Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.8%

Time: 9.4s

Alternatives: 5

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (- (/ p x))
   (sqrt (+ 0.5 (* 0.5 (/ x (hypot x (* p 2.0))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -(p / x);
	} else {
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p * 2.0))))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -(p / x);
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (x / Math.hypot(x, (p * 2.0))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0:
		tmp = -(p / x)
	else:
		tmp = math.sqrt((0.5 + (0.5 * (x / math.hypot(x, (p * 2.0))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(-Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(x / hypot(x, Float64(p * 2.0))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0)
		tmp = -(p / x);
	else
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], (-N[(p / x), $MachinePrecision]), N[Sqrt[N[(0.5 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 12.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 12.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 12.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 12.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 12.6%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 12.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   6. Taylor expanded in x around -inf 47.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 47.7%

         \[\leadsto \color{blue}{-\frac{p}{x}} \]

   8. Simplified 47.7%

      \[\leadsto \color{blue}{-\frac{p}{x}} \]

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 98.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 99.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]

Alternative 2: 64.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := -\frac{p}{x}\\ \mathbf{if}\;x \leq -14000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (- (/ p x))))
   (if (<= x -14000000.0)
     t_0
     (if (<= x -8e-54)
       (sqrt 0.5)
       (if (<= x -1.05e-81)
         t_0
         (if (<= x 1.45)
           (sqrt 0.5)
           (if (<= x 3.95e+87) 1.0 (if (<= x 1.9e+101) (sqrt 0.5) 1.0))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -(p / x);
	double tmp;
	if (x <= -14000000.0) {
		tmp = t_0;
	} else if (x <= -8e-54) {
		tmp = sqrt(0.5);
	} else if (x <= -1.05e-81) {
		tmp = t_0;
	} else if (x <= 1.45) {
		tmp = sqrt(0.5);
	} else if (x <= 3.95e+87) {
		tmp = 1.0;
	} else if (x <= 1.9e+101) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(p / x)
    if (x <= (-14000000.0d0)) then
        tmp = t_0
    else if (x <= (-8d-54)) then
        tmp = sqrt(0.5d0)
    else if (x <= (-1.05d-81)) then
        tmp = t_0
    else if (x <= 1.45d0) then
        tmp = sqrt(0.5d0)
    else if (x <= 3.95d+87) then
        tmp = 1.0d0
    else if (x <= 1.9d+101) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -(p / x);
	double tmp;
	if (x <= -14000000.0) {
		tmp = t_0;
	} else if (x <= -8e-54) {
		tmp = Math.sqrt(0.5);
	} else if (x <= -1.05e-81) {
		tmp = t_0;
	} else if (x <= 1.45) {
		tmp = Math.sqrt(0.5);
	} else if (x <= 3.95e+87) {
		tmp = 1.0;
	} else if (x <= 1.9e+101) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -(p / x)
	tmp = 0
	if x <= -14000000.0:
		tmp = t_0
	elif x <= -8e-54:
		tmp = math.sqrt(0.5)
	elif x <= -1.05e-81:
		tmp = t_0
	elif x <= 1.45:
		tmp = math.sqrt(0.5)
	elif x <= 3.95e+87:
		tmp = 1.0
	elif x <= 1.9e+101:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(-Float64(p / x))
	tmp = 0.0
	if (x <= -14000000.0)
		tmp = t_0;
	elseif (x <= -8e-54)
		tmp = sqrt(0.5);
	elseif (x <= -1.05e-81)
		tmp = t_0;
	elseif (x <= 1.45)
		tmp = sqrt(0.5);
	elseif (x <= 3.95e+87)
		tmp = 1.0;
	elseif (x <= 1.9e+101)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -(p / x);
	tmp = 0.0;
	if (x <= -14000000.0)
		tmp = t_0;
	elseif (x <= -8e-54)
		tmp = sqrt(0.5);
	elseif (x <= -1.05e-81)
		tmp = t_0;
	elseif (x <= 1.45)
		tmp = sqrt(0.5);
	elseif (x <= 3.95e+87)
		tmp = 1.0;
	elseif (x <= 1.9e+101)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = (-N[(p / x), $MachinePrecision])}, If[LessEqual[x, -14000000.0], t$95$0, If[LessEqual[x, -8e-54], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, -1.05e-81], t$95$0, If[LessEqual[x, 1.45], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, 3.95e+87], 1.0, If[LessEqual[x, 1.9e+101], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e7 or -8.0000000000000002e-54 < x < -1.05e-81

   1. Initial program 36.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 36.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 36.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 36.6%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 36.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   6. Taylor expanded in x around -inf 46.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 46.8%

         \[\leadsto \color{blue}{-\frac{p}{x}} \]

   8. Simplified 46.8%

      \[\leadsto \color{blue}{-\frac{p}{x}} \]

   if -1.4e7 < x < -8.0000000000000002e-54 or -1.05e-81 < x < 1.44999999999999996 or 3.9499999999999998e87 < x < 1.8999999999999999e101

