v * (v * (-0.10204081632653061 / H))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -1.9e-136)
		tmp = atan(-1.0);
	elseif (v <= 1.1e-128)
		tmp = atan(Float64(v * Float64(v * Float64(-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -1.9e-136)
		tmp = atan(-1.0);
	elseif (v <= 1.1e-128)
		tmp = atan((v * (v * (-0.10204081632653061 / H))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -1.9e-136], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.1e-128], N[ArcTan[N[(v * N[(v * N[(-0.10204081632653061 / H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.9000000000000001e-136

   1. Initial program 57.8%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 76.2%

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

   if -1.9000000000000001e-136 < v < 1.10000000000000005e-128

   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

   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. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in v around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 31.5

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

   8. Applied rewrites 31.5%

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\left(v \cdot v\right) \cdot -0.10204081632653061}{H}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if 1.10000000000000005e-128 < v

   1. Initial program 59.4%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 89.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.9 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.1 \cdot 10^{-128}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \left(v \cdot \frac{-0.10204081632653061}{H}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.7 \cdot 10^{-306}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 1.7e-306) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 1.7e-306) {
		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 <= 1.7d-306) 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 <= 1.7e-306) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 1.7e-306:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 1.7e-306)
		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 <= 1.7e-306)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 1.7e-306], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.6999999999999999e-306

   1. Initial program 68.3%

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

   3. Taylor expanded in v around -inf

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

   5. Applied rewrites 58.1%

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

   if 1.6999999999999999e-306 < v

   1. Initial program 67.1%

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

   3. Taylor expanded in v around inf

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

   5. Applied rewrites 73.5%

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

Alternative 10: 34.3% 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 67.7%

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

3. Taylor expanded in v around -inf

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

5. Applied rewrites 28.2%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))