Optimal throwing angle

Percentage Accurate: 65.8% → 99.5%

Time: 8.8s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 10^{+116}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{1}{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan -1.0)
   (if (<= v 1e+116)
     (atan (* v (sqrt (/ 1.0 (- (* v v) (* 19.6 H))))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan(-1.0);
	} else if (v <= 1e+116) {
		tmp = atan((v * sqrt((1.0 / ((v * v) - (19.6 * H))))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 1d+116) then
        tmp = atan((v * sqrt((1.0d0 / ((v * v) - (19.6d0 * h))))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1e+116) {
		tmp = Math.atan((v * Math.sqrt((1.0 / ((v * v) - (19.6 * H))))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+154:
		tmp = math.atan(-1.0)
	elif v <= 1e+116:
		tmp = math.atan((v * math.sqrt((1.0 / ((v * v) - (19.6 * H))))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+116)
		tmp = atan(Float64(v * sqrt(Float64(1.0 / Float64(Float64(v * v) - Float64(19.6 * H))))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 1e+116)
		tmp = atan((v * sqrt((1.0 / ((v * v) - (19.6 * H))))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1e+116], N[ArcTan[N[(v * N[Sqrt[N[(1.0 / N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   1. Initial program 3.1%

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

      1. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -2.00000000000000007e154 < v < 1.00000000000000002e116

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around 0 99.8%

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - 19.6 \cdot H}}\right)} \]
   6. Step-by-step derivation

      1. pow2 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 1.00000000000000002e116 < v

   1. Initial program 27.8%

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

      1. metadata-eval 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - 19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2e+154)
   (atan -1.0)
   (if (<= v 2e+117) (atan (/ v (sqrt (- (* v v) (* 19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = atan(-1.0);
	} else if (v <= 2e+117) {
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2d+154)) then
        tmp = atan((-1.0d0))
    else if (v <= 2d+117) then
        tmp = atan((v / sqrt(((v * v) - (19.6d0 * h)))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2e+154) {
		tmp = Math.atan(-1.0);
	} else if (v <= 2e+117) {
		tmp = Math.atan((v / Math.sqrt(((v * v) - (19.6 * H)))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2e+154:
		tmp = math.atan(-1.0)
	elif v <= 2e+117:
		tmp = math.atan((v / math.sqrt(((v * v) - (19.6 * H)))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+117)
		tmp = atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2e+154)
		tmp = atan(-1.0);
	elseif (v <= 2e+117)
		tmp = atan((v / sqrt(((v * v) - (19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 2e+117], N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.00000000000000007e154

   1. Initial program 3.1%

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

      1. metadata-eval 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in v around -inf 100.0%

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

   if -2.00000000000000007e154 < v < 2.0000000000000001e117

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   if 2.0000000000000001e117 < v

   1. Initial program 27.8%

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

      1. metadata-eval 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in v around inf 100.0%

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

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4.5 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4.5e-41)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (if (<= v 7.6e-90)
     (atan (* v (sqrt (/ -0.05102040816326531 H))))
     (atan (/ v (+ v (* -9.8 (/ H v))))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4.5e-41) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 7.6e-90) {
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-4.5d-41)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else if (v <= 7.6d-90) then
        tmp = atan((v * sqrt(((-0.05102040816326531d0) / h))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -4.5e-41) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 7.6e-90) {
		tmp = Math.atan((v * Math.sqrt((-0.05102040816326531 / H))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -4.5e-41:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	elif v <= 7.6e-90:
		tmp = math.atan((v * math.sqrt((-0.05102040816326531 / H))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4.5e-41)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))));
	elseif (v <= 7.6e-90)
		tmp = atan(Float64(v * sqrt(Float64(-0.05102040816326531 / H))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -4.5e-41)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	elseif (v <= 7.6e-90)
		tmp = atan((v * sqrt((-0.05102040816326531 / H))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4.5e-41], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 7.6e-90], N[ArcTan[N[(v * N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.5e-41

   1. Initial program 57.4%

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

      1. metadata-eval 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in v around -inf 94.9%

