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

Percentage Accurate: 88.3% → 99.4%

Time: 11.5s

Alternatives: 8

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: 88.3% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-246}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-246) 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-246) {
		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-246)) 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-246) {
		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-246:
		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-246)
		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-246)
		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-246], 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))) < -1.99999999999999991e-246 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.99999999999999991e-246 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 8.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. --lowering--.f64 N/A

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

      19. /-lowering-/.f64 100.0%

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

   5. 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: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-68}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+171}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -2.05e+61)
     t_0
     (if (<= y -1.25e-68)
       (+ x y)
       (if (<= y 6e+21)
         (* x (/ z (- z y)))
         (if (<= y 8.5e+171) (/ y (- 1.0 (/ y z))) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2.05e+61) {
		tmp = t_0;
	} else if (y <= -1.25e-68) {
		tmp = x + y;
	} else if (y <= 6e+21) {
		tmp = x * (z / (z - y));
	} else if (y <= 8.5e+171) {
		tmp = 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 = z * ((-1.0d0) - (x / y))
    if (y <= (-2.05d+61)) then
        tmp = t_0
    else if (y <= (-1.25d-68)) then
        tmp = x + y
    else if (y <= 6d+21) then
        tmp = x * (z / (z - y))
    else if (y <= 8.5d+171) then
        tmp = 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 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2.05e+61) {
		tmp = t_0;
	} else if (y <= -1.25e-68) {
		tmp = x + y;
	} else if (y <= 6e+21) {
		tmp = x * (z / (z - y));
	} else if (y <= 8.5e+171) {
		tmp = y / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -2.05e+61:
		tmp = t_0
	elif y <= -1.25e-68:
		tmp = x + y
	elif y <= 6e+21:
		tmp = x * (z / (z - y))
	elif y <= 8.5e+171:
		tmp = y / (1.0 - (y / z))
	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 <= -2.05e+61)
		tmp = t_0;
	elseif (y <= -1.25e-68)
		tmp = Float64(x + y);
	elseif (y <= 6e+21)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	elseif (y <= 8.5e+171)
		tmp = Float64(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 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -2.05e+61)
		tmp = t_0;
	elseif (y <= -1.25e-68)
		tmp = x + y;
	elseif (y <= 6e+21)
		tmp = x * (z / (z - y));
	elseif (y <= 8.5e+171)
		tmp = y / (1.0 - (y / z));
	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, -2.05e+61], t$95$0, If[LessEqual[y, -1.25e-68], N[(x + y), $MachinePrecision], If[LessEqual[y, 6e+21], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+171], N[(y / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.04999999999999986e61 or 8.4999999999999995e171 < y

   1. Initial program 63.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. --lowering--.f64 N/A

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

      19. /-lowering-/.f64 88.0%

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

   5. Simplified 88.0%

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

   if -2.04999999999999986e61 < y < -1.24999999999999993e-68

   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 64.8%

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

   5. Simplified 64.8%

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

   if -1.24999999999999993e-68 < y < 6e21

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 87.8%

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

   5. Simplified 87.8%

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

      1. clear-num N/A

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

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

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

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

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 87.7%

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

   7. Applied egg-rr 87.7%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1 \cdot \frac{y}{x} + \frac{z}{x}}{z}\right)}\right) \]
   9. Step-by-step derivation

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 71.3%

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

   10. Simplified 71.3%

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

       1. clear-num N/A

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

       2. sub-div N/A

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

       3. associate-/r/ N/A

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

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

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

       5. /-lowering-/.f64 N/A

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

       6. --lowering--.f64 88.0%

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

   12. Applied egg-rr 88.0%

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

   if 6e21 < y < 8.4999999999999995e171

   1. Initial program 87.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 81.2%

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

   5. Simplified 81.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-68}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+171}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-67}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 580000:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -2.9e+60)
     t_0
     (if (<= y -1.02e-67)
       (+ x y)
       (if (<= y 580000.0) (* x (/ z (- z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -2.9e+60) {
		tmp = t_0;
	} else if (y <= -1.02e-67) {
		tmp = x + y;
	} else if (y <= 580000.0) {
		tmp = x * (z / (z - 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 <= (-2.9d+60)) then
        tmp = t_0
    else if (y <= (-1.02d-67)) then
        tmp = x + y
    else if (y <= 580000.0d0) then
        tmp = x * (z / (z - 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 <= -2.9e+60) {
		tmp = t_0;
	} else if (y <= -1.02e-67) {
		tmp = x + y;
	} else if (y <= 580000.0) {
		tmp = x * (z / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -2.9e+60:
		tmp = t_0
	elif y <= -1.02e-67:
		tmp = x + y
	elif y <= 580000.0:
		tmp = x * (z / (z - 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 <= -2.9e+60)
		tmp = t_0;
	elseif (y <= -1.02e-67)
		tmp = Float64(x + y);
	elseif (y <= 580000.0)
		tmp = Float64(x * Float64(z / Float64(z - 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 <= -2.9e+60)
		tmp = t_0;
	elseif (y <= -1.02e-67)
		tmp = x + y;
	elseif (y <= 580000.0)
		tmp = x * (z / (z - 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, -2.9e+60], t$95$0, If[LessEqual[y, -1.02e-67], N[(x + y), $MachinePrecision], If[LessEqual[y, 580000.0], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e60 or 5.8e5 < y

