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

Percentage Accurate: 89.1% → 99.5%

Time: 9.9s

Alternatives: 11

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: 89.1% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-254}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 + \frac{-1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-254)
     t_0
     (if (<= t_0 0.0) (* z (+ -1.0 (/ -1.0 (/ y x)))) 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-254) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 + (-1.0 / (y / x)));
	} 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-254)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) + ((-1.0d0) / (y / x)))
    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-254) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 + (-1.0 / (y / x)));
	} 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-254:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = z * (-1.0 + (-1.0 / (y / x)))
	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-254)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 + Float64(-1.0 / Float64(y / x))));
	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-254)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 + (-1.0 / (y / x)));
	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-254], t$95$0, If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 + N[(-1.0 / N[(y / x), $MachinePrecision]), $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))) < -1.9999999999999998e-254 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 -1.9999999999999998e-254 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 11.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num N/A

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

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

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

      3. /-lowering-/.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-254}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 + \frac{-1}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-254}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-254) 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-254) {
		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-254)) 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-254) {
		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-254:
		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-254)
		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-254)
		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-254], 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))) < -1.9999999999999998e-254 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 -1.9999999999999998e-254 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 11.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 99.9%

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

   5. Simplified 99.9%

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

Alternative 3: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2300:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (fma (/ -1.0 y) x -1.0))))
   (if (<= y -9e-17)
     t_0
     (if (<= y 2300.0)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 4.9e+101) (* z (/ y (- z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * fma((-1.0 / y), x, -1.0);
	double tmp;
	if (y <= -9e-17) {
		tmp = t_0;
	} else if (y <= 2300.0) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 4.9e+101) {
		tmp = z * (y / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * fma(Float64(-1.0 / y), x, -1.0))
	tmp = 0.0
	if (y <= -9e-17)
		tmp = t_0;
	elseif (y <= 2300.0)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 4.9e+101)
		tmp = Float64(z * Float64(y / Float64(z - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(-1.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e-17], t$95$0, If[LessEqual[y, 2300.0], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e+101], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999957e-17 or 4.89999999999999983e101 < y

   1. Initial program 71.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 83.5%

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

   5. Simplified 83.5%

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. +-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. div-inv N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 83.5%

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

   9. Applied egg-rr 83.5%

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

   if -8.99999999999999957e-17 < y < 2300

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 80.0%

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

   if 2300 < y < 4.89999999999999983e101

   1. Initial program 89.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 76.7%

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

   5. Simplified 76.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-17}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \mathbf{elif}\;y \leq 2300:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4500:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+102}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (fma (/ -1.0 y) x -1.0))))
   (if (<= y -4.2e-16)
     t_0
     (if (<= y 4500.0)
       (* x (/ z (- z y)))
       (if (<= y 1.12e+102) (* z (/ y (- z y))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * fma((-1.0 / y), x, -1.0);
	double tmp;
	if (y <= -4.2e-16) {
		tmp = t_0;
	} else if (y <= 4500.0) {
		tmp = x * (z / (z - y));
	} else if (y <= 1.12e+102) {
		tmp = z * (y / (z - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(z * fma(Float64(-1.0 / y), x, -1.0))
	tmp = 0.0
	if (y <= -4.2e-16)
		tmp = t_0;
	elseif (y <= 4500.0)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	elseif (y <= 1.12e+102)
		tmp = Float64(z * Float64(y / Float64(z - y)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(-1.0 / y), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.2e-16], t$95$0, If[LessEqual[y, 4500.0], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+102], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2000000000000002e-16 or 1.11999999999999992e102 < y

   1. Initial program 71.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 83.5%

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

   5. Simplified 83.5%

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. +-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. div-inv N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. *-commutative N/A

