Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.4% → 99.1%

Time: 7.0s

Alternatives: 7

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-232) t_0 (if (<= t_0 0.0) (* z (- -1.0 (/ x y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-2d-232)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -2e-232:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -2e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -2e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-232], t$95$0, If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2.00000000000000005e-232 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -2.00000000000000005e-232 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 26.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

         \[\leadsto -1 \cdot z - \color{blue}{\left(\frac{x \cdot z}{y} - \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{\mathsf{neg}\left({z}^{2}\right)}{\color{blue}{y}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{-1 \cdot {z}^{2}}{y}\right) \]

      7. div-sub N/A

         \[\leadsto -1 \cdot z - \frac{x \cdot z - -1 \cdot {z}^{2}}{\color{blue}{y}} \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot z\right), \color{blue}{\left(\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z - -1 \cdot {z}^{2}\right), \color{blue}{y}\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot {z}^{2}\right), y\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + 1 \cdot {z}^{2}\right), y\right)\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + {z}^{2}\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left({z}^{2} + x \cdot z\right), y\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot z + x \cdot z\right), y\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot \left(z + x\right)\right), y\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(z + x\right)\right), y\right)\right) \]

      20. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, x\right)\right), y\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0 - z\right) - \frac{z \cdot \left(z + x\right)}{y}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + \frac{x}{y}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \left(1 + \frac{x}{y}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y}}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 - \color{blue}{\frac{x}{y}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      8. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{z \cdot \left(-1 - \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -5.8e+31)
     t_0
     (if (<= y -5.7e-95)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 1e+122) (+ x y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.8e+31) {
		tmp = t_0;
	} else if (y <= -5.7e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-5.8d+31)) then
        tmp = t_0
    else if (y <= (-5.7d-95)) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 1d+122) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -5.8e+31) {
		tmp = t_0;
	} else if (y <= -5.7e-95) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -5.8e+31:
		tmp = t_0
	elif y <= -5.7e-95:
		tmp = x / (1.0 - (y / z))
	elif y <= 1e+122:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -5.8e+31)
		tmp = t_0;
	elseif (y <= -5.7e-95)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1e+122)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -5.8e+31)
		tmp = t_0;
	elseif (y <= -5.7e-95)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 1e+122)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+31], t$95$0, If[LessEqual[y, -5.7e-95], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+122], N[(x + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8000000000000001e31 or 1.00000000000000001e122 < y

   1. Initial program 76.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

         \[\leadsto -1 \cdot z - \color{blue}{\left(\frac{x \cdot z}{y} - \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{\mathsf{neg}\left({z}^{2}\right)}{\color{blue}{y}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{-1 \cdot {z}^{2}}{y}\right) \]

      7. div-sub N/A

         \[\leadsto -1 \cdot z - \frac{x \cdot z - -1 \cdot {z}^{2}}{\color{blue}{y}} \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot z\right), \color{blue}{\left(\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z - -1 \cdot {z}^{2}\right), \color{blue}{y}\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot {z}^{2}\right), y\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + 1 \cdot {z}^{2}\right), y\right)\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + {z}^{2}\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left({z}^{2} + x \cdot z\right), y\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot z + x \cdot z\right), y\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot \left(z + x\right)\right), y\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(z + x\right)\right), y\right)\right) \]

      20. +-lowering-+.f64 71.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, x\right)\right), y\right)\right) \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\left(0 - z\right) - \frac{z \cdot \left(z + x\right)}{y}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + \frac{x}{y}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \left(1 + \frac{x}{y}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y}}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 - \color{blue}{\frac{x}{y}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      8. /-lowering-/.f64 83.6%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 83.6%

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

   if -5.8000000000000001e31 < y < -5.7e-95

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 74.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 74.1%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]

   if -5.7e-95 < y < 1.00000000000000001e122

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-lowering-+.f64 79.6%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 79.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+29}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.2e+29)
   (- 0.0 z)
   (if (<= y 2.35e-39) x (if (<= y 1e+122) y (- 0.0 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+29) {
		tmp = 0.0 - z;
	} else if (y <= 2.35e-39) {
		tmp = x;
	} else if (y <= 1e+122) {
		tmp = y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.2d+29)) then
        tmp = 0.0d0 - z
    else if (y <= 2.35d-39) then
        tmp = x
    else if (y <= 1d+122) then
        tmp = y
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.2e+29) {
		tmp = 0.0 - z;
	} else if (y <= 2.35e-39) {
		tmp = x;
	} else if (y <= 1e+122) {
		tmp = y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.2e+29:
		tmp = 0.0 - z
	elif y <= 2.35e-39:
		tmp = x
	elif y <= 1e+122:
		tmp = y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.2e+29)
		tmp = Float64(0.0 - z);
	elseif (y <= 2.35e-39)
		tmp = x;
	elseif (y <= 1e+122)
		tmp = y;
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.2e+29)
		tmp = 0.0 - z;
	elseif (y <= 2.35e-39)
		tmp = x;
	elseif (y <= 1e+122)
		tmp = y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.2e+29], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 2.35e-39], x, If[LessEqual[y, 1e+122], y, N[(0.0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e29 or 1.00000000000000001e122 < y

