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

Percentage Accurate: 87.9% → 99.8%

Time: 4.2s

Alternatives: 6

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-294} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-294) (not (<= t_0 0.0)))
     t_0
     (fma
      (/ (+ (fma z (/ (fma z x (* z z)) y) (* z x)) (* z z)) y)
      -1.0
      (- 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-294) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = fma(((fma(z, (fma(z, x, (z * z)) / y), (z * x)) + (z * z)) / y), -1.0, -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-294) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(fma(z, Float64(fma(z, x, Float64(z * z)) / y), Float64(z * x)) + Float64(z * z)) / y), -1.0, Float64(-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-294], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[(N[(z * N[(N[(z * x + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.0000000000000003e-294 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -5.0000000000000003e-294 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.9%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-294} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-294} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-298}\right):\\ \;\;\;\;\frac{x + y}{\left({y}^{-1} - {z}^{-1}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-294) (not (<= t_0 4e-298)))
     (/ (+ x y) (* (- (pow y -1.0) (pow z -1.0)) y))
     (fma
      (/ (+ (fma z (/ (fma z x (* z z)) y) (* z x)) (* z z)) y)
      -1.0
      (- 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-294) || !(t_0 <= 4e-298)) {
		tmp = (x + y) / ((pow(y, -1.0) - pow(z, -1.0)) * y);
	} else {
		tmp = fma(((fma(z, (fma(z, x, (z * z)) / y), (z * x)) + (z * z)) / y), -1.0, -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-294) || !(t_0 <= 4e-298))
		tmp = Float64(Float64(x + y) / Float64(Float64((y ^ -1.0) - (z ^ -1.0)) * y));
	else
		tmp = fma(Float64(Float64(fma(z, Float64(fma(z, x, Float64(z * z)) / y), Float64(z * x)) + Float64(z * z)) / y), -1.0, Float64(-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-294], N[Not[LessEqual[t$95$0, 4e-298]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / N[(N[(N[Power[y, -1.0], $MachinePrecision] - N[Power[z, -1.0], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z * N[(N[(z * x + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.0000000000000003e-294 or 3.99999999999999965e-298 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -5.0000000000000003e-294 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 3.99999999999999965e-298

   1. Initial program 11.1%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-294} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 4 \cdot 10^{-298}\right):\\ \;\;\;\;\frac{x + y}{\left({y}^{-1} - {z}^{-1}\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\left({y}^{-1} - {z}^{-0.5} \cdot {z}^{-0.5}\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e+20)
   (+ (fma y (/ (+ y x) z) y) x)
   (if (<= z 4.5e-104)
     (fma (/ (+ (fma z (/ (fma z x (* z z)) y) (* z x)) (* z z)) y) -1.0 (- z))
     (/ (+ x y) (* (- (pow y -1.0) (* (pow z -0.5) (pow z -0.5))) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+20) {
		tmp = fma(y, ((y + x) / z), y) + x;
	} else if (z <= 4.5e-104) {
		tmp = fma(((fma(z, (fma(z, x, (z * z)) / y), (z * x)) + (z * z)) / y), -1.0, -z);
	} else {
		tmp = (x + y) / ((pow(y, -1.0) - (pow(z, -0.5) * pow(z, -0.5))) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e+20)
		tmp = Float64(fma(y, Float64(Float64(y + x) / z), y) + x);
	elseif (z <= 4.5e-104)
		tmp = fma(Float64(Float64(fma(z, Float64(fma(z, x, Float64(z * z)) / y), Float64(z * x)) + Float64(z * z)) / y), -1.0, Float64(-z));
	else
		tmp = Float64(Float64(x + y) / Float64(Float64((y ^ -1.0) - Float64((z ^ -0.5) * (z ^ -0.5))) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e+20], N[(N[(y * N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.5e-104], N[(N[(N[(N[(z * N[(N[(z * x + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + (-z)), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(N[(N[Power[y, -1.0], $MachinePrecision] - N[(N[Power[z, -0.5], $MachinePrecision] * N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e20

   1. Initial program 100.0%

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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -5.2e20 < z < 4.4999999999999997e-104

   1. Initial program 69.7%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 80.4%

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

   if 4.4999999999999997e-104 < z

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 97.6

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

   5. Applied rewrites 97.6%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 97.6

         \[\leadsto \frac{x + y}{\left({y}^{-1} - {z}^{-0.5} \cdot {z}^{-0.5}\right) \cdot y} \]

   7. Applied rewrites 97.6%

      \[\leadsto \frac{x + y}{\left({y}^{-1} - {z}^{-0.5} \cdot {z}^{-0.5}\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\left({y}^{-1} - {z}^{-0.5} \cdot {z}^{-0.5}\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 1.8 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+20) (not (<= z 1.8e+26)))
   (+ (fma y (/ (+ y x) z) y) x)
   (fma (/ (+ (fma z (/ (fma z x (* z z)) y) (* z x)) (* z z)) y) -1.0 (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+20) || !(z <= 1.8e+26)) {
		tmp = fma(y, ((y + x) / z), y) + x;
	} else {
		tmp = fma(((fma(z, (fma(z, x, (z * z)) / y), (z * x)) + (z * z)) / y), -1.0, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+20) || !(z <= 1.8e+26))
		tmp = Float64(fma(y, Float64(Float64(y + x) / z), y) + x);
	else
		tmp = fma(Float64(Float64(fma(z, Float64(fma(z, x, Float64(z * z)) / y), Float64(z * x)) