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

Percentage Accurate: 87.5% → 99.8%

Time: 7.9s

Alternatives: 9

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.5% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-297} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{z \cdot \left(\mathsf{fma}\left(\frac{z}{y}, x + z, x\right) + z\right)}{y} + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-297) (not (<= t_0 0.0)))
     t_0
     (- (+ (/ (* z (+ (fma (/ z y) (+ x z) x) z)) y) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-297) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -(((z * (fma((z / y), (x + z), x) + z)) / y) + z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-297) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-Float64(Float64(Float64(z * Float64(fma(Float64(z / y), Float64(x + z), x) + z)) / y) + z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-297], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(N[(N[(z * N[(N[(N[(z / y), $MachinePrecision] * N[(x + z), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-297 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

   if -5e-297 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-297} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{z \cdot \left(\mathsf{fma}\left(\frac{z}{y}, x + z, x\right) + z\right)}{y} + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-261} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-261) (not (<= t_0 0.0))) t_0 (- (fma z (/ x y) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-261) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-261) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-261], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999981e-261 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

   if -4.99999999999999981e-261 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 11.3%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 78.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+66}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{+33}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+84}:\\ \;\;\;\;-\frac{z \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e+66)
   (- z)
   (if (<= y 1.14e+33) (+ y x) (if (<= y 1.46e+84) (- (/ (* z x) y)) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+66) {
		tmp = -z;
	} else if (y <= 1.14e+33) {
		tmp = y + x;
	} else if (y <= 1.46e+84) {
		tmp = -((z * x) / y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6d+66)) then
        tmp = -z
    else if (y <= 1.14d+33) then
        tmp = y + x
    else if (y <= 1.46d+84) then
        tmp = -((z * x) / y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+66) {
		tmp = -z;
	} else if (y <= 1.14e+33) {
		tmp = y + x;
	} else if (y <= 1.46e+84) {
		tmp = -((z * x) / y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e+66:
		tmp = -z
	elif y <= 1.14e+33:
		tmp = y + x
	elif y <= 1.46e+84:
		tmp = -((z * x) / y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e+66)
		tmp = Float64(-z);
	elseif (y <= 1.14e+33)
		tmp = Float64(y + x);
	elseif (y <= 1.46e+84)
		tmp = Float64(-Float64(Float64(z * x) / y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6e+66)
		tmp = -z;
	elseif (y <= 1.14e+33)
		tmp = y + x;
	elseif (y <= 1.46e+84)
		tmp = -((z * x) / y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e+66], (-z), If[LessEqual[y, 1.14e+33], N[(y + x), $MachinePrecision], If[LessEqual[y, 1.46e+84], (-N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.00000000000000005e66 or 1.46e84 < y

   1. Initial program 73.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.9

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

   5. Applied rewrites 63.9%

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

   if -6.00000000000000005e66 < y < 1.14e33

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 30.8%

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 70.5

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

   10. Applied rewrites 70.5%

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

   if 1.14e33 < y < 1.46e84

   1. Initial program 82.8%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.2%

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

Alternative 4: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154}:\\ \;\;\;\;\left(y + x\right) \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.45e+154)
   (* (+ y x) (+ (/ y z) 1.0))
   (if (<= z 1.4e-8) (- (fma z (/ x y) z)) (+ y (fma (/ x z) y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.45e+154) {
		tmp = (y + x) * ((y / z) + 1.0);
	} else if (z <= 1.4e-8) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = y + fma((x / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.45e+154)
		tmp = Float64(Float64(y + x) * Float64(Float64(y / z) + 1.0));
	elseif (z <= 1.4e-8)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(y + fma(Float64(x / z), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.45e+154], N[(N[(y + x), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e-8], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(y + N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4500000000000001e154

   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. difference-of-squares N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if -2.4500000000000001e154 < z < 1.4e-8

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.6%

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

   if 1.4e-8 < 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. fp-cancel-sub-sign-inv N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 5: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.45e+154)
   (+ y x)
   (if (<= z 1.4e-8) (- (fma z (/ x y) z)) (+ y (fma (/ x z) y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.45e+154) {
		tmp = y + x;
	} else if (z <= 1.4e-8) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = y + fma((x / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.45e+154)
		tmp = Float64(y + x);
	elseif (z <= 1.4e-8)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(y + fma(Float64(x / z), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.45e+154], N[(y + x), $MachinePrecision], If[LessEqual[z, 1.4e-8], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(y + N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4500000000000001e154

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 7.8%

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

   7. Applied rewrites 7.8%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 93.6

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

   10. Applied rewrites 93.6%

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

   if -2.4500000000000001e154 < z < 1.4e-8

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.6%

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

   if 1.4e-8 < 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. fp-cancel-sub-sign-inv N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 6: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.45e+154) (not (<= z 1.4e-8))) (+ y x) (- (fma z (/ x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.45e+154) || !(z <= 1.4e-8)) {
		tmp = y + x;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.45e+154) || !(z <= 1.4e-8))
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.45e+154], N[Not[LessEqual[z, 1.4e-8]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -2.4500000000000001e154 or 1.4e-8 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 21.4%

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

   7. Applied rewrites 21.4%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.2

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

   10. Applied rewrites 79.2%

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

   if -2.4500000000000001e154 < z < 1.4e-8

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 74.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 74.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{z}{y}, x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.45e+154) (not (<= z 1.4e-8))) (+ y x) (- (fma (/ z y) x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.45e+154) || !(z <= 1.4e-8)) {
		tmp = y + x;
	} else {
		tmp = -fma((z / y), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.45e+154) || !(z <= 1.4e-8))
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(Float64(z / y), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.45e+154], N[Not[LessEqual[z, 1.4e-8]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(N[(z / y), $MachinePrecision] * x + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -2.4500000000000001e154 or 1.4e-8 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 21.4%

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

   7. Applied rewrites 21.4%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.2

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

   10. Applied rewrites 79.2%

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

   if -2.4500000000000001e154 < z < 1.4e-8

   1. Initial program 78.5%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+154} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{z}{y}, x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+66} \lor \neg \left(y \leq 260000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6e+66) (not (<= y 260000.0))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6e+66) || !(y <= 260000.0)) {
		tmp = -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 ((y <= (-6d+66)) .or. (.not. (y <= 260000.0d0))) then
        tmp = -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 ((y <= -6e+66) || !(y <= 260000.0)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6e+66) or not (y <= 260000.0):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6e+66) || !(y <= 260000.0))
		tmp = Float64(-z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6e+66) || ~((y <= 260000.0)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6e+66], N[Not[LessEqual[y, 260000.0]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.00000000000000005e66 or 2.6e5 < y

   1. Initial program 76.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 58.1

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

   5. Applied rewrites 58.1%

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

   if -6.00000000000000005e66 < y < 2.6e5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. div-add N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-lft-out N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

   5. Applied rewrites 28.1%

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 73.2

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

   10. Applied rewrites 73.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+66} \lor \neg \left(y \leq 260000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.4% 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 87.4%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 37.0

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

5. Applied rewrites 37.0%

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

Developer Target 1: 93.6% 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 2024339 
(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))))