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

Percentage Accurate: 88.0% → 99.6%

Time: 6.2s

Alternatives: 6

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: 88.0% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-268}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-268) t_0 (if (<= t_0 0.0) (- (/ (* (- x) z) y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-268) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((-x * z) / 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) / (1.0d0 - (y / z))
    if (t_0 <= (-1d-268)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((-x * z) / 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) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-268) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((-x * z) / y) - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -1e-268:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = ((-x * z) / y) - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -1e-268)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(-x) * z) / y) - z);
	else
		tmp = t_0;
	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 <= -1e-268)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((-x * z) / y) - z;
	else
		tmp = t_0;
	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, -1e-268], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[((-x) * z), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -9.99999999999999958e-269 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -9.99999999999999958e-269 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 6.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 6.2

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

   4. Applied rewrites 6.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   6. 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-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + x, \frac{y}{z}, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+62)
   (+ y x)
   (if (<= z 2.6e-11) (- (/ (* (- x) z) y) z) (+ (fma (+ y x) (/ y z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+62) {
		tmp = y + x;
	} else if (z <= 2.6e-11) {
		tmp = ((-x * z) / y) - z;
	} else {
		tmp = fma((y + x), (y / z), y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+62)
		tmp = Float64(y + x);
	elseif (z <= 2.6e-11)
		tmp = Float64(Float64(Float64(Float64(-x) * z) / y) - z);
	else
		tmp = Float64(fma(Float64(y + x), Float64(y / z), y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+62], N[(y + x), $MachinePrecision], If[LessEqual[z, 2.6e-11], N[(N[(N[((-x) * z), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] * N[(y / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8e62

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.8

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 18.1

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

   7. Applied rewrites 18.1%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 84.1

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

   10. Applied rewrites 84.1%

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

   if -4.8e62 < z < 2.6000000000000001e-11

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 78.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   6. 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-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.9%

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

   if 2.6000000000000001e-11 < z

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + x, \frac{y}{z}, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+62)
   (+ y x)
   (if (<= z 2.6e-11) (- (/ (* (- x) z) y) z) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+62) {
		tmp = y + x;
	} else if (z <= 2.6e-11) {
		tmp = ((-x * z) / 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 <= (-4.8d+62)) then
        tmp = y + x
    else if (z <= 2.6d-11) then
        tmp = ((-x * z) / 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 <= -4.8e+62) {
		tmp = y + x;
	} else if (z <= 2.6e-11) {
		tmp = ((-x * z) / y) - z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e+62:
		tmp = y + x
	elif z <= 2.6e-11:
		tmp = ((-x * z) / y) - z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+62)
		tmp = Float64(y + x);
	elseif (z <= 2.6e-11)
		tmp = Float64(Float64(Float64(Float64(-x) * z) / y) - z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e+62)
		tmp = y + x;
	elseif (z <= 2.6e-11)
		tmp = ((-x * z) / y) - z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+62], N[(y + x), $MachinePrecision], If[LessEqual[z, 2.6e-11], N[(N[(N[((-x) * z), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e62 or 2.6000000000000001e-11 < 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. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 17.2

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if -4.8e62 < z < 2.6000000000000001e-11

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 78.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) - \frac{{z}^{2}}{y}} \]
   6. 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-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. distribute-neg-out N/A

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

      8. *-lft-identity N/A

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

      9. metadata-eval N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. distribute-neg-frac N/A

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

      12. mul-1-neg N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

      15. lower--.f64 N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(-x\right) \cdot z}{y} - z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+62)
   (+ y x)
   (if (<= z 2.6e-11) (* (- -1.0 (/ x y)) z) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+62) {
		tmp = y + x;
	} else if (z <= 2.6e-11) {
		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 <= (-4.8d+62)) then
        tmp = y + x
    else if (z <= 2.6d-11) 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 <= -4.8e+62) {
		tmp = y + x;
	} else if (z <= 2.6e-11) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e+62:
		tmp = y + x
	elif z <= 2.6e-11:
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+62)
		tmp = Float64(y + x);
	elseif (z <= 2.6e-11)
		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 <= -4.8e+62)
		tmp = y + x;
	elseif (z <= 2.6e-11)
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+62], N[(y + x), $MachinePrecision], If[LessEqual[z, 2.6e-11], 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 < -4.8e62 or 2.6000000000000001e-11 < 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. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 17.2

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 83.8

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

   10. Applied rewrites 83.8%

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

   if -4.8e62 < z < 2.6000000000000001e-11

   1. Initial program 79.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+137}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+93}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e+137) (- z) (if (<= y 7.5e+93) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e+137) {
		tmp = -z;
	} else if (y <= 7.5e+93) {
		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 <= (-4d+137)) then
        tmp = -z
    else if (y <= 7.5d+93) 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 <= -4e+137) {
		tmp = -z;
	} else if (y <= 7.5e+93) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e+137:
		tmp = -z
	elif y <= 7.5e+93:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e+137)
		tmp = Float64(-z);
	elseif (y <= 7.5e+93)
		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 <= -4e+137)
		tmp = -z;
	elseif (y <= 7.5e+93)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e+137], (-z), If[LessEqual[y, 7.5e+93], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000001e137 or 7.5000000000000002e93 < y

   1. Initial program 67.3%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.4

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

   5. Applied rewrites 68.4%

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

   if -4.0000000000000001e137 < y < 7.5000000000000002e93

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 31.6

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

   7. Applied rewrites 31.6%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 65.9

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

   10. Applied rewrites 65.9%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+137}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+93}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 88.1%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 32.0

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

5. Applied rewrites 32.0%

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

Developer Target 1: 93.2% 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 2024331 
(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))))