Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left|y - x\right|}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ (fabs (- y x)) 2.0)))

C

double code(double x, double y) {
	return x + (fabs((y - x)) / 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (abs((y - x)) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return x + (Math.abs((y - x)) / 2.0);
}

Python

def code(x, y):
	return x + (math.fabs((y - x)) / 2.0)

Julia

function code(x, y)
	return Float64(x + Float64(abs(Float64(y - x)) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (abs((y - x)) / 2.0);
end

Wolfram

code[x_, y_] := N[(x + N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]

2. Final simplification 99.9%

   \[\leadsto x + \frac{\left|y - x\right|}{2} \]

Alternative 2: 82.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + y\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x y))))
   (if (<= x -9e-19)
     t_0
     (if (<= x 5.2e-87) (* (fabs (- y x)) 0.5) (+ x t_0)))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -9e-19) {
		tmp = t_0;
	} else if (x <= 5.2e-87) {
		tmp = fabs((y - x)) * 0.5;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x + y)
    if (x <= (-9d-19)) then
        tmp = t_0
    else if (x <= 5.2d-87) then
        tmp = abs((y - x)) * 0.5d0
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -9e-19) {
		tmp = t_0;
	} else if (x <= 5.2e-87) {
		tmp = Math.abs((y - x)) * 0.5;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x + y)
	tmp = 0
	if x <= -9e-19:
		tmp = t_0
	elif x <= 5.2e-87:
		tmp = math.fabs((y - x)) * 0.5
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x + y))
	tmp = 0.0
	if (x <= -9e-19)
		tmp = t_0;
	elseif (x <= 5.2e-87)
		tmp = Float64(abs(Float64(y - x)) * 0.5);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x + y);
	tmp = 0.0;
	if (x <= -9e-19)
		tmp = t_0;
	elseif (x <= 5.2e-87)
		tmp = abs((y - x)) * 0.5;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-19], t$95$0, If[LessEqual[x, 5.2e-87], N[(N[Abs[N[(y - x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000026e-19

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 28.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 26.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 26.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 26.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 26.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 16.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 16.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 16.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 16.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 16.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 19.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 89.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around 0 89.6%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot x + 0.5 \cdot y\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 89.6%

         \[\leadsto \color{blue}{\left(x + -0.5 \cdot x\right) + 0.5 \cdot y} \]

      2. distribute-rgt1-in 89.6%

         \[\leadsto \color{blue}{\left(-0.5 + 1\right) \cdot x} + 0.5 \cdot y \]

      3. metadata-eval 89.6%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      4. distribute-lft-out 89.6%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   8. Simplified 89.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   if -9.00000000000000026e-19 < x < 5.20000000000000005e-87

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around 0 81.7%

      \[\leadsto \color{blue}{0.5 \cdot \left|y - x\right|} \]

   if 5.20000000000000005e-87 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 92.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 80.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 80.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 80.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 80.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 17.0%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 27.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 30.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 31.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 31.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   6. Step-by-step derivation

      1. fma-udef 31.9%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 0.5 + x} \]

      2. *-commutative 31.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(y - x\right)} + x \]

      3. sub-neg 31.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(-x\right)\right)} + x \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) + x \]

      5. sqrt-unprod 54.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) + x \]

      6. sqr-neg 54.6%

         \[\leadsto 0.5 \cdot \left(y + \sqrt{\color{blue}{x \cdot x}}\right) + x \]

      7. sqrt-unprod 84.2%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + x \]

      8. add-sqr-sqrt 84.4%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{x}\right) + x \]

   7. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right) + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-87}:\\ \;\;\;\;\left|y - x\right| \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 3: 58.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9e-47) (* x 0.5) (if (<= x 4.5e-81) (* y 0.5) (* x 1.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-47) {
		tmp = x * 0.5;
	} else if (x <= 4.5e-81) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.9d-47)) then
        tmp = x * 0.5d0
    else if (x <= 4.5d-81) then
        tmp = y * 0.5d0
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.9e-47) {
		tmp = x * 0.5;
	} else if (x <= 4.5e-81) {
		tmp = y * 0.5;
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.9e-47:
		tmp = x * 0.5
	elif x <= 4.5e-81:
		tmp = y * 0.5
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9e-47)
		tmp = Float64(x * 0.5);
	elseif (x <= 4.5e-81)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.9e-47)
		tmp = x * 0.5;
	elseif (x <= 4.5e-81)
		tmp = y * 0.5;
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.9e-47], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 4.5e-81], N[(y * 0.5), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e-47

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 34.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 29.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 29.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 15.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 15.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 15.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 15.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 15.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 20.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 85.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around 0 67.7%

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

      1. distribute-rgt1-in 67.7%

         \[\leadsto \color{blue}{\left(-0.5 + 1\right) \cdot x} \]

