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

Percentage Accurate: 88.2% → 99.0%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

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: 88.2% 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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -6 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\left(\frac{\left(\left(\frac{\left(z + x\right) \cdot z}{y} + x\right) + z\right) \cdot z}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -6e-232)
     t_0
     (if (<= t_0 0.0)
       (- (+ (/ (* (+ (+ (/ (* (+ z x) z) y) x) z) z) y) z))
       (* (/ z (- z y)) (+ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -6e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    if (t_0 <= (-6d-232)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z)
    else
        tmp = (z / (z - y)) * (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -6e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -6e-232:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z)
	else:
		tmp = (z / (z - y)) * (y + x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -6e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + x) * z) / y) + x) + z) * z) / y) + z));
	else
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -6e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	else
		tmp = (z / (z - y)) * (y + x);
	end
	tmp_2 = 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, -6e-232], t$95$0, If[LessEqual[t$95$0, 0.0], (-N[(N[(N[(N[(N[(N[(N[(N[(z + x), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.99999999999999979e-232

   1. Initial program 99.9%

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

   if -5.99999999999999979e-232 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 11.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 100.0%

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

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

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lower-pow.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. *-inverses N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. clear-num N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -6 \cdot 10^{-232}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\left(\frac{\left(\left(\frac{\left(z + x\right) \cdot z}{y} + x\right) + z\right) \cdot z}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -6 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -6e-232)
     t_0
     (if (<= t_0 0.0) (* (- -1.0 (/ x y)) z) (* (/ z (- z y)) (+ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -6e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    if (t_0 <= (-6d-232)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = (z / (z - y)) * (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -6e-232) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -6e-232:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = (z / (z - y)) * (y + x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -6e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -6e-232)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = (z / (z - y)) * (y + x);
	end
	tmp_2 = 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, -6e-232], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.99999999999999979e-232

   1. Initial program 99.9%

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

   if -5.99999999999999979e-232 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 11.2%

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

   3. Taylor expanded in z around 0

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

      1. 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%

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

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

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lower-pow.f64 81.6

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

   4. Applied rewrites 81.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. *-inverses N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. clear-num N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -6 \cdot 10^{-232}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y + x}{z - y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))) (t_1 (* (/ z (- z y)) (+ y x))))
   (if (<= t_0 -2e-115) t_1 (if (<= t_0 0.0) (* (/ (+ y x) (- z y)) z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -2e-115) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((y + x) / (z - y)) * z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    t_1 = (z / (z - y)) * (y + x)
    if (t_0 <= (-2d-115)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((y + x) / (z - y)) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -2e-115) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((y + x) / (z - y)) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	t_1 = (z / (z - y)) * (y + x)
	tmp = 0
	if t_0 <= -2e-115:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = ((y + x) / (z - y)) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(z / Float64(z - y)) * Float64(y + x))
	tmp = 0.0
	if (t_0 <= -2e-115)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(y + x) / Float64(z - y)) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	t_1 = (z / (z - y)) * (y + x);
	tmp = 0.0;
	if (t_0 <= -2e-115)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = ((y + x) / (z - y)) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-115], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2.0000000000000001e-115 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lower-pow.f64 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. *-inverses N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. clear-num N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -2.0000000000000001e-115 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 39.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lower-pow.f64 18.2

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

   4. Applied rewrites 18.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. *-inverses N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. clear-num N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 43.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-115}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{y + x}{z - y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, y, x\right) + y\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (fma (/ x z) y x) y)))
   (if (<= z -1.75e-26) t_0 (if (<= z 7e-61) (* (- -1.0 (/ x y)) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), y, x) + y;
	double tmp;
	if (z <= -1.75e-26) {
		tmp = t_0;
	} else if (z <= 7e-61) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(x / z), y, x) + y)
	tmp = 0.0
	if (z <= -1.75e-26)
		tmp = t_0;
	elseif (z <= 7e-61)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[z, -1.75e-26], t$95$0, If[LessEqual[z, 7e-61], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999992e-26 or 7.0000000000000006e-61 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. 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 79.9

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

   5. Applied rewrites 79.9%

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

   if -1.74999999999999992e-26 < z < 7.0000000000000006e-61

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right) + y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right) + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-26}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.75e-26)
   (+ y x)
   (if (<= z 2.9e-62) (* (- -1.0 (/ x y)) z) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.75e-26) {
		tmp = y + x;
	} else if (z <= 2.9e-62) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.75d-26)) then
        tmp = y + x
    else if (z <= 2.9d-62) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.75e-26) {
		tmp = y + x;
	} else if (z <= 2.9e-62) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.75e-26:
		tmp = y + x
	elif z <= 2.9e-62:
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.75e-26)
		tmp = Float64(y + x);
	elseif (z <= 2.9e-62)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.75e-26)
		tmp = y + x;
	elseif (z <= 2.9e-62)
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.75e-26], N[(y + x), $MachinePrecision], If[LessEqual[z, 2.9e-62], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999992e-26 or 2.89999999999999986e-62 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -1.74999999999999992e-26 < z < 2.89999999999999986e-62

   1. Initial program 68.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 76.8

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

   5. Applied rewrites 76.8%

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

Alternative 6: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+243}:\\ \;\;\;\;\frac{z}{z - y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{z - y} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e+243) (* (/ z (- z y)) x) (* (/ (+ y x) (- z y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+243) {
		tmp = (z / (z - y)) * x;
	} else {
		tmp = ((y + x) / (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 (x <= (-7.6d+243)) then
        tmp = (z / (z - y)) * x
    else
        tmp = ((y + x) / (z - y)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+243) {
		tmp = (z / (z - y)) * x;
	} else {
		tmp = ((y + x) / (z - y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.6e+243:
		tmp = (z / (z - y)) * x
	else:
		tmp = ((y + x) / (z - y)) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.6e+243)
		tmp = Float64(Float64(z / Float64(z - y)) * x);
	else
		tmp = Float64(Float64(Float64(y + x) / Float64(z - y)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.6e+243)
		tmp = (z / (z - y)) * x;
	else
		tmp = ((y + x) / (z - y)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.6e+243], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999996e243

   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. *-inverses N/A

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

      3. div-sub N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.8%

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

   if -7.59999999999999996e243 < x

   1. Initial program 87.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. pow2 N/A

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

      17. lower-pow.f64 67.1

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

   4. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-- N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. *-inverses N/A

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

      17. div-sub N/A

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

      18. lift--.f64 N/A

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

      19. clear-num N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.1

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

   8. Applied rewrites 96.1%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+243}:\\ \;\;\;\;\frac{z}{z - y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{z - y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+191}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+135}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.45e+191) (- z) (if (<= y 1.16e+135) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.45e+191) {
		tmp = -z;
	} else if (y <= 1.16e+135) {
		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 <= (-2.45d+191)) then
        tmp = -z
    else if (y <= 1.16d+135) 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 <= -2.45e+191) {
		tmp = -z;
	} else if (y <= 1.16e+135) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.45e+191:
		tmp = -z
	elif y <= 1.16e+135:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.45e+191)
		tmp = Float64(-z);
	elseif (y <= 1.16e+135)
		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 <= -2.45e+191)
		tmp = -z;
	elseif (y <= 1.16e+135)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.45e+191], (-z), If[LessEqual[y, 1.16e+135], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -2.45e191 or 1.16000000000000005e135 < y

   1. Initial program 63.4%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -2.45e191 < y < 1.16000000000000005e135

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 8: 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 87.8%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 31.9

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

5. Applied rewrites 31.9%

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

Developer Target 1: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024243 
(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))))