Optimal throwing angle

Percentage Accurate: 66.4% → 99.7%

Time: 3.1s

Alternatives: 5

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    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: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    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: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-1 \cdot \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, 1\right) \cdot v\right)}\right)\\ \mathbf{elif}\;v \leq 3 \cdot 10^{+119}:\\ \;\;\;\;\tan^{-1} \left({\left({\left(\mathsf{fma}\left({v}^{1}, {v}^{1}, -19.6 \cdot H\right)\right)}^{-1}\right)}^{0.5} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, -48.02, -9.8 \cdot {v}^{-1}\right), H, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+142)
   (atan (/ v (* -1.0 (* (fma -9.8 (/ H (* v v)) 1.0) v))))
   (if (<= v 3e+119)
     (atan
      (* (pow (pow (fma (pow v 1.0) (pow v 1.0) (* -19.6 H)) -1.0) 0.5) v))
     (atan
      (/
       v
       (fma (fma (/ H (* (* v v) v)) -48.02 (* -9.8 (pow v -1.0))) H v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+142) {
		tmp = atan((v / (-1.0 * (fma(-9.8, (H / (v * v)), 1.0) * v))));
	} else if (v <= 3e+119) {
		tmp = atan((pow(pow(fma(pow(v, 1.0), pow(v, 1.0), (-19.6 * H)), -1.0), 0.5) * v));
	} else {
		tmp = atan((v / fma(fma((H / ((v * v) * v)), -48.02, (-9.8 * pow(v, -1.0))), H, v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+142)
		tmp = atan(Float64(v / Float64(-1.0 * Float64(fma(-9.8, Float64(H / Float64(v * v)), 1.0) * v))));
	elseif (v <= 3e+119)
		tmp = atan(Float64(((fma((v ^ 1.0), (v ^ 1.0), Float64(-19.6 * H)) ^ -1.0) ^ 0.5) * v));
	else
		tmp = atan(Float64(v / fma(fma(Float64(H / Float64(Float64(v * v) * v)), -48.02, Float64(-9.8 * (v ^ -1.0))), H, v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+142], N[ArcTan[N[(v / N[(-1.0 * N[(N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 3e+119], N[ArcTan[N[(N[Power[N[Power[N[(N[Power[v, 1.0], $MachinePrecision] * N[Power[v, 1.0], $MachinePrecision] + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(N[(N[(H / N[(N[(v * v), $MachinePrecision] * v), $MachinePrecision]), $MachinePrecision] * -48.02 + N[(-9.8 * N[Power[v, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * H + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.0000000000000001e142

   1. Initial program 7.7%

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

   3. Taylor expanded in v around -inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -5.0000000000000001e142 < v < 3.00000000000000001e119

   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. Taylor expanded in v around 0

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

   5. Applied rewrites 99.8%

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

   if 3.00000000000000001e119 < v

   1. Initial program 21.3%

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

   3. Taylor expanded in H around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + H \cdot \left(\frac{-2401}{50} \cdot \frac{H}{{v}^{3}} - \frac{49}{5} \cdot \frac{1}{v}\right)}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{H \cdot \left(\frac{-2401}{50} \cdot \frac{H}{{v}^{3}} - \frac{49}{5} \cdot \frac{1}{v}\right) + \color{blue}{v}}\right) \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\left(\frac{-2401}{50} \cdot \frac{H}{{v}^{3}} - \frac{49}{5} \cdot \frac{1}{v}\right) \cdot H + v}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-2401}{50} \cdot \frac{H}{{v}^{3}} - \frac{49}{5} \cdot \frac{1}{v}, \color{blue}{H}, v\right)}\right) \]

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

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{-2401}{50} \cdot \frac{H}{{v}^{3}} + \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}, H, v\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{H}{{v}^{3}} \cdot \frac{-2401}{50} + \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}, H, v\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{{v}^{3}}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{{v}^{3}}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      8. unpow3 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      9. pow2 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{{v}^{2} \cdot v}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{{v}^{2} \cdot v}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \left(\mathsf{neg}\left(\frac{49}{5}\right)\right) \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      13. metadata-eval N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \frac{-49}{5} \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \frac{-49}{5} \cdot \frac{1}{v}\right), H, v\right)}\right) \]

      15. inv-pow N/A

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, \frac{-2401}{50}, \frac{-49}{5} \cdot {v}^{-1}\right), H, v\right)}\right) \]

