Optimal throwing angle

Percentage Accurate: 66.3% → 99.6%

Time: 2.8s

Alternatives: 6

Speedup: 2.6×

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: 66.3% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \begin{array}{l} \mathbf{if}\;v\_m \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\tan^{-1} \left(\frac{v\_m}{\sqrt{\mathsf{fma}\left(v\_m, v\_m, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H)
 :precision binary64
 (*
  v_s
  (if (<= v_m 5e+122)
    (atan (/ v_m (sqrt (fma v_m v_m (* -19.6 H)))))
    (atan 1.0))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	double tmp;
	if (v_m <= 5e+122) {
		tmp = atan((v_m / sqrt(fma(v_m, v_m, (-19.6 * H)))));
	} else {
		tmp = atan(1.0);
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	tmp = 0.0
	if (v_m <= 5e+122)
		tmp = atan(Float64(v_m / sqrt(fma(v_m, v_m, Float64(-19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := N[(v$95$s * If[LessEqual[v$95$m, 5e+122], N[ArcTan[N[(v$95$m / N[Sqrt[N[(v$95$m * v$95$m + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if v < 4.99999999999999989e122

   1. Initial program 99.6%

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

      1. 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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

   if 4.99999999999999989e122 < v

   1. Initial program 18.7%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 99.5%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{v\_m}{\mathsf{fma}\left(\frac{H}{v\_m}, -9.8, v\_m\right)}\right)\\ t_1 := \tan^{-1} \left(\frac{v\_m}{\sqrt{v\_m \cdot v\_m - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H)
 :precision binary64
 (let* ((t_0 (atan (/ v_m (fma (/ H v_m) -9.8 v_m))))
        (t_1 (atan (/ v_m (sqrt (- (* v_m v_m) (* (* 2.0 9.8) H)))))))
   (*
    v_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 4e-5)
        (atan (* (sqrt (/ -0.05102040816326531 H)) v_m))
        t_0)))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	double t_0 = atan((v_m / fma((H / v_m), -9.8, v_m)));
	double t_1 = atan((v_m / sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 4e-5) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v_m));
	} else {
		tmp = t_0;
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	t_0 = atan(Float64(v_m / fma(Float64(H / v_m), -9.8, v_m)))
	t_1 = atan(Float64(v_m / sqrt(Float64(Float64(v_m * v_m) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 4e-5)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v_m));
	else
		tmp = t_0;
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := Block[{t$95$0 = N[ArcTan[N[(v$95$m / N[(N[(H / v$95$m), $MachinePrecision] * -9.8 + v$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(v$95$m / N[Sqrt[N[(N[(v$95$m * v$95$m), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(v$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 4e-5], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v$95$m), $MachinePrecision]], $MachinePrecision], t$95$0]]), $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 4.00000000000000033e-5 < (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 53.3%

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

   2. Taylor expanded in H around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   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))))) < 4.00000000000000033e-5

   1. Initial program 99.5%

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

   2. Taylor expanded in v around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sqrt-div N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in v around 0

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

      1. lower-/.f64 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 3: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{v\_m}{\sqrt{v\_m \cdot v\_m - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\tan^{-1} 1\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(\frac{H}{v\_m \cdot v\_m}, 9.8, 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H)
 :precision binary64
 (let* ((t_0 (atan (/ v_m (sqrt (- (* v_m v_m) (* (* 2.0 9.8) H)))))))
   (*
    v_s
    (if (<= t_0 0.0)
      (atan 1.0)
      (if (<= t_0 0.6)
        (atan (* (sqrt (/ -0.05102040816326531 H)) v_m))
        (atan (fma (/ H (* v_m v_m)) 9.8 1.0)))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	double t_0 = atan((v_m / sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = atan(1.0);
	} else if (t_0 <= 0.6) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v_m));
	} else {
		tmp = atan(fma((H / (v_m * v_m)), 9.8, 1.0));
	}
	return v_s * tmp;
}

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	t_0 = atan(Float64(v_m / sqrt(Float64(Float64(v_m * v_m) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = atan(1.0);
	elseif (t_0 <= 0.6)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v_m));
	else
		tmp = atan(fma(Float64(H / Float64(v_m * v_m)), 9.8, 1.0));
	end
	return Float64(v_s * tmp)
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := Block[{t$95$0 = N[ArcTan[N[(v$95$m / N[Sqrt[N[(N[(v$95$m * v$95$m), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(v$95$s * If[LessEqual[t$95$0, 0.0], N[ArcTan[1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v$95$m), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(H / N[(v$95$m * v$95$m), $MachinePrecision]), $MachinePrecision] * 9.8 + 1.0), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 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

   1. Initial program 12.0%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 91.0%

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

   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))))) < 0.599999999999999978

   1. Initial program 99.5%

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

   2. Taylor expanded in v around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sqrt-div N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in v around 0

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

      1. lower-/.f64 98.7

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

   9. Applied rewrites 98.7%

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

   if 0.599999999999999978 < (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. Taylor expanded in v around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 4: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{v\_m}{\sqrt{v\_m \cdot v\_m - \left(2 \cdot 9.8\right) \cdot H}}\right)\\ v\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\tan^{-1} 1\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H)
 :precision binary64
 (let* ((t_0 (atan (/ v_m (sqrt (- (* v_m v_m) (* (* 2.0 9.8) H)))))))
   (*
    v_s
    (if (<= t_0 0.0)
      (atan 1.0)
      (if (<= t_0 0.6)
        (atan (* (sqrt (/ -0.05102040816326531 H)) v_m))
        (atan 1.0))))))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	double t_0 = atan((v_m / sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = atan(1.0);
	} else if (t_0 <= 0.6) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v_m));
	} else {
		tmp = atan(1.0);
	}
	return v_s * tmp;
}

