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

Percentage Accurate: 88.4% → 99.3%

Time: 4.4s

Alternatives: 9

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: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{y}, z \cdot x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-250)
     t_0
     (if (<= t_0 0.0) (fma (/ -1.0 y) (* z x) (- z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-250) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = fma((-1.0 / y), (z * x), -z);
	} else {
		tmp = t_0;
	}
	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 <= -5e-250)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = fma(Float64(-1.0 / y), Float64(z * x), Float64(-z));
	else
		tmp = t_0;
	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, -5e-250], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 / y), $MachinePrecision] * N[(z * x), $MachinePrecision] + (-z)), $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))) < -5.00000000000000027e-250 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -5.00000000000000027e-250 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 22.1%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{y}, z \cdot x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{z - y}, z, \frac{x}{1 - \frac{y}{z}}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

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

   2. *-inverses N/A

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

   3. div-sub N/A

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

   4. associate-/r/ N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. associate-*r/ N/A

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

   11. *-rgt-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 97.8

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

5. Applied rewrites 97.8%

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

Alternative 3: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+188}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1200:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{+136}:\\ \;\;\;\;\frac{z}{z - y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -1.15e+188)
     (- z)
     (if (<= y -1200.0)
       (/ y t_0)
       (if (<= y 1.3e-87)
         (/ x t_0)
         (if (<= y 1.72e+136) (* (/ z (- z y)) y) (- (fma (/ x y) z z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -1.15e+188) {
		tmp = -z;
	} else if (y <= -1200.0) {
		tmp = y / t_0;
	} else if (y <= 1.3e-87) {
		tmp = x / t_0;
	} else if (y <= 1.72e+136) {
		tmp = (z / (z - y)) * y;
	} else {
		tmp = -fma((x / y), z, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -1.15e+188)
		tmp = Float64(-z);
	elseif (y <= -1200.0)
		tmp = Float64(y / t_0);
	elseif (y <= 1.3e-87)
		tmp = Float64(x / t_0);
	elseif (y <= 1.72e+136)
		tmp = Float64(Float64(z / Float64(z - y)) * y);
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.15e+188], (-z), If[LessEqual[y, -1200.0], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 1.3e-87], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 1.72e+136], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.15000000000000006e188

   1. Initial program 63.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   if -1.15000000000000006e188 < y < -1200

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -1200 < y < 1.30000000000000001e-87

   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. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 1.30000000000000001e-87 < y < 1.71999999999999989e136

   1. Initial program 95.3%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.4%

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

   if 1.71999999999999989e136 < y

   1. Initial program 74.5%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.2%

      \[\leadsto -\mathsf{fma}\left(\frac{x}{y}, z, z\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{z - y} \cdot y\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+188}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z (- z y)) y)))
   (if (<= y -1.15e+188)
     (- z)
     (if (<= y -1200.0)
       t_0
       (if (<= y 1.3e-87)
         (/ x (- 1.0 (/ y z)))
         (if (<= y 1.72e+136) t_0 (- (fma (/ x y) z z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (z / (z - y)) * y;
	double tmp;
	if (y <= -1.15e+188) {
		tmp = -z;
	} else if (y <= -1200.0) {
		tmp = t_0;
	} else if (y <= 1.3e-87) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.72e+136) {
		tmp = t_0;
	} else {
		tmp = -fma((x / y), z, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / Float64(z - y)) * y)
	tmp = 0.0
	if (y <= -1.15e+188)
		tmp = Float64(-z);
	elseif (y <= -1200.0)
		tmp = t_0;
	elseif (y <= 1.3e-87)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.72e+136)
		tmp = t_0;
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.15e+188], (-z), If[LessEqual[y, -1200.0], t$95$0, If[LessEqual[y, 1.3e-87], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e+136], t$95$0, (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.15000000000000006e188

   1. Initial program 63.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   if -1.15000000000000006e188 < y < -1200 or 1.30000000000000001e-87 < y < 1.71999999999999989e136

   1. Initial program 94.4%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.3%

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

   if -1200 < y < 1.30000000000000001e-87

   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. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 1.71999999999999989e136 < y

