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

Percentage Accurate: 88.2% → 99.0%

Time: 6.1s

Alternatives: 7

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-223} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-223) (not (<= t_0 0.0))) t_0 (* (- -1.0 (/ x y)) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-223) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (x / y)) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-223)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((-1.0d0) - (x / y)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-223) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 - (x / y)) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-223) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = (-1.0 - (x / y)) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-223) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-223) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = (-1.0 - (x / y)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-223], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.00000000000000024e-223 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 -5.00000000000000024e-223 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 16.7%

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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-223} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-55} \lor \neg \left(y \leq 4.7 \cdot 10^{+45}\right):\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-55) (not (<= y 4.7e+45)))
   (* (- -1.0 (/ x y)) z)
   (/ x (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-55) || !(y <= 4.7e+45)) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d-55)) .or. (.not. (y <= 4.7d+45))) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = x / (1.0d0 - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-55) || !(y <= 4.7e+45)) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e-55) or not (y <= 4.7e+45):
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-55) || !(y <= 4.7e+45))
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e-55) || ~((y <= 4.7e+45)))
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e-55], N[Not[LessEqual[y, 4.7e+45]], $MachinePrecision]], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999995e-56 or 4.70000000000000002e45 < y

   1. Initial program 75.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -9.99999999999999995e-56 < y < 4.70000000000000002e45

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-55} \lor \neg \left(y \leq 4.7 \cdot 10^{+45}\right):\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{y}{z} + 1\right) \cdot \left(x + y\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;-\left(\frac{z \cdot \left(x + z\right)}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5e+43)
   (* (+ (/ y z) 1.0) (+ x y))
   (if (<= z 1.8e-47) (- (+ (/ (* z (+ x z)) y) z)) (+ y (fma (/ x z) y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e+43) {
		tmp = ((y / z) + 1.0) * (x + y);
	} else if (z <= 1.8e-47) {
		tmp = -(((z * (x + z)) / y) + z);
	} else {
		tmp = y + fma((x / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5e+43)
		tmp = Float64(Float64(Float64(y / z) + 1.0) * Float64(x + y));
	elseif (z <= 1.8e-47)
		tmp = Float64(-Float64(Float64(Float64(z * Float64(x + z)) / y) + z));
	else
		tmp = Float64(y + fma(Float64(x / z), y, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5e+43], N[(N[(N[(y / z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-47], (-N[(N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / 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 < -5.0000000000000004e43

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 81.6%

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

   if -5.0000000000000004e43 < z < 1.79999999999999995e-47

   1. Initial program 76.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 76.1

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

      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 76.1

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

   4. Applied rewrites 76.1%

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

   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. sub-neg N/A

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

      2. distribute-lft-out N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-out N/A

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

      5. remove-double-neg N/A

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

      6. sub-neg N/A

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

      7. associate-+r- N/A

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

      8. distribute-frac-neg N/A

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

      9. mul-1-neg N/A

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

      10. div-sub N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-out 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 77.9%

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

   if 1.79999999999999995e-47 < z

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(\frac{y}{z} + 1\right) \cdot \left(x + y\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-47}:\\ \;\;\;\;-\left(\frac{z \cdot \left(x + z\right)}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;y + \mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-55} \lor \neg \left(y \leq 3.5 \cdot 10^{+85}\right):\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e-55) (not (<= y 3.5e+85))) (* (- -1.0 (/ x y)) z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-55) || !(y <= 3.5e+85)) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d-55)) .or. (.not. (y <= 3.5d+85))) then
        tmp = ((-1.0d0) - (x / y)) * z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e-55) || !(y <= 3.5e+85)) {
		tmp = (-1.0 - (x / y)) * z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e-55) or not (y <= 3.5e+85):
		tmp = (-1.0 - (x / y)) * z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e-55) || !(y <= 3.5e+85))
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e-55) || ~((y <= 3.5e+85)))
		tmp = (-1.0 - (x / y)) * z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e-55], N[Not[LessEqual[y, 3.5e+85]], $MachinePrecision]], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999995e-56 or 3.50000000000000005e85 < y

   1. Initial program 74.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

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

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -9.99999999999999995e-56 < y < 3.50000000000000005e85

   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 y around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 75.8

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

   10. Applied rewrites 75.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-55} \lor \neg \left(y \leq 3.5 \cdot 10^{+85}\right):\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;\left(-1 - \frac{z}{y}\right) \cdot z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+87}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+129)
   (* (- -1.0 (/ z y)) z)
   (if (<= y 5.6e+87) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+129) {
		tmp = (-1.0 - (z / y)) * z;
	} else if (y <= 5.6e+87) {
		tmp = 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 <= (-1.4d+129)) then
        tmp = ((-1.0d0) - (z / y)) * z
    else if (y <= 5.6d+87) then
        tmp = 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 <= -1.4e+129) {
		tmp = (-1.0 - (z / y)) * z;
	} else if (y <= 5.6e+87) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+129:
		tmp = (-1.0 - (z / y)) * z
	elif y <= 5.6e+87:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+129)
		tmp = Float64(Float64(-1.0 - Float64(z / y)) * z);
	elseif (y <= 5.6e+87)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e+129)
		tmp = (-1.0 - (z / y)) * z;
	elseif (y <= 5.6e+87)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+129], N[(N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 5.6e+87], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999987e129

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 75.7%

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

   if -1.39999999999999987e129 < y < 5.6000000000000003e87

   1. Initial program 97.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 97.2

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

      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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 68.7

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

   10. Applied rewrites 68.7%

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

   if 5.6000000000000003e87 < y

   1. Initial program 65.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 74.2

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

   5. Applied rewrites 74.2%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;\left(-1 - \frac{z}{y}\right) \cdot z\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+87}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129} \lor \neg \left(y \leq 5.6 \cdot 10^{+87}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+129) (not (<= y 5.6e+87))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+129) || !(y <= 5.6e+87)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.4d+129)) .or. (.not. (y <= 5.6d+87))) then
        tmp = -z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+129) || !(y <= 5.6e+87)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e+129) or not (y <= 5.6e+87):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+129) || !(y <= 5.6e+87))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.4e+129) || ~((y <= 5.6e+87)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+129], N[Not[LessEqual[y, 5.6e+87]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999987e129 or 5.6000000000000003e87 < y

   1. Initial program 65.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   if -1.39999999999999987e129 < y < 5.6000000000000003e87

   1. Initial program 97.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 97.2

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

      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 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. distribute-neg-frac2 N/A

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

      5. mul-1-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   8. Taylor expanded in z around inf

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

      1. lower-+.f64 68.7

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

   10. Applied rewrites 68.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+129} \lor \neg \left(y \leq 5.6 \cdot 10^{+87}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% 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 88.2%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

Developer Target 1: 93.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024314 
(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))))