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

      1. pow2 57.5%

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

   7. Applied egg-rr 94.9%

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

   if -4.5e-41 < v < 7.6e-90

   1. Initial program 99.4%

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

      1. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in v around 0 95.3%

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

   if 7.6e-90 < v

   1. Initial program 59.0%

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

      1. sqr-neg 59.0%

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

      2. sqr-neg 59.0%

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

      3. fma-neg 59.0%

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

      4. *-commutative 59.0%

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

      5. distribute-rgt-neg-in 59.0%

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

      6. metadata-eval 59.0%

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

      7. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in H around 0 88.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4.5 \cdot 10^{-41}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 7.6 \cdot 10^{-90}:\\ \;\;\;\;\tan^{-1} \left(v \cdot \sqrt{\frac{-0.05102040816326531}{H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7.6 \cdot 10^{-125}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 1.32 \cdot 10^{-96}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -7.6e-125)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (if (<= v 1.32e-96)
     (atan (* -0.10204081632653061 (/ v (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -7.6e-125) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 1.32e-96) {
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-7.6d-125)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else if (v <= 1.32d-96) then
        tmp = atan(((-0.10204081632653061d0) * (v / (h / v))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -7.6e-125) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else if (v <= 1.32e-96) {
		tmp = Math.atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -7.6e-125:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	elif v <= 1.32e-96:
		tmp = math.atan((-0.10204081632653061 * (v / (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -7.6e-125)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))));
	elseif (v <= 1.32e-96)
		tmp = atan(Float64(-0.10204081632653061 * Float64(v / Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -7.6e-125)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	elseif (v <= 1.32e-96)
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -7.6e-125], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 1.32e-96], N[ArcTan[N[(-0.10204081632653061 * N[(v / N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -7.6000000000000002e-125

   1. Initial program 61.4%

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

      1. metadata-eval 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in v around -inf 89.0%

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

      1. pow2 61.4%

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

   7. Applied egg-rr 89.0%

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

   if -7.6000000000000002e-125 < v < 1.32e-96

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. sqr-neg 99.4%

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

      3. fma-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in H around 0 29.9%

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

   6. Taylor expanded in v around 0 29.9%

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

      1. *-un-lft-identity 29.9%

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

      2. times-frac 29.9%

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

      3. metadata-eval 29.9%

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

   8. Applied egg-rr 29.9%

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

   if 1.32e-96 < v

   1. Initial program 59.0%

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

      1. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in v around inf 88.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7.6 \cdot 10^{-125}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{elif}\;v \leq 1.32 \cdot 10^{-96}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.8 \cdot 10^{-175}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 1.32 \cdot 10^{-96}:\\ \;\;\;\;\tan^{-1} \left(-0.10204081632653061 \cdot \frac{v}{\frac{H}{v}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.8e-175)
   (atan -1.0)
   (if (<= v 1.32e-96)
     (atan (* -0.10204081632653061 (/ v (/ H v))))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.8e-175) {
		tmp = atan(-1.0);
	} else if (v <= 1.32e-96) {
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-6.8d-175)) then
        tmp = atan((-1.0d0))
    else if (v <= 1.32d-96) then
        tmp = atan(((-0.10204081632653061d0) * (v / (h / v))))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -6.8e-175) {
		tmp = Math.atan(-1.0);
	} else if (v <= 1.32e-96) {
		tmp = Math.atan((-0.10204081632653061 * (v / (H / v))));
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -6.8e-175:
		tmp = math.atan(-1.0)
	elif v <= 1.32e-96:
		tmp = math.atan((-0.10204081632653061 * (v / (H / v))))
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.8e-175)
		tmp = atan(-1.0);
	elseif (v <= 1.32e-96)
		tmp = atan(Float64(-0.10204081632653061 * Float64(v / Float64(H / v))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -6.8e-175)
		tmp = atan(-1.0);
	elseif (v <= 1.32e-96)
		tmp = atan((-0.10204081632653061 * (v / (H / v))));
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.8e-175], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 1.32e-96], N[ArcTan[N[(-0.10204081632653061 * N[(v / N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.8e-175