Fortran

v\_m =     private
v\_s =     private
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_s, v_m, h)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: v_m
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan((v_m / sqrt(((v_m * v_m) - ((2.0d0 * 9.8d0) * h)))))
    if (t_0 <= 0.0d0) then
        tmp = atan(1.0d0)
    else if (t_0 <= 0.6d0) then
        tmp = atan((sqrt(((-0.05102040816326531d0) / h)) * v_m))
    else
        tmp = atan(1.0d0)
    end if
    code = v_s * tmp
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double v_m, double H) {
	double t_0 = Math.atan((v_m / Math.sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.atan(1.0);
	} else if (t_0 <= 0.6) {
		tmp = Math.atan((Math.sqrt((-0.05102040816326531 / H)) * v_m));
	} else {
		tmp = Math.atan(1.0);
	}
	return v_s * tmp;
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, v_m, H):
	t_0 = math.atan((v_m / math.sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.atan(1.0)
	elif t_0 <= 0.6:
		tmp = math.atan((math.sqrt((-0.05102040816326531 / H)) * v_m))
	else:
		tmp = math.atan(1.0)
	return v_s * tmp

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	t_0 = atan(Float64(v_m / sqrt(Float64(Float64(v_m * v_m) - Float64(Float64(2.0 * 9.8) * H)))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = atan(1.0);
	elseif (t_0 <= 0.6)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v_m));
	else
		tmp = atan(1.0);
	end
	return Float64(v_s * tmp)
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp_2 = code(v_s, v_m, H)
	t_0 = atan((v_m / sqrt(((v_m * v_m) - ((2.0 * 9.8) * H)))));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = atan(1.0);
	elseif (t_0 <= 0.6)
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v_m));
	else
		tmp = atan(1.0);
	end
	tmp_2 = v_s * tmp;
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := Block[{t$95$0 = N[ArcTan[N[(v$95$m / N[Sqrt[N[(N[(v$95$m * v$95$m), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(v$95$s * If[LessEqual[t$95$0, 0.0], N[ArcTan[1.0], $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v$95$m), $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 0.599999999999999978 < (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 53.1%

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

   2. Taylor expanded in v around inf

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

   4. Applied rewrites 94.1%

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

   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))))) < 0.599999999999999978

   1. Initial program 99.5%

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

   2. Taylor expanded in v around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sqrt-div N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in v around 0

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

      1. lower-/.f64 98.7

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

   9. Applied rewrites 98.7%

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

Alternative 5: 68.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \tan^{-1} 1 \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H) :precision binary64 (* v_s (atan 1.0)))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	return v_s * atan(1.0);
}

Fortran

v\_m =     private
v\_s =     private
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_s, v_m, h)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: v_m
    real(8), intent (in) :: h
    code = v_s * atan(1.0d0)
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double v_m, double H) {
	return v_s * Math.atan(1.0);
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, v_m, H):
	return v_s * math.atan(1.0)

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	return Float64(v_s * atan(1.0))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, v_m, H)
	tmp = v_s * atan(1.0);
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := N[(v$95$s * N[ArcTan[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in v around inf

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

4. Applied rewrites 68.7%

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

Alternative 6: 1.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} v\_m = \left|v\right| \\ v\_s = \mathsf{copysign}\left(1, v\right) \\ v\_s \cdot \tan^{-1} -1 \end{array} \]

FPCore

v\_m = (fabs.f64 v)
v\_s = (copysign.f64 #s(literal 1 binary64) v)
(FPCore (v_s v_m H) :precision binary64 (* v_s (atan -1.0)))

C

v\_m = fabs(v);
v\_s = copysign(1.0, v);
double code(double v_s, double v_m, double H) {
	return v_s * atan(-1.0);
}

Fortran

v\_m =     private
v\_s =     private
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_s, v_m, h)
use fmin_fmax_functions
    real(8), intent (in) :: v_s
    real(8), intent (in) :: v_m
    real(8), intent (in) :: h
    code = v_s * atan((-1.0d0))
end function

Java

v\_m = Math.abs(v);
v\_s = Math.copySign(1.0, v);
public static double code(double v_s, double v_m, double H) {
	return v_s * Math.atan(-1.0);
}

Python

v\_m = math.fabs(v)
v\_s = math.copysign(1.0, v)
def code(v_s, v_m, H):
	return v_s * math.atan(-1.0)

Julia

v\_m = abs(v)
v\_s = copysign(1.0, v)
function code(v_s, v_m, H)
	return Float64(v_s * atan(-1.0))
end

MATLAB

v\_m = abs(v);
v\_s = sign(v) * abs(1.0);
function tmp = code(v_s, v_m, H)
	tmp = v_s * atan(-1.0);
end

Wolfram

v\_m = N[Abs[v], $MachinePrecision]
v\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[v]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[v$95$s_, v$95$m_, H_] := N[(v$95$s * N[ArcTan[-1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.3%

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

2. Taylor expanded in v around -inf

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

4. Applied rewrites 1.8%

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

Reproduce

herbie shell --seed 2025121 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))