   1. Initial program 76.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 64.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 64.7%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{-z} \]

   if -1.2e29 < y < 2.3500000000000001e-39

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 63.3%

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

   if 2.3500000000000001e-39 < y < 1.00000000000000001e122

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 72.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 50.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+29}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -2.1e+31) t_0 (if (<= y 1e+122) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2.1e+31) {
		tmp = t_0;
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-2.1d+31)) then
        tmp = t_0
    else if (y <= 1d+122) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2.1e+31) {
		tmp = t_0;
	} else if (y <= 1e+122) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -2.1e+31:
		tmp = t_0
	elif y <= 1e+122:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -2.1e+31)
		tmp = t_0;
	elseif (y <= 1e+122)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -2.1e+31)
		tmp = t_0;
	elseif (y <= 1e+122)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e+31], t$95$0, If[LessEqual[y, 1e+122], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999979e31 or 1.00000000000000001e122 < y

   1. Initial program 76.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

         \[\leadsto -1 \cdot z - \color{blue}{\left(\frac{x \cdot z}{y} - \left(\mathsf{neg}\left(\frac{{z}^{2}}{y}\right)\right)\right)} \]

      5. distribute-frac-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{\mathsf{neg}\left({z}^{2}\right)}{\color{blue}{y}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{-1 \cdot {z}^{2}}{y}\right) \]

      7. div-sub N/A

         \[\leadsto -1 \cdot z - \frac{x \cdot z - -1 \cdot {z}^{2}}{\color{blue}{y}} \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot z\right), \color{blue}{\left(\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \left(\frac{\color{blue}{x \cdot z - -1 \cdot {z}^{2}}}{y}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z - -1 \cdot {z}^{2}\right), \color{blue}{y}\right)\right) \]

      13. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot {z}^{2}\right), y\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + 1 \cdot {z}^{2}\right), y\right)\right) \]

      15. *-lft-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(x \cdot z + {z}^{2}\right), y\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left({z}^{2} + x \cdot z\right), y\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot z + x \cdot z\right), y\right)\right) \]

      18. distribute-rgt-out N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\left(z \cdot \left(z + x\right)\right), y\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(z + x\right)\right), y\right)\right) \]

      20. +-lowering-+.f64 71.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{+.f64}\left(z, x\right)\right), y\right)\right) \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\left(0 - z\right) - \frac{z \cdot \left(z + x\right)}{y}} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + \frac{x}{y}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \left(1 + \frac{x}{y}\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\left(1 + \frac{x}{y}\right)\right)\right)}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{y}}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(-1 - \color{blue}{\frac{x}{y}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]

      8. /-lowering-/.f64 83.6%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Simplified 83.6%

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

   if -2.09999999999999979e31 < y < 1.00000000000000001e122

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-lowering-+.f64 75.6%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 75.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+31}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 10^{+122}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+31}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+123}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+31) (- 0.0 z) (if (<= y 1.16e+123) (+ x y) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+31) {
		tmp = 0.0 - z;
	} else if (y <= 1.16e+123) {
		tmp = x + y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.2d+31)) then
        tmp = 0.0d0 - z
    else if (y <= 1.16d+123) then
        tmp = x + y
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+31) {
		tmp = 0.0 - z;
	} else if (y <= 1.16e+123) {
		tmp = x + y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+31:
		tmp = 0.0 - z
	elif y <= 1.16e+123:
		tmp = x + y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+31)
		tmp = Float64(0.0 - z);
	elseif (y <= 1.16e+123)
		tmp = Float64(x + y);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+31)
		tmp = 0.0 - z;
	elseif (y <= 1.16e+123)
		tmp = x + y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+31], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 1.16e+123], N[(x + y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.19999999999999958e31 or 1.16e123 < y

   1. Initial program 76.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 64.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 64.7%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 64.7%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{-z} \]

   if -4.19999999999999958e31 < y < 1.16e123

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. +-lowering-+.f64 75.6%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 75.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+31}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+123}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-55}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-55) x (if (<= x 2.7e-150) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-55) {
		tmp = x;
	} else if (x <= 2.7e-150) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.1d-55)) then
        tmp = x
    else if (x <= 2.7d-150) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-55) {
		tmp = x;
	} else if (x <= 2.7e-150) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e-55:
		tmp = x
	elif x <= 2.7e-150:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-55)
		tmp = x;
	elseif (x <= 2.7e-150)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e-55)
		tmp = x;
	elseif (x <= 2.7e-150)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e-55], x, If[LessEqual[x, 2.7e-150], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1000000000000002e-55 or 2.7000000000000001e-150 < x

   1. Initial program 93.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 53.4%

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

   if -2.1000000000000002e-55 < x < 2.7000000000000001e-150

   1. Initial program 89.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 78.1%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{y}{1 - \frac{y}{z}}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y} \]
   7. Step-by-step derivation

   8. Simplified 44.3%

      \[\leadsto \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.1% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 91.8%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 38.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

  (/ (+ x y) (- 1.0 (/ y z))))