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

Percentage Accurate: 87.9% → 99.9%

Time: 7.9s

Alternatives: 8

Speedup: 0.6×

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: 87.9% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z - y}, z, z \cdot \frac{y}{z - y}\right)\\ \mathbf{if}\;y \leq -50000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x (- z y)) z (* z (/ y (- z y))))))
   (if (<= y -50000000.0)
     t_0
     (if (<= y 5e-13) (/ (+ y x) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / (z - y)), z, (z * (y / (z - y))));
	double tmp;
	if (y <= -50000000.0) {
		tmp = t_0;
	} else if (y <= 5e-13) {
		tmp = (y + x) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / Float64(z - y)), z, Float64(z * Float64(y / Float64(z - y))))
	tmp = 0.0
	if (y <= -50000000.0)
		tmp = t_0;
	elseif (y <= 5e-13)
		tmp = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * z + N[(z * N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -50000000.0], t$95$0, If[LessEqual[y, 5e-13], N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e7 or 4.9999999999999999e-13 < y

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -5e7 < y < 4.9999999999999999e-13

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -50000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z - y}, z, z \cdot \frac{y}{z - y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z - y}, z, z \cdot \frac{y}{z - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-301}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-301)
     t_0
     (if (<= t_0 0.0) (- (fma z (/ x y) z)) (* (/ z (- z y)) (+ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-301) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -1e-301)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-301], t$95$0, If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -1.00000000000000007e-301

   1. Initial program 99.9%

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

   if -1.00000000000000007e-301 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.2%

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

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

      23. lower-neg.f64 N/A

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

   5. Applied rewrites 99.9%

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

   if 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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-301}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))) (t_1 (* (/ z (- z y)) (+ y x))))
   (if (<= t_0 -1e-301) t_1 (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -1e-301) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(z / Float64(z - y)) * Float64(y + x))
	tmp = 0.0
	if (t_0 <= -1e-301)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-301], t$95$1, If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -1.00000000000000007e-301 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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 87.9

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

   if -1.00000000000000007e-301 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.2%

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

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

      23. lower-neg.f64 N/A

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+171}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-224}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.6e+171)
   (- z)
   (if (<= y 1.95e-224) (+ y x) (if (<= y 6e+50) (* x (/ z (- z y))) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+171) {
		tmp = -z;
	} else if (y <= 1.95e-224) {
		tmp = y + x;
	} else if (y <= 6e+50) {
		tmp = x * (z / (z - y));
	} else {
		tmp = -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 <= (-6.6d+171)) then
        tmp = -z
    else if (y <= 1.95d-224) then
        tmp = y + x
    else if (y <= 6d+50) then
        tmp = x * (z / (z - y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+171) {
		tmp = -z;
	} else if (y <= 1.95e-224) {
		tmp = y + x;
	} else if (y <= 6e+50) {
		tmp = x * (z / (z - y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.6e+171:
		tmp = -z
	elif y <= 1.95e-224:
		tmp = y + x
	elif y <= 6e+50:
		tmp = x * (z / (z - y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.6e+171)
		tmp = Float64(-z);
	elseif (y <= 1.95e-224)
		tmp = Float64(y + x);
	elseif (y <= 6e+50)
		tmp = Float64(x * Float64(z / Float64(z - y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.6e+171)
		tmp = -z;
	elseif (y <= 1.95e-224)
		tmp = y + x;
	elseif (y <= 6e+50)
		tmp = x * (z / (z - y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.6e+171], (-z), If[LessEqual[y, 1.95e-224], N[(y + x), $MachinePrecision], If[LessEqual[y, 6e+50], N[(x * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.59999999999999982e171 or 5.9999999999999996e50 < y

   1. Initial program 64.6%

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

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

      2. lower-neg.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -6.59999999999999982e171 < y < 1.9499999999999999e-224

   1. Initial program 98.3%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 1.9499999999999999e-224 < y < 5.9999999999999996e50

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-sub N/A

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

      3. associate-/r/ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-inverses N/A

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

      8. div-sub N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 56.8%

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

   10. Applied rewrites 67.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+171}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-224}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 1 + \frac{x}{z}, x\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (+ 1.0 (/ x z)) x)))
   (if (<= z -3.5e+27) t_0 (if (<= z 2.5e-25) (- (fma z (/ x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (1.0 + (x / z)), x);
	double tmp;
	if (z <= -3.5e+27) {
		tmp = t_0;
	} else if (z <= 2.5e-25) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(1.0 + Float64(x / z)), x)
	tmp = 0.0
	if (z <= -3.5e+27)
		tmp = t_0;
	elseif (z <= 2.5e-25)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(x / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -3.5e+27], t$95$0, If[LessEqual[z, 2.5e-25], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000002e27 or 2.49999999999999981e-25 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -3.5000000000000002e27 < z < 2.49999999999999981e-25

   1. Initial program 76.6%

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

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

      23. lower-neg.f64 N/A

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

   5. Applied rewrites 72.8%

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

Alternative 6: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.5e+27)
   (+ y x)
   (if (<= z 2.5e-25) (- (fma z (/ x y) z)) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.5e+27) {
		tmp = y + x;
	} else if (z <= 2.5e-25) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.5e+27)
		tmp = Float64(y + x);
	elseif (z <= 2.5e-25)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.5e+27], N[(y + x), $MachinePrecision], If[LessEqual[z, 2.5e-25], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000002e27 or 2.49999999999999981e-25 < z

   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

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

      2. lower-+.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -3.5000000000000002e27 < z < 2.49999999999999981e-25

   1. Initial program 76.6%

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

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

      2. associate-/l* N/A

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

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

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

      4. distribute-neg-frac N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. unsub-neg N/A

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

      9. div-sub N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

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

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. *-commutative N/A

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

      20. unsub-neg N/A

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

      21. mul-1-neg N/A

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

      22. distribute-neg-out N/A

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

      23. lower-neg.f64 N/A

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

   5. Applied rewrites 72.8%

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

Alternative 7: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+171}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.96 \cdot 10^{+80}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.6e+171) (- z) (if (<= y 1.96e+80) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+171) {
		tmp = -z;
	} else if (y <= 1.96e+80) {
		tmp = y + x;
	} else {
		tmp = -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 <= (-6.6d+171)) then
        tmp = -z
    else if (y <= 1.96d+80) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+171) {
		tmp = -z;
	} else if (y <= 1.96e+80) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.6e+171:
		tmp = -z
	elif y <= 1.96e+80:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.6e+171)
		tmp = Float64(-z);
	elseif (y <= 1.96e+80)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.6e+171)
		tmp = -z;
	elseif (y <= 1.96e+80)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.6e+171], (-z), If[LessEqual[y, 1.96e+80], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -6.59999999999999982e171 or 1.9599999999999999e80 < y

   1. Initial program 64.6%

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

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

      2. lower-neg.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -6.59999999999999982e171 < y < 1.9599999999999999e80

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 68.6

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

   5. Applied rewrites 68.6%

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

Alternative 8: 34.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 88.5%

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

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

   2. lower-neg.f64 31.1

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

5. Applied rewrites 31.1%

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

Developer Target 1: 93.7% 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 2024219 
(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))))