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

Percentage Accurate: 87.9% → 99.7%

Time: 5.7s

Alternatives: 7

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-266} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-296}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-266) (not (<= t_0 4e-296)))
     t_0
     (* (- -1.0 (/ x y)) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-266) || !(t_0 <= 4e-296)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (x / y)) * 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-266)) .or. (.not. (t_0 <= 4d-296))) then
        tmp = t_0
    else
        tmp = ((-1.0d0) - (x / y)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-266) || !(t_0 <= 4e-296)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (x / y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-266) or not (t_0 <= 4e-296):
		tmp = t_0
	else:
		tmp = (-1.0 - (x / y)) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-266) || !(t_0 <= 4e-296))
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-266) || ~((t_0 <= 4e-296)))
		tmp = t_0;
	else
		tmp = (-1.0 - (x / y)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-266], N[Not[LessEqual[t$95$0, 4e-296]], $MachinePrecision]], t$95$0, N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999992e-266 or 4e-296 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.99999999999999992e-266 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 4e-296

   1. Initial program 12.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-266} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 4 \cdot 10^{-296}\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1750000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, \left(-x\right) - z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{z + y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.05e+89)
   (+ x y)
   (if (<= z 1750000.0)
     (fma (/ z y) (- (- x) z) (- z))
     (* (+ x y) (/ (+ z y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+89) {
		tmp = x + y;
	} else if (z <= 1750000.0) {
		tmp = fma((z / y), (-x - z), -z);
	} else {
		tmp = (x + y) * ((z + y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.05e+89)
		tmp = Float64(x + y);
	elseif (z <= 1750000.0)
		tmp = fma(Float64(z / y), Float64(Float64(-x) - z), Float64(-z));
	else
		tmp = Float64(Float64(x + y) * Float64(Float64(z + y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.05e+89], N[(x + y), $MachinePrecision], If[LessEqual[z, 1750000.0], N[(N[(z / y), $MachinePrecision] * N[((-x) - z), $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(N[(z + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.04999999999999993e89

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

   9. Taylor expanded in z around inf

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

       1. lower-+.f64 89.5

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

   11. Applied rewrites 89.5%

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

   if -1.04999999999999993e89 < z < 1.75e6

   1. Initial program 81.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 77.2

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

   8. Applied rewrites 77.2%

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

   if 1.75e6 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.9%

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

Alternative 3: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1750000:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{z + y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.05e+89)
   (+ x y)
   (if (<= z 1750000.0) (* (- -1.0 (/ x y)) z) (* (+ x y) (/ (+ z y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+89) {
		tmp = x + y;
	} else if (z <= 1750000.0) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = (x + y) * ((z + y) / 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 (z <= (-1.05d+89)) then
        tmp = x + y
    else if (z <= 1750000.0d0) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = (x + y) * ((z + y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+89) {
		tmp = x + y;
	} else if (z <= 1750000.0) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = (x + y) * ((z + y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.05e+89:
		tmp = x + y
	elif z <= 1750000.0:
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = (x + y) * ((z + y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.05e+89)
		tmp = Float64(x + y);
	elseif (z <= 1750000.0)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = Float64(Float64(x + y) * Float64(Float64(z + y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.05e+89)
		tmp = x + y;
	elseif (z <= 1750000.0)
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = (x + y) * ((z + y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.05e+89], N[(x + y), $MachinePrecision], If[LessEqual[z, 1750000.0], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(N[(z + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.04999999999999993e89

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 13.3

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

   8. Applied rewrites 13.3%

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

   9. Taylor expanded in z around inf

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

       1. lower-+.f64 89.5

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

   11. Applied rewrites 89.5%

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

   if -1.04999999999999993e89 < z < 1.75e6

   1. Initial program 81.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if 1.75e6 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.9%