   1. Initial program 74.5%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 91.2%

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

Alternative 5: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-29}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e-29)
   (- (fma x (/ z y) z))
   (if (<= y 1.35e+45) (fma x (/ y z) (+ y x)) (- (fma (/ x y) z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e-29) {
		tmp = -fma(x, (z / y), z);
	} else if (y <= 1.35e+45) {
		tmp = fma(x, (y / z), (y + x));
	} else {
		tmp = -fma((x / y), z, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e-29)
		tmp = Float64(-fma(x, Float64(z / y), z));
	elseif (y <= 1.35e+45)
		tmp = fma(x, Float64(y / z), Float64(y + x));
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.7e-29], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, 1.35e+45], N[(x * N[(y / z), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6999999999999998e-29

   1. Initial program 84.0%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.3%

      \[\leadsto -\mathsf{fma}\left(\frac{x}{y}, z, z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

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

   if -4.6999999999999998e-29 < y < 1.34999999999999992e45

   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. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 76.4%

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

   if 1.34999999999999992e45 < y

   1. Initial program 82.1%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.4%

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

Alternative 6: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-29}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+45}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{x}{y}, z, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e-29)
   (- (fma x (/ z y) z))
   (if (<= y 1.35e+45) (+ y x) (- (fma (/ x y) z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e-29) {
		tmp = -fma(x, (z / y), z);
	} else if (y <= 1.35e+45) {
		tmp = y + x;
	} else {
		tmp = -fma((x / y), z, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e-29)
		tmp = Float64(-fma(x, Float64(z / y), z));
	elseif (y <= 1.35e+45)
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(Float64(x / y), z, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.7e-29], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, 1.35e+45], N[(y + x), $MachinePrecision], (-N[(N[(x / y), $MachinePrecision] * z + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6999999999999998e-29

   1. Initial program 84.0%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.3%

      \[\leadsto -\mathsf{fma}\left(\frac{x}{y}, z, z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

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

   if -4.6999999999999998e-29 < y < 1.34999999999999992e45

   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. +-commutative N/A

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if 1.34999999999999992e45 < y

   1. Initial program 82.1%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.4%

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

Alternative 7: 72.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+34}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-62}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e+34) (+ y x) (if (<= z 1.85e-62) (- (fma x (/ z y) z)) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e+34) {
		tmp = y + x;
	} else if (z <= 1.85e-62) {
		tmp = -fma(x, (z / y), z);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e+34)
		tmp = Float64(y + x);
	elseif (z <= 1.85e-62)
		tmp = Float64(-fma(x, Float64(z / y), z));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e+34], N[(y + x), $MachinePrecision], If[LessEqual[z, 1.85e-62], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision]), N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.00000000000000018e34 or 1.8499999999999999e-62 < z

   1. Initial program 100.0%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.1

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

   8. Applied rewrites 79.1%

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

   if -3.00000000000000018e34 < z < 1.8499999999999999e-62

   1. Initial program 84.5%

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

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.9%

      \[\leadsto -\mathsf{fma}\left(\frac{x}{y}, z, z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.3%

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

Alternative 8: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+17}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+45}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.7e+17) (- z) (if (<= y 1.7e+45) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.7e+17) {
		tmp = -z;
	} else if (y <= 1.7e+45) {
		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 <= (-4.7d+17)) then
        tmp = -z
    else if (y <= 1.7d+45) 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 <= -4.7e+17) {
		tmp = -z;
	} else if (y <= 1.7e+45) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.7e+17:
		tmp = -z
	elif y <= 1.7e+45:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.7e+17)
		tmp = Float64(-z);
	elseif (y <= 1.7e+45)
		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 <= -4.7e+17)
		tmp = -z;
	elseif (y <= 1.7e+45)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.7e+17], (-z), If[LessEqual[y, 1.7e+45], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e17 or 1.7e45 < y

   1. Initial program 82.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 64.8

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

   5. Applied rewrites 64.8%

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

   if -4.7e17 < y < 1.7e45

   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. +-commutative N/A

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

      2. *-inverses N/A

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

      3. div-sub N/A

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

      4. associate-/r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.8

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

   8. Applied rewrites 73.8%

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

Alternative 9: 34.0% 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 92.3%

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

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

   2. lower-neg.f64 33.4

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

5. Applied rewrites 33.4%

   \[\leadsto \color{blue}{-z} \]
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 2024308 
(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))))