   1. Initial program 63.0%

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

      1. metadata-eval 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in v around -inf 84.8%

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

   if -6.8e-175 < v < 1.32e-96

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. sqr-neg 99.4%

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

      3. fma-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in H around 0 32.4%

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

   6. Taylor expanded in v around 0 32.4%

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

      1. *-un-lft-identity 32.4%

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

      2. times-frac 32.4%

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

      3. metadata-eval 32.4%

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

   8. Applied egg-rr 32.4%

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

   if 1.32e-96 < v

   1. Initial program 59.0%

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

      1. metadata-eval 59.0%

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

   3. Simplified 59.0%

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

   5. Taylor expanded in v around inf 88.4%

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

Alternative 6: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -7.2 \cdot 10^{-125}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -7.2e-125)
   (atan (+ -1.0 (* -9.8 (/ H (* v v)))))
   (atan (/ v (+ v (* -9.8 (/ H v)))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -7.2e-125) {
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-7.2d-125)) then
        tmp = atan(((-1.0d0) + ((-9.8d0) * (h / (v * v)))))
    else
        tmp = atan((v / (v + ((-9.8d0) * (h / v)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -7.2e-125) {
		tmp = Math.atan((-1.0 + (-9.8 * (H / (v * v)))));
	} else {
		tmp = Math.atan((v / (v + (-9.8 * (H / v)))));
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -7.2e-125:
		tmp = math.atan((-1.0 + (-9.8 * (H / (v * v)))))
	else:
		tmp = math.atan((v / (v + (-9.8 * (H / v)))))
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -7.2e-125)
		tmp = atan(Float64(-1.0 + Float64(-9.8 * Float64(H / Float64(v * v)))));
	else
		tmp = atan(Float64(v / Float64(v + Float64(-9.8 * Float64(H / v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -7.2e-125)
		tmp = atan((-1.0 + (-9.8 * (H / (v * v)))));
	else
		tmp = atan((v / (v + (-9.8 * (H / v)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -7.2e-125], N[ArcTan[N[(-1.0 + N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(v + N[(-9.8 * N[(H / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -7.2000000000000004e-125

   1. Initial program 61.4%

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

      1. metadata-eval 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in v around -inf 89.0%

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

      1. pow2 61.4%

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

   7. Applied egg-rr 89.0%

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

   if -7.2000000000000004e-125 < v

   1. Initial program 74.8%

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

      1. sqr-neg 74.8%

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

      2. sqr-neg 74.8%

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

      3. fma-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-rgt-neg-in 74.8%

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

      6. metadata-eval 74.8%

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

      7. metadata-eval 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in H around 0 65.8%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -7.2 \cdot 10^{-125}:\\ \;\;\;\;\tan^{-1} \left(-1 + -9.8 \cdot \frac{H}{v \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{v + -9.8 \cdot \frac{H}{v}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 3.4 \cdot 10^{-302}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (if (<= v 3.4e-302) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= 3.4e-302) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= 3.4d-302) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= 3.4e-302) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= 3.4e-302:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= 3.4e-302)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= 3.4e-302)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, 3.4e-302], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.4e-302

   1. Initial program 68.5%

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

      1. metadata-eval 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in v around -inf 72.6%

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

   if 3.4e-302 < v

   1. Initial program 69.8%

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

      1. metadata-eval 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in v around inf 65.9%

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

Alternative 8: 34.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} -1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(v, h)
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((-1.0d0))
end function

Java

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

Python

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

Julia

function code(v, H)
	return atan(-1.0)
end

MATLAB

function tmp = code(v, H)
	tmp = atan(-1.0);
end

Wolfram

code[v_, H_] := N[ArcTan[-1.0], $MachinePrecision]

Derivation

1. Initial program 69.2%

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

   1. metadata-eval 69.2%

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

3. Simplified 69.2%

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

5. Taylor expanded in v around -inf 38.6%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))