   1. Initial program 86.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

   2. Taylor expanded in x around 0 72.6%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]

   if 1.44999999999999996 < x < 3.9499999999999998e87 or 1.8999999999999999e101 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. distribute-rgt-in 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{1 \cdot 0.5 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}}\right)\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{0.5} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}\right)\right) \]

      4. +-commutative 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}} \cdot 0.5}\right)\right) \]

      5. add-sqr-sqrt 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}} \cdot 0.5}\right)\right) \]

      6. hypot-def 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}} \cdot 0.5}\right)\right) \]

      7. associate-*l* 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)} \cdot 0.5}\right)\right) \]

      8. sqrt-prod 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)} \cdot 0.5}\right)\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)} \cdot 0.5}\right)\right) \]

      10. sqrt-unprod 44.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)} \cdot 0.5}\right)\right) \]

      11. add-sqr-sqrt 99.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)} \cdot 0.5}\right)\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

   4. Taylor expanded in x around inf 78.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14000000:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-81}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 3.95 \cdot 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 64.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := -\frac{p}{x}\\ \mathbf{if}\;x \leq -950000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (- (/ p x))))
   (if (<= x -950000.0)
     t_0
     (if (<= x -1.7e-54)
       (sqrt (+ 0.5 (/ (* x 0.25) p)))
       (if (<= x -7e-81)
         t_0
         (if (<= x 0.5)
           (sqrt 0.5)
           (if (<= x 1e+87) 1.0 (if (<= x 5.4e+101) (sqrt 0.5) 1.0))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -(p / x);
	double tmp;
	if (x <= -950000.0) {
		tmp = t_0;
	} else if (x <= -1.7e-54) {
		tmp = sqrt((0.5 + ((x * 0.25) / p)));
	} else if (x <= -7e-81) {
		tmp = t_0;
	} else if (x <= 0.5) {
		tmp = sqrt(0.5);
	} else if (x <= 1e+87) {
		tmp = 1.0;
	} else if (x <= 5.4e+101) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(p / x)
    if (x <= (-950000.0d0)) then
        tmp = t_0
    else if (x <= (-1.7d-54)) then
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p)))
    else if (x <= (-7d-81)) then
        tmp = t_0
    else if (x <= 0.5d0) then
        tmp = sqrt(0.5d0)
    else if (x <= 1d+87) then
        tmp = 1.0d0
    else if (x <= 5.4d+101) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -(p / x);
	double tmp;
	if (x <= -950000.0) {
		tmp = t_0;
	} else if (x <= -1.7e-54) {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p)));
	} else if (x <= -7e-81) {
		tmp = t_0;
	} else if (x <= 0.5) {
		tmp = Math.sqrt(0.5);
	} else if (x <= 1e+87) {
		tmp = 1.0;
	} else if (x <= 5.4e+101) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -(p / x)
	tmp = 0
	if x <= -950000.0:
		tmp = t_0
	elif x <= -1.7e-54:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p)))
	elif x <= -7e-81:
		tmp = t_0
	elif x <= 0.5:
		tmp = math.sqrt(0.5)
	elif x <= 1e+87:
		tmp = 1.0
	elif x <= 5.4e+101:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(-Float64(p / x))
	tmp = 0.0
	if (x <= -950000.0)
		tmp = t_0;
	elseif (x <= -1.7e-54)
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p)));
	elseif (x <= -7e-81)
		tmp = t_0;
	elseif (x <= 0.5)
		tmp = sqrt(0.5);
	elseif (x <= 1e+87)
		tmp = 1.0;
	elseif (x <= 5.4e+101)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -(p / x);
	tmp = 0.0;
	if (x <= -950000.0)
		tmp = t_0;
	elseif (x <= -1.7e-54)
		tmp = sqrt((0.5 + ((x * 0.25) / p)));
	elseif (x <= -7e-81)
		tmp = t_0;
	elseif (x <= 0.5)
		tmp = sqrt(0.5);
	elseif (x <= 1e+87)
		tmp = 1.0;
	elseif (x <= 5.4e+101)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = (-N[(p / x), $MachinePrecision])}, If[LessEqual[x, -950000.0], t$95$0, If[LessEqual[x, -1.7e-54], N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -7e-81], t$95$0, If[LessEqual[x, 0.5], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, 1e+87], 1.0, If[LessEqual[x, 5.4e+101], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.5e5 or -1.69999999999999994e-54 < x < -6.99999999999999973e-81

   1. Initial program 36.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 36.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 36.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 36.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 36.6%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 36.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   6. Taylor expanded in x around -inf 46.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 46.8%

         \[\leadsto \color{blue}{-\frac{p}{x}} \]

   8. Simplified 46.8%

      \[\leadsto \color{blue}{-\frac{p}{x}} \]

   if -9.5e5 < x < -1.69999999999999994e-54

   1. Initial program 69.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 68.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 68.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 68.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 68.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 69.2%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 69.2%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   6. Taylor expanded in x around 0 65.6%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.6%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.25 \cdot x}{p}}} \]