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

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

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 83.5%

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

   9. Applied egg-rr 83.5%

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

   if -4.2000000000000002e-16 < y < 4500

   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 68.2%

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

   5. Simplified 68.2%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 79.9%

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

   7. Applied egg-rr 79.9%

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

   if 4500 < y < 1.11999999999999992e102

   1. Initial program 89.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 76.7%

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

   5. Simplified 76.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \mathbf{elif}\;y \leq 4500:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+102}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\frac{-1}{y}, x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 960:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -2.7e-16)
     t_0
     (if (<= y 960.0)
       (* x (/ z (- z y)))
       (if (<= y 1.7e+102) (* z (/ y (- 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 <= -2.7e-16) {
		tmp = t_0;
	} else if (y <= 960.0) {
		tmp = x * (z / (z - y));
	} else if (y <= 1.7e+102) {
		tmp = z * (y / (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 <= (-2.7d-16)) then
        tmp = t_0
    else if (y <= 960.0d0) then
        tmp = x * (z / (z - y))
    else if (y <= 1.7d+102) then
        tmp = z * (y / (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 <= -2.7e-16) {
		tmp = t_0;
	} else if (y <= 960.0) {
		tmp = x * (z / (z - y));
	} else if (y <= 1.7e+102) {
		tmp = z * (y / (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 <= -2.7e-16:
		tmp = t_0
	elif y <= 960.0:
		tmp = x * (z / (z - y))
	elif y <= 1.7e+102:
		tmp = z * (y / (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 <= -2.7e-16)
		tmp = t_0;
	elseif (y <= 960.0)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	elseif (y <= 1.7e+102)
		tmp = Float64(z * Float64(y / 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 <= -2.7e-16)
		tmp = t_0;
	elseif (y <= 960.0)
		tmp = x * (z / (z - y));
	elseif (y <= 1.7e+102)
		tmp = z * (y / (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, -2.7e-16], t$95$0, If[LessEqual[y, 960.0], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+102], N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.69999999999999999e-16 or 1.7e102 < y

   1. Initial program 71.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 83.5%

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

   5. Simplified 83.5%

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

   if -2.69999999999999999e-16 < y < 960

   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 68.2%

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

   5. Simplified 68.2%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 79.9%

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

   7. Applied egg-rr 79.9%

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

   if 960 < y < 1.7e102

   1. Initial program 89.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 76.7%

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

   5. Simplified 76.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-16}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 960:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+102}:\\ \;\;\;\;z \cdot \frac{y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 390000000:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -5.5e-19) t_0 (if (<= y 390000000.0) (* 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 <= -5.5e-19) {
		tmp = t_0;
	} else if (y <= 390000000.0) {
		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 <= (-5.5d-19)) then
        tmp = t_0
    else if (y <= 390000000.0d0) 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 <= -5.5e-19) {
		tmp = t_0;
	} else if (y <= 390000000.0) {
		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 <= -5.5e-19:
		tmp = t_0
	elif y <= 390000000.0:
		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 <= -5.5e-19)
		tmp = t_0;
	elseif (y <= 390000000.0)
		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 <= -5.5e-19)
		tmp = t_0;
	elseif (y <= 390000000.0)
		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, -5.5e-19], t$95$0, If[LessEqual[y, 390000000.0], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999996e-19 or 3.9e8 < y

   1. Initial program 73.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 80.2%

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

   5. Simplified 80.2%

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

   if -5.4999999999999996e-19 < y < 3.9e8

   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 67.3%

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

   5. Simplified 67.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 78.8%

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

   7. Applied egg-rr 78.8%

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

Alternative 7: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.6e-19)
   (- 0.0 z)
   (if (<= y 7.2e+34) (* x (/ z (- z y))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.6e-19) {
		tmp = 0.0 - z;
	} else if (y <= 7.2e+34) {
		tmp = x * (z / (z - 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 <= (-5.6d-19)) then
        tmp = 0.0d0 - z
    else if (y <= 7.2d+34) then
        tmp = x * (z / (z - 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 <= -5.6e-19) {
		tmp = 0.0 - z;
	} else if (y <= 7.2e+34) {
		tmp = x * (z / (z - y));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.6e-19:
		tmp = 0.0 - z
	elif y <= 7.2e+34:
		tmp = x * (z / (z - y))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.6e-19)
		tmp = Float64(0.0 - z);
	elseif (y <= 7.2e+34)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.6e-19)
		tmp = 0.0 - z;
	elseif (y <= 7.2e+34)
		tmp = x * (z / (z - y));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.6e-19], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 7.2e+34], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.60000000000000005e-19 or 7.2000000000000001e34 < y