      16. lower-pow.f64 100.0

          \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, -48.02, -9.8 \cdot {v}^{-1}\right), H, v\right)}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{H}{\left(v \cdot v\right) \cdot v}, -48.02, -9.8 \cdot {v}^{-1}\right), H, v\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+142}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{-1 \cdot \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, 1\right) \cdot v\right)}\right)\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} \left({\left({\left(\mathsf{fma}\left({v}^{1}, {v}^{1}, -19.6 \cdot H\right)\right)}^{-1}\right)}^{0.5} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -5e+142)
   (atan (/ v (* -1.0 (* (fma -9.8 (/ H (* v v)) 1.0) v))))
   (if (<= v 1.4e+154)
     (atan
      (* (pow (pow (fma (pow v 1.0) (pow v 1.0) (* -19.6 H)) -1.0) 0.5) v))
     (atan
      (*
       (pow
        (/ (- (* (/ (* v v) H) -0.002603082049146189) 0.05102040816326531) H)
        0.5)
       v)))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -5e+142) {
		tmp = atan((v / (-1.0 * (fma(-9.8, (H / (v * v)), 1.0) * v))));
	} else if (v <= 1.4e+154) {
		tmp = atan((pow(pow(fma(pow(v, 1.0), pow(v, 1.0), (-19.6 * H)), -1.0), 0.5) * v));
	} else {
		tmp = atan((pow((((((v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H), 0.5) * v));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -5e+142)
		tmp = atan(Float64(v / Float64(-1.0 * Float64(fma(-9.8, Float64(H / Float64(v * v)), 1.0) * v))));
	elseif (v <= 1.4e+154)
		tmp = atan(Float64(((fma((v ^ 1.0), (v ^ 1.0), Float64(-19.6 * H)) ^ -1.0) ^ 0.5) * v));
	else
		tmp = atan(Float64((Float64(Float64(Float64(Float64(Float64(v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H) ^ 0.5) * v));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -5e+142], N[ArcTan[N[(v / N[(-1.0 * N[(N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.4e+154], N[ArcTan[N[(N[Power[N[Power[N[(N[Power[v, 1.0], $MachinePrecision] * N[Power[v, 1.0], $MachinePrecision] + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Power[N[(N[(N[(N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision] * -0.002603082049146189), $MachinePrecision] - 0.05102040816326531), $MachinePrecision] / H), $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.0000000000000001e142

   1. Initial program 7.7%

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

   3. Taylor expanded in v around -inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if -5.0000000000000001e142 < v < 1.4e154

   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. Taylor expanded in v around 0

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

   5. Applied rewrites 99.8%

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

   if 1.4e154 < v

   1. Initial program 3.1%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in H around inf

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      6. pow2 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      7. lift-*.f64 18.8

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]

   8. Applied rewrites 18.8%

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left({\left({\left(\mathsf{fma}\left({v}^{1}, {v}^{1}, -19.6 \cdot H\right)\right)}^{-1}\right)}^{0.5} \cdot v\right)\\ t_1 := \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-250}:\\ \;\;\;\;\tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0
         (atan
          (*
           (pow (pow (fma (pow v 1.0) (pow v 1.0) (* -19.6 H)) -1.0) 0.5)
           v)))
        (t_1 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H)))))))
   (if (<= t_1 -4e-115)
     t_0
     (if (<= t_1 5e-250)
       (atan
        (*
         (pow
          (/ (- (* (/ (* v v) H) -0.002603082049146189) 0.05102040816326531) H)
          0.5)
         v))
       t_0))))

C

double code(double v, double H) {
	double t_0 = atan((pow(pow(fma(pow(v, 1.0), pow(v, 1.0), (-19.6 * H)), -1.0), 0.5) * v));
	double t_1 = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_1 <= -4e-115) {
		tmp = t_0;
	} else if (t_1 <= 5e-250) {
		tmp = atan((pow((((((v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H), 0.5) * v));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = atan(Float64(((fma((v ^ 1.0), (v ^ 1.0), Float64(-19.6 * H)) ^ -1.0) ^ 0.5) * v))
	t_1 = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_1 <= -4e-115)
		tmp = t_0;
	elseif (t_1 <= 5e-250)
		tmp = atan(Float64((Float64(Float64(Float64(Float64(Float64(v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H) ^ 0.5) * v));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[ArcTan[N[(N[Power[N[Power[N[(N[Power[v, 1.0], $MachinePrecision] * N[Power[v, 1.0], $MachinePrecision] + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e-115], t$95$0, If[LessEqual[t$95$1, 5e-250], N[ArcTan[N[(N[Power[N[(N[(N[(N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision] * -0.002603082049146189), $MachinePrecision] - 0.05102040816326531), $MachinePrecision] / H), $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))) < -4.0000000000000002e-115 or 5.00000000000000027e-250 < (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))))

   1. Initial program 99.8%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 99.8%

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

   if -4.0000000000000002e-115 < (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))) < 5.00000000000000027e-250