      2. metadata-eval 67.7%

         \[\leadsto \color{blue}{0.5} \cdot x \]

      3. *-commutative 67.7%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   8. Simplified 67.7%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if -2.9e-47 < x < 4.5e-81

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 95.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 49.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 49.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 49.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 49.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 34.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 34.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 35.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 35.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 62.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 65.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 65.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around inf 54.3%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 4.5e-81 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 92.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 80.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 80.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 80.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 80.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 15.4%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 15.4%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 26.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 26.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 28.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 30.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 30.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   6. Step-by-step derivation

      1. fma-udef 30.4%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 0.5 + x} \]

      2. *-commutative 30.4%

         \[\leadsto \color{blue}{0.5 \cdot \left(y - x\right)} + x \]

      3. sub-neg 30.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(-x\right)\right)} + x \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) + x \]

      5. sqrt-unprod 53.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) + x \]

      6. sqr-neg 53.7%

         \[\leadsto 0.5 \cdot \left(y + \sqrt{\color{blue}{x \cdot x}}\right) + x \]

      7. sqrt-unprod 83.9%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + x \]

      8. add-sqr-sqrt 84.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{x}\right) + x \]

   7. Applied egg-rr 84.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right) + x} \]

   8. Taylor expanded in y around 0 69.3%

      \[\leadsto \color{blue}{x + 0.5 \cdot x} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 69.3%

         \[\leadsto \color{blue}{\left(0.5 + 1\right) \cdot x} \]

      2. metadata-eval 69.3%

         \[\leadsto \color{blue}{1.5} \cdot x \]

      3. *-commutative 69.3%

         \[\leadsto \color{blue}{x \cdot 1.5} \]

   10. Simplified 69.3%

       \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-47}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 4: 74.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x + y\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ x y)))) (if (<= x -2e-264) t_0 (+ x t_0))))

C

double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -2e-264) {
		tmp = t_0;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x + y)
    if (x <= (-2d-264)) then
        tmp = t_0
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 * (x + y);
	double tmp;
	if (x <= -2e-264) {
		tmp = t_0;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 * (x + y)
	tmp = 0
	if x <= -2e-264:
		tmp = t_0
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 * Float64(x + y))
	tmp = 0.0
	if (x <= -2e-264)
		tmp = t_0;
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 * (x + y);
	tmp = 0.0;
	if (x <= -2e-264)
		tmp = t_0;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-264], t$95$0, N[(x + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-264

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 56.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 36.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 36.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 36.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 36.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 22.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 22.8%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 22.8%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 22.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 39.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 81.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around 0 81.9%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot x + 0.5 \cdot y\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 81.9%

         \[\leadsto \color{blue}{\left(x + -0.5 \cdot x\right) + 0.5 \cdot y} \]

      2. distribute-rgt1-in 81.9%

         \[\leadsto \color{blue}{\left(-0.5 + 1\right) \cdot x} + 0.5 \cdot y \]

      3. metadata-eval 81.9%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      4. distribute-lft-out 81.9%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   8. Simplified 81.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   if -2e-264 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 93.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 70.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 70.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 21.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 21.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 29.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 29.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 38.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   6. Step-by-step derivation

      1. fma-udef 38.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 0.5 + x} \]

      2. *-commutative 38.8%

         \[\leadsto \color{blue}{0.5 \cdot \left(y - x\right)} + x \]

      3. sub-neg 38.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(-x\right)\right)} + x \]

      4. add-sqr-sqrt 3.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) + x \]

      5. sqrt-unprod 55.8%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) + x \]

      6. sqr-neg 55.8%

         \[\leadsto 0.5 \cdot \left(y + \sqrt{\color{blue}{x \cdot x}}\right) + x \]

      7. sqrt-unprod 73.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + x \]

      8. add-sqr-sqrt 77.4%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{x}\right) + x \]

   7. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \left(x + y\right)\\ \end{array} \]

Alternative 5: 68.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7e-55) (* 0.5 (+ x y)) (* x 1.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7e-55) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.7d-55) then
        tmp = 0.5d0 * (x + y)
    else
        tmp = x * 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.7e-55) {
		tmp = 0.5 * (x + y);
	} else {
		tmp = x * 1.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.7e-55:
		tmp = 0.5 * (x + y)
	else:
		tmp = x * 1.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7e-55)
		tmp = Float64(0.5 * Float64(x + y));
	else
		tmp = Float64(x * 1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.7e-55)
		tmp = 0.5 * (x + y);
	else
		tmp = x * 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7e-55], N[(0.5 * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x * 1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.70000000000000004e-55

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 40.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 40.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 40.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 40.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 26.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 26.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 27.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 27.1%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 27.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 43.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 73.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around 0 73.3%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot x + 0.5 \cdot y\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 73.3%

         \[\leadsto \color{blue}{\left(x + -0.5 \cdot x\right) + 0.5 \cdot y} \]