   8. Simplified 65.6%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{0.25 \cdot x}{p}}} \]

   if -6.99999999999999973e-81 < x < 0.5 or 9.9999999999999996e86 < x < 5.40000000000000012e101

   1. Initial program 92.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]

   2. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]

   if 0.5 < x < 9.9999999999999996e86 or 5.40000000000000012e101 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. distribute-rgt-in 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{1 \cdot 0.5 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}}\right)\right) \]

      3. metadata-eval 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{0.5} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}\right)\right) \]

      4. +-commutative 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}} \cdot 0.5}\right)\right) \]

      5. add-sqr-sqrt 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}} \cdot 0.5}\right)\right) \]

      6. hypot-def 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}} \cdot 0.5}\right)\right) \]

      7. associate-*l* 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)} \cdot 0.5}\right)\right) \]

      8. sqrt-prod 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)} \cdot 0.5}\right)\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)} \cdot 0.5}\right)\right) \]

      10. sqrt-unprod 44.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)} \cdot 0.5}\right)\right) \]

      11. add-sqr-sqrt 99.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)} \cdot 0.5}\right)\right) \]

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

   4. Taylor expanded in x around inf 78.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -950000:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-81}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;x \leq 0.5:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq 10^{+87}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 55.7% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-146}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -8.5e-146) (- (/ p x)) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -8.5e-146) {
		tmp = -(p / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-8.5d-146)) then
        tmp = -(p / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -8.5e-146) {
		tmp = -(p / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -8.5e-146:
		tmp = -(p / x)
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -8.5e-146)
		tmp = Float64(-Float64(p / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -8.5e-146)
		tmp = -(p / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -8.5e-146], (-N[(p / x), $MachinePrecision]), 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999997e-146

   1. Initial program 54.3%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 53.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 53.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 53.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 53.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

      2. expm1-log1p 54.3%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}} \]

      3. *-commutative 54.3%

         \[\leadsto \sqrt{0.5 + \color{blue}{0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} \]

   6. Taylor expanded in x around -inf 26.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   7. Step-by-step derivation

      1. mul-1-neg 26.6%

         \[\leadsto \color{blue}{-\frac{p}{x}} \]

   8. Simplified 26.6%

      \[\leadsto \color{blue}{-\frac{p}{x}} \]

   if -8.4999999999999997e-146 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. distribute-rgt-in 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{1 \cdot 0.5 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}}\right)\right) \]

      3. metadata-eval 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{0.5} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}\right)\right) \]

      4. +-commutative 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}} \cdot 0.5}\right)\right) \]

      5. add-sqr-sqrt 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}} \cdot 0.5}\right)\right) \]

      6. hypot-def 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}} \cdot 0.5}\right)\right) \]

      7. associate-*l* 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)} \cdot 0.5}\right)\right) \]

      8. sqrt-prod 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)} \cdot 0.5}\right)\right) \]

      9. metadata-eval 99.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)} \cdot 0.5}\right)\right) \]

      10. sqrt-unprod 45.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)} \cdot 0.5}\right)\right) \]

      11. add-sqr-sqrt 99.2%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)} \cdot 0.5}\right)\right) \]

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

   4. Taylor expanded in x around inf 58.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-146}:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 36.1% accurate, 215.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ 1 \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 1.0)

C

p = abs(p);
double code(double p, double x) {
	return 1.0;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	return 1.0;
}

Python

p = abs(p)
def code(p, x):
	return 1.0

Julia

p = abs(p)
function code(p, x)
	return 1.0
end

MATLAB

p = abs(p)
function tmp = code(p, x)
	tmp = 1.0;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := 1.0

Derivation

1. Initial program 76.1%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Step-by-step derivation

   1. expm1-log1p-u 75.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

   2. distribute-rgt-in 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{1 \cdot 0.5 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}}\right)\right) \]

   3. metadata-eval 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{0.5} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5}\right)\right) \]

   4. +-commutative 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}} \cdot 0.5}\right)\right) \]

   5. add-sqr-sqrt 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}} \cdot 0.5}\right)\right) \]

   6. hypot-def 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}} \cdot 0.5}\right)\right) \]

   7. associate-*l* 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)} \cdot 0.5}\right)\right) \]

   8. sqrt-prod 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)} \cdot 0.5}\right)\right) \]

   9. metadata-eval 75.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)} \cdot 0.5}\right)\right) \]

   10. sqrt-unprod 34.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)} \cdot 0.5}\right)\right) \]

   11. add-sqr-sqrt 75.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)} \cdot 0.5}\right)\right) \]

3. Applied egg-rr 75.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5}\right)\right)} \]

4. Taylor expanded in x around inf 34.2%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 34.2%

   \[\leadsto 1 \]

Developer target: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023301 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))