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

Alternative 4: 72.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+89} \lor \neg \left(z \leq 5.2 \cdot 10^{-13}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05e+89) (not (<= z 5.2e-13)))
   (+ x y)
   (* (- -1.0 (/ x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+89) || !(z <= 5.2e-13)) {
		tmp = x + y;
	} else {
		tmp = (-1.0 - (x / y)) * 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 ((z <= (-1.05d+89)) .or. (.not. (z <= 5.2d-13))) then
        tmp = x + y
    else
        tmp = ((-1.0d0) - (x / y)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+89) || !(z <= 5.2e-13)) {
		tmp = x + y;
	} else {
		tmp = (-1.0 - (x / y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05e+89) or not (z <= 5.2e-13):
		tmp = x + y
	else:
		tmp = (-1.0 - (x / y)) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05e+89) || !(z <= 5.2e-13))
		tmp = Float64(x + y);
	else
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.05e+89) || ~((z <= 5.2e-13)))
		tmp = x + y;
	else
		tmp = (-1.0 - (x / y)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05e+89], N[Not[LessEqual[z, 5.2e-13]], $MachinePrecision]], N[(x + y), $MachinePrecision], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999993e89 or 5.2000000000000001e-13 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 20.0

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

   8. Applied rewrites 20.0%

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

   9. Taylor expanded in z around inf

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

       1. lower-+.f64 81.3

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

   11. Applied rewrites 81.3%

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

   if -1.04999999999999993e89 < z < 5.2000000000000001e-13

   1. Initial program 81.1%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+89} \lor \neg \left(z \leq 5.2 \cdot 10^{-13}\right):\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+145}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 0.0135:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.2e+145) (- z) (if (<= y 0.0135) (+ x y) (* z (- -1.0 (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+145) {
		tmp = -z;
	} else if (y <= 0.0135) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (z / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.2d+145)) then
        tmp = -z
    else if (y <= 0.0135d0) then
        tmp = x + y
    else
        tmp = z * ((-1.0d0) - (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.2e+145) {
		tmp = -z;
	} else if (y <= 0.0135) {
		tmp = x + y;
	} else {
		tmp = z * (-1.0 - (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.2e+145:
		tmp = -z
	elif y <= 0.0135:
		tmp = x + y
	else:
		tmp = z * (-1.0 - (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.2e+145)
		tmp = Float64(-z);
	elseif (y <= 0.0135)
		tmp = Float64(x + y);
	else
		tmp = Float64(z * Float64(-1.0 - Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.2e+145)
		tmp = -z;
	elseif (y <= 0.0135)
		tmp = x + y;
	else
		tmp = z * (-1.0 - (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.2e+145], (-z), If[LessEqual[y, 0.0135], N[(x + y), $MachinePrecision], N[(z * N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000009e145

   1. Initial program 65.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

   if -2.20000000000000009e145 < y < 0.0134999999999999998

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 32.9

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-+.f64 67.8

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

   11. Applied rewrites 67.8%

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

   if 0.0134999999999999998 < y

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 27.0%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 74.0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \frac{{z}^{2}}{y} - \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.4%

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

Alternative 6: 66.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+145} \lor \neg \left(y \leq 0.0135\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e+145) (not (<= y 0.0135))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e+145) || !(y <= 0.0135)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.2d+145)) .or. (.not. (y <= 0.0135d0))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e+145) || !(y <= 0.0135)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e+145) or not (y <= 0.0135):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e+145) || !(y <= 0.0135))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e+145) || ~((y <= 0.0135)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e+145], N[Not[LessEqual[y, 0.0135]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000009e145 or 0.0134999999999999998 < y

   1. Initial program 76.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -2.20000000000000009e145 < y < 0.0134999999999999998

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 32.9

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in z around inf

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

       1. lower-+.f64 67.8

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

   11. Applied rewrites 67.8%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+145} \lor \neg \left(y \leq 0.0135\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.6% 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 89.6%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 36.9

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

5. Applied rewrites 36.9%

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

Developer Target 1: 94.0% 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 2024318 
(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))))