      2. distribute-rgt1-in 73.3%

         \[\leadsto \color{blue}{\left(-0.5 + 1\right) \cdot x} + 0.5 \cdot y \]

      3. metadata-eval 73.3%

         \[\leadsto \color{blue}{0.5} \cdot x + 0.5 \cdot y \]

      4. distribute-lft-out 73.3%

         \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y\right)} \]

   if 2.70000000000000004e-55 < x

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 92.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 83.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 83.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 83.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 83.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 13.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 13.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 25.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 25.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 27.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 28.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 28.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]
   6. Step-by-step derivation

      1. fma-udef 28.8%

         \[\leadsto \color{blue}{\left(y - x\right) \cdot 0.5 + x} \]

      2. *-commutative 28.8%

         \[\leadsto \color{blue}{0.5 \cdot \left(y - x\right)} + x \]

      3. sub-neg 28.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(-x\right)\right)} + x \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) + x \]

      5. sqrt-unprod 51.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) + x \]

      6. sqr-neg 51.1%

         \[\leadsto 0.5 \cdot \left(y + \sqrt{\color{blue}{x \cdot x}}\right) + x \]

      7. sqrt-unprod 83.7%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) + x \]

      8. add-sqr-sqrt 83.9%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{x}\right) + x \]

   7. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right) + x} \]

   8. Taylor expanded in y around 0 71.1%

      \[\leadsto \color{blue}{x + 0.5 \cdot x} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 71.1%

         \[\leadsto \color{blue}{\left(0.5 + 1\right) \cdot x} \]

      2. metadata-eval 71.1%

         \[\leadsto \color{blue}{1.5} \cdot x \]

      3. *-commutative 71.1%

         \[\leadsto \color{blue}{x \cdot 1.5} \]

   10. Simplified 71.1%

       \[\leadsto \color{blue}{x \cdot 1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-55}:\\ \;\;\;\;0.5 \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1.5\\ \end{array} \]

Alternative 6: 32.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.3e-239) x (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-239) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-239) then
        tmp = x
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-239) {
		tmp = x;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-239:
		tmp = x
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-239)
		tmp = x;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-239)
		tmp = x;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-239], x, N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.30000000000000001e-239

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]

   2. Taylor expanded in x around inf 13.3%

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

   if 1.30000000000000001e-239 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 80.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 62.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 62.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 62.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 62.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 42.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 42.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 46.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 46.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 60.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 78.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around inf 57.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 46.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.4e-27) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-27) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.4d-27) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.4e-27) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.4e-27:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.4e-27)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.4e-27)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.4e-27], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4e-27

   1. Initial program 99.9%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 74.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 45.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 1.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 1.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 6.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 6.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 6.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 21.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 43.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 43.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around 0 37.2%

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

      1. distribute-rgt1-in 37.2%

         \[\leadsto \color{blue}{\left(-0.5 + 1\right) \cdot x} \]

      2. metadata-eval 37.2%

         \[\leadsto \color{blue}{0.5} \cdot x \]

      3. *-commutative 37.2%

         \[\leadsto \color{blue}{x \cdot 0.5} \]

   8. Simplified 37.2%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

   if 1.4e-27 < y

   1. Initial program 100.0%

      \[x + \frac{\left|y - x\right|}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 82.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)\right)} \]

      2. expm1-udef 77.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x + \frac{\left|y - x\right|}{2}\right)} - 1} \]

      3. +-commutative 77.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left|y - x\right|}{2} + x}\right)} - 1 \]

      4. div-inv 77.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|y - x\right| \cdot \frac{1}{2}} + x\right)} - 1 \]

      5. fma-def 77.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\left|y - x\right|, \frac{1}{2}, x\right)}\right)} - 1 \]

      6. add-sqr-sqrt 68.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\left|\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}\right|, \frac{1}{2}, x\right)\right)} - 1 \]

      7. fabs-sqr 68.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{\sqrt{y - x} \cdot \sqrt{y - x}}, \frac{1}{2}, x\right)\right)} - 1 \]

      8. add-sqr-sqrt 70.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\color{blue}{y - x}, \frac{1}{2}, x\right)\right)} - 1 \]

      9. metadata-eval 70.5%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, \color{blue}{0.5}, x\right)\right)} - 1 \]

   3. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 74.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(y - x, 0.5, x\right)\right)\right)} \]

      2. expm1-log1p 91.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   5. Simplified 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, 0.5, x\right)} \]

   6. Taylor expanded in y around inf 77.5%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8: 11.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

   \[x + \frac{\left|y - x\right|}{2} \]

2. Taylor expanded in x around inf 11.8%

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

3. Final simplification 11.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderSpotLegend from Chart-1.5.3"
  :precision binary64
  (+ x (/ (fabs (- y x)) 2.0)))