   1. Initial program 36.6%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in H around inf

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      6. pow2 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      7. lift-*.f64 47.5

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]

   8. Applied rewrites 47.5%

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25}\\ t_1 := \tan^{-1} \left(\left(t\_0 \cdot t\_0\right) \cdot v\right)\\ t_2 := \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (pow (pow (fma -19.6 H (* v v)) -1.0) 0.25))
        (t_1 (atan (* (* t_0 t_0) v)))
        (t_2 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H)))))))
   (if (<= t_2 -1e-41)
     t_1
     (if (<= t_2 5e-26)
       (atan
        (*
         (pow
          (/ (- (* (/ (* v v) H) -0.002603082049146189) 0.05102040816326531) H)
          0.5)
         v))
       t_1))))

C

double code(double v, double H) {
	double t_0 = pow(pow(fma(-19.6, H, (v * v)), -1.0), 0.25);
	double t_1 = atan(((t_0 * t_0) * v));
	double t_2 = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_2 <= -1e-41) {
		tmp = t_1;
	} else if (t_2 <= 5e-26) {
		tmp = atan((pow((((((v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H), 0.5) * v));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(v, H)
	t_0 = (fma(-19.6, H, Float64(v * v)) ^ -1.0) ^ 0.25
	t_1 = atan(Float64(Float64(t_0 * t_0) * v))
	t_2 = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_2 <= -1e-41)
		tmp = t_1;
	elseif (t_2 <= 5e-26)
		tmp = atan(Float64((Float64(Float64(Float64(Float64(Float64(v * v) / H) * -0.002603082049146189) - 0.05102040816326531) / H) ^ 0.5) * v));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[Power[N[Power[N[(-19.6 * H + N[(v * v), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1e-41], t$95$1, If[LessEqual[t$95$2, 5e-26], N[ArcTan[N[(N[Power[N[(N[(N[(N[(N[(v * v), $MachinePrecision] / H), $MachinePrecision] * -0.002603082049146189), $MachinePrecision] - 0.05102040816326531), $MachinePrecision] / H), $MachinePrecision], 0.5], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))) < -1.00000000000000001e-41 or 5.00000000000000019e-26 < (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H)))))

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\tan^{-1} \left({\left({\left(\mathsf{fma}\left({v}^{1}, {v}^{1}, -19.6 \cdot H\right)\right)}^{-1}\right)}^{0.5} \cdot v\right)} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. sqr-pow N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 99.7%

      \[\leadsto \tan^{-1} \left(\left({\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25}\right) \cdot v\right) \]

   if -1.00000000000000001e-41 < (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))) < 5.00000000000000019e-26

   1. Initial program 53.9%

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

   3. Taylor expanded in v around 0

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in H around inf

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      2. lower--.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{-25}{9604} \cdot \frac{{v}^{2}}{H} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      3. *-commutative N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{{v}^{2}}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      6. pow2 N/A

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot \frac{-25}{9604} - \frac{5}{98}}{H}\right)}^{\frac{1}{2}} \cdot v\right) \]

      7. lift-*.f64 61.9

         \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]

   8. Applied rewrites 61.9%

      \[\leadsto \tan^{-1} \left({\left(\frac{\frac{v \cdot v}{H} \cdot -0.002603082049146189 - 0.05102040816326531}{H}\right)}^{0.5} \cdot v\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25}\\ \tan^{-1} \left(\left(t\_0 \cdot t\_0\right) \cdot v\right) \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0 (pow (pow (fma -19.6 H (* v v)) -1.0) 0.25)))
   (atan (* (* t_0 t_0) v))))

C

double code(double v, double H) {
	double t_0 = pow(pow(fma(-19.6, H, (v * v)), -1.0), 0.25);
	return atan(((t_0 * t_0) * v));
}

Julia

function code(v, H)
	t_0 = (fma(-19.6, H, Float64(v * v)) ^ -1.0) ^ 0.25
	return atan(Float64(Float64(t_0 * t_0) * v))
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[Power[N[Power[N[(-19.6 * H + N[(v * v), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], 0.25], $MachinePrecision]}, N[ArcTan[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in v around 0

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

5. Applied rewrites 70.0%

   \[\leadsto \color{blue}{\tan^{-1} \left({\left({\left(\mathsf{fma}\left({v}^{1}, {v}^{1}, -19.6 \cdot H\right)\right)}^{-1}\right)}^{0.5} \cdot v\right)} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. sqr-pow N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 69.8%

   \[\leadsto \tan^{-1} \left(\left({\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25} \cdot {\left({\left(\mathsf{fma}\left(-19.6, H, v \cdot v\right)\right)}^{-1}\right)}^{0.25}\right) \cdot v\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))