   1. Initial program 69.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. --lowering--.f64 N/A

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

      19. /-lowering-/.f64 82.3%

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

   5. Simplified 82.3%

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

   if -2.9e60 < y < -1.01999999999999993e-67

   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 64.8%

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

   5. Simplified 64.8%

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

   if -1.01999999999999993e-67 < y < 5.8e5

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 87.8%

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

   5. Simplified 87.8%

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

      1. clear-num N/A

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

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

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

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

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 87.6%

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

   7. Applied egg-rr 87.6%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-1 \cdot \frac{y}{x} + \frac{z}{x}}{z}\right)}\right) \]
   9. Step-by-step derivation

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 71.0%

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

   10. Simplified 71.0%

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

       1. clear-num N/A

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

       2. sub-div N/A

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

       3. associate-/r/ N/A

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

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

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

       5. /-lowering-/.f64 N/A

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

       6. --lowering--.f64 87.9%

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

   12. Applied egg-rr 87.9%

       \[\leadsto \color{blue}{\frac{z}{z - y} \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-67}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 580000:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-50}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -3e+60) t_0 (if (<= y 7.2e-50) (+ 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 <= -3e+60) {
		tmp = t_0;
	} else if (y <= 7.2e-50) {
		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 <= (-3d+60)) then
        tmp = t_0
    else if (y <= 7.2d-50) 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 <= -3e+60) {
		tmp = t_0;
	} else if (y <= 7.2e-50) {
		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 <= -3e+60:
		tmp = t_0
	elif y <= 7.2e-50:
		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 <= -3e+60)
		tmp = t_0;
	elseif (y <= 7.2e-50)
		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 <= -3e+60)
		tmp = t_0;
	elseif (y <= 7.2e-50)
		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, -3e+60], t$95$0, If[LessEqual[y, 7.2e-50], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9999999999999998e60 or 7.19999999999999958e-50 < y

   1. Initial program 71.8%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. associate-*r/ N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. div-sub N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lft-mult-inverse N/A

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

      17. metadata-eval N/A

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

      18. --lowering--.f64 N/A

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

      19. /-lowering-/.f64 81.1%

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

   5. Simplified 81.1%

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

   if -2.9999999999999998e60 < y < 7.19999999999999958e-50

   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 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+60}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-50}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+61}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 8500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e+61) (- 0.0 z) (if (<= y 8500.0) (+ x y) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.05e+61) {
		tmp = 0.0 - z;
	} else if (y <= 8500.0) {
		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 <= (-2.05d+61)) then
        tmp = 0.0d0 - z
    else if (y <= 8500.0d0) 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 <= -2.05e+61) {
		tmp = 0.0 - z;
	} else if (y <= 8500.0) {
		tmp = x + y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.05e+61:
		tmp = 0.0 - z
	elif y <= 8500.0:
		tmp = x + y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.05e+61)
		tmp = Float64(0.0 - z);
	elseif (y <= 8500.0)
		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 <= -2.05e+61)
		tmp = 0.0 - z;
	elseif (y <= 8500.0)
		tmp = x + y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.05e+61], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 8500.0], N[(x + y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.04999999999999986e61 or 8500 < y

   1. Initial program 69.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 69.5%

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

   5. Simplified 69.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 69.5%

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

   7. Applied egg-rr 69.5%

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

   if -2.04999999999999986e61 < y < 8500

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 74.4%

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

   5. Simplified 74.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+61}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 8500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-26}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 46:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-26) (- 0.0 z) (if (<= y 46.0) x (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-26) {
		tmp = 0.0 - z;
	} else if (y <= 46.0) {
		tmp = x;
	} 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 <= (-1.6d-26)) then
        tmp = 0.0d0 - z
    else if (y <= 46.0d0) then
        tmp = x
    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 <= -1.6e-26) {
		tmp = 0.0 - z;
	} else if (y <= 46.0) {
		tmp = x;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e-26:
		tmp = 0.0 - z
	elif y <= 46.0:
		tmp = x
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e-26)
		tmp = Float64(0.0 - z);
	elseif (y <= 46.0)
		tmp = x;
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e-26)
		tmp = 0.0 - z;
	elseif (y <= 46.0)
		tmp = x;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e-26], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 46.0], x, N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e-26 or 46 < y

   1. Initial program 73.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 -1.6000000000000001e-26 < y < 46

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 67.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-26}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 46:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1200000000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1200000000.0) y (if (<= y 4e+46) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1200000000.0) {
		tmp = y;
	} else if (y <= 4e+46) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 <= (-1200000000.0d0)) then
        tmp = y
    else if (y <= 4d+46) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1200000000.0) {
		tmp = y;
	} else if (y <= 4e+46) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1200000000.0:
		tmp = y
	elif y <= 4e+46:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1200000000.0)
		tmp = y;
	elseif (y <= 4e+46)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1200000000.0)
		tmp = y;
	elseif (y <= 4e+46)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1200000000.0], y, If[LessEqual[y, 4e+46], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e9 or 4e46 < y

   1. Initial program 71.8%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. /-lowering-/.f64 22.2%

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

   5. Simplified 22.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 19.8%

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

   if -1.2e9 < y < 4e46

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 64.5%

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

Alternative 8: 34.7% 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 86.3%

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

3. Taylor expanded in y around 0

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

5. Simplified 36.1%

   \[\leadsto \color{blue}{x} \]
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 2024192 
(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))))