Optimal throwing angle

Percentage Accurate: 68.0% → 99.8%

Time: 3.7s

Alternatives: 5

Speedup: 0.7×

Specification

Math

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

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: 68.0% accurate, 1.0× speedup

Math

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

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: 99.8% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, v\right) \cdot \begin{array}{l} \mathbf{if}\;\left|v\right| \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\tan^{-1} \left({\left(\mathsf{fma}\left(H, -19.6, \left|v\right| \cdot \left|v\right|\right)\right)}^{-0.5} \cdot \left|v\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (*
  (copysign 1.0 v)
  (if (<= (fabs v) 2e+150)
    (atan (* (pow (fma H -19.6 (* (fabs v) (fabs v))) -0.5) (fabs v)))
    (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (fabs(v) <= 2e+150) {
		tmp = atan((pow(fma(H, -19.6, (fabs(v) * fabs(v))), -0.5) * fabs(v)));
	} else {
		tmp = atan(1.0);
	}
	return copysign(1.0, v) * tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (abs(v) <= 2e+150)
		tmp = atan(Float64((fma(H, -19.6, Float64(abs(v) * abs(v))) ^ -0.5) * abs(v)));
	else
		tmp = atan(1.0);
	end
	return Float64(copysign(1.0, v) * tmp)
end

Wolfram

code[v_, H_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[v], $MachinePrecision], 2e+150], N[ArcTan[N[(N[Power[N[(H * -19.6 + N[(N[Abs[v], $MachinePrecision] * N[Abs[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Abs[v], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if v < 1.99999999999999996e150

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-abs-rev N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. sqr-neg-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sqr-neg-rev N/A

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

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

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

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

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

      13. *-commutative N/A

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

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

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

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

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

      16. sqr-neg-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 68.0%

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

   7. Applied rewrites 68.0%

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

   if 1.99999999999999996e150 < v

   1. Initial program 68.0%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 34.6%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, H_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[v], $MachinePrecision], 2e+150], N[ArcTan[N[(N[Abs[v], $MachinePrecision] / N[Sqrt[N[(H * -19.6 + N[(N[Abs[v], $MachinePrecision] * N[Abs[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if v < 1.99999999999999996e150

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-abs-rev N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. sqr-neg-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 68.0%

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

   3. Applied rewrites 68.0%

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

   if 1.99999999999999996e150 < v

   1. Initial program 68.0%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 34.6%

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

Alternative 3: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{\left|v\right|}{\sqrt{\left|v\right| \cdot \left|v\right| - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ \mathsf{copysign}\left(1, v\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\tan^{-1} 1\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot \left|v\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (let* ((t_0
         (atan
          (/ (fabs v) (sqrt (- (* (fabs v) (fabs v)) (* (* 2.0 9.8) H)))))))
   (*
    (copysign 1.0 v)
    (if (<= t_0 0.0)
      (atan 1.0)
      (if (<= t_0 0.1)
        (atan (* (sqrt (/ -0.05102040816326531 H)) (fabs v)))
        (atan 1.0))))))

C

double code(double v, double H) {
	double t_0 = atan((fabs(v) / sqrt(((fabs(v) * fabs(v)) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = atan(1.0);
	} else if (t_0 <= 0.1) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * fabs(v)));
	} else {
		tmp = atan(1.0);
	}
	return copysign(1.0, v) * tmp;
}

Java

public static double code(double v, double H) {
	double t_0 = Math.atan((Math.abs(v) / Math.sqrt(((Math.abs(v) * Math.abs(v)) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.atan(1.0);
	} else if (t_0 <= 0.1) {
		tmp = Math.atan((Math.sqrt((-0.05102040816326531 / H)) * Math.abs(v)));
	} else {
		tmp = Math.atan(1.0);
	}
	return Math.copySign(1.0, v) * tmp;
}

Python

def code(v, H):
	t_0 = math.atan((math.fabs(v) / math.sqrt(((math.fabs(v) * math.fabs(v)) - ((2.0 * 9.8) * H)))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.atan(1.0)
	elif t_0 <= 0.1:
		tmp = math.atan((math.sqrt((-0.05102040816326531 / H)) * math.fabs(v)))
	else:
		tmp = math.atan(1.0)
	return math.copysign(1.0, v) * tmp

Julia

function code(v, H)
	t_0 = atan(Float64(abs(v) / sqrt(Float64(Float64(abs(v) * abs(v)) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = atan(1.0);
	elseif (t_0 <= 0.1)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * abs(v)));
	else
		tmp = atan(1.0);
	end
	return Float64(copysign(1.0, v) * tmp)
end

MATLAB

function tmp_2 = code(v, H)
	t_0 = atan((abs(v) / sqrt(((abs(v) * abs(v)) - ((2.0 * 9.8) * H)))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = atan(1.0);
	elseif (t_0 <= 0.1)
		tmp = atan((sqrt((-0.05102040816326531 / H)) * abs(v)));
	else
		tmp = atan(1.0);
	end
	tmp_2 = (sign(v) * abs(1.0)) * tmp;
end

Wolfram

code[v_, H_] := Block[{t$95$0 = N[ArcTan[N[(N[Abs[v], $MachinePrecision] / N[Sqrt[N[(N[(N[Abs[v], $MachinePrecision] * N[Abs[v], $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[ArcTan[1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.1], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * N[Abs[v], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]), $MachinePrecision]]

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))))) < 0.0 or 0.10000000000000001 < (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 68.0%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 34.6%

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

   if 0.0 < (atan.f64 (/.f64 v (sqrt.f64 (-.f64 (*.f64 v v) (*.f64 (*.f64 #s(literal 2 binary64) #s(literal 49/5 binary64)) H))))) < 0.10000000000000001

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-abs-rev N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. distribute-rgt-neg-out N/A

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

      12. sqr-neg-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 68.0%

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

   3. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. lift-*.f64 N/A

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

   5. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 68.0%

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. lift-*.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-fma.f64 68.0%

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in v around 0

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

      1. lower-/.f64 39.8%

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

   10. Applied rewrites 39.8%

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

Alternative 4: 67.1% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, v\right) \cdot \tan^{-1} 1 \]

FPCore

(FPCore (v H) :precision binary64 (* (copysign 1.0 v) (atan 1.0)))

C

double code(double v, double H) {
	return copysign(1.0, v) * atan(1.0);
}

Java

public static double code(double v, double H) {
	return Math.copySign(1.0, v) * Math.atan(1.0);
}

Python

def code(v, H):
	return math.copysign(1.0, v) * math.atan(1.0)

Julia

function code(v, H)
	return Float64(copysign(1.0, v) * atan(1.0))
end

MATLAB

function tmp = code(v, H)
	tmp = (sign(v) * abs(1.0)) * atan(1.0);
end

Wolfram

code[v_, H_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[ArcTan[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

2. Taylor expanded in v around inf

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

4. Applied rewrites 34.6%

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

Alternative 5: 34.3% accurate, 2.6× speedup

Math

\[\tan^{-1} -1 \]

FPCore

(FPCore (v H) :precision binary64 (atan -1.0))

C

double code(double v, double H) {
	return atan(-1.0);
}

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((-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 68.0%

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

2. Taylor expanded in v around -inf

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

4. Applied rewrites 34.3%

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

Reproduce

herbie shell --seed 2025183 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))