   1. Initial program 72.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 64.4%

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

   5. Simplified 64.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 64.4%

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

   7. Applied egg-rr 64.4%

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

   if -5.60000000000000005e-19 < y < 7.2000000000000001e34

   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. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. --lowering--.f64 66.4%

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

   5. Simplified 66.4%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 77.5%

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

   7. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{x \cdot \frac{z}{z - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+36}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.3e-7) (- 0.0 z) (if (<= y 2.35e+36) (+ x y) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.3e-7) {
		tmp = 0.0 - z;
	} else if (y <= 2.35e+36) {
		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 <= (-4.3d-7)) then
        tmp = 0.0d0 - z
    else if (y <= 2.35d+36) 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 <= -4.3e-7) {
		tmp = 0.0 - z;
	} else if (y <= 2.35e+36) {
		tmp = x + y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.3e-7:
		tmp = 0.0 - z
	elif y <= 2.35e+36:
		tmp = x + y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.3e-7)
		tmp = Float64(0.0 - z);
	elseif (y <= 2.35e+36)
		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 <= -4.3e-7)
		tmp = 0.0 - z;
	elseif (y <= 2.35e+36)
		tmp = x + y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.3e-7], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 2.35e+36], N[(x + y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.3000000000000001e-7 or 2.34999999999999994e36 < y

   1. Initial program 71.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 64.6%

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

   5. Simplified 64.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 64.6%

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

   7. Applied egg-rr 64.6%

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

   if -4.3000000000000001e-7 < y < 2.34999999999999994e36

   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 74.3%

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

   5. Simplified 74.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+36}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-16}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 22500000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.5e-16) (- 0.0 z) (if (<= y 22500000000000.0) x (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.5e-16) {
		tmp = 0.0 - z;
	} else if (y <= 22500000000000.0) {
		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 <= (-8.5d-16)) then
        tmp = 0.0d0 - z
    else if (y <= 22500000000000.0d0) 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 <= -8.5e-16) {
		tmp = 0.0 - z;
	} else if (y <= 22500000000000.0) {
		tmp = x;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.5e-16:
		tmp = 0.0 - z
	elif y <= 22500000000000.0:
		tmp = x
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.5e-16)
		tmp = Float64(0.0 - z);
	elseif (y <= 22500000000000.0)
		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 <= -8.5e-16)
		tmp = 0.0 - z;
	elseif (y <= 22500000000000.0)
		tmp = x;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.5e-16], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 22500000000000.0], x, N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000001e-16 or 2.25e13 < y

   1. Initial program 73.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 63.4%

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

   5. Simplified 63.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 63.4%

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

   7. Applied egg-rr 63.4%

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

   if -8.5000000000000001e-16 < y < 2.25e13

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 61.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-16}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 22500000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-197}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-67}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e-197) x (if (<= x 1.22e-67) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e-197) {
		tmp = x;
	} else if (x <= 1.22e-67) {
		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 <= (-8.2d-197)) then
        tmp = x
    else if (x <= 1.22d-67) 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 <= -8.2e-197) {
		tmp = x;
	} else if (x <= 1.22e-67) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e-197:
		tmp = x
	elif x <= 1.22e-67:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e-197)
		tmp = x;
	elseif (x <= 1.22e-67)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e-197)
		tmp = x;
	elseif (x <= 1.22e-67)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e-197], x, If[LessEqual[x, 1.22e-67], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2e-197 or 1.22e-67 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 46.1%

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

   if -8.2e-197 < x < 1.22e-67

   1. Initial program 81.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. /-lowering-/.f64 42.9%

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

   5. Simplified 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 29.4%

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

Alternative 11: 35.9% accurate, 29.0× 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 87.2%

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

3. Taylor expanded in y around 0

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

5. Simplified 38.2%

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

Developer Target 1: 94.1% accurate, 0.7× 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 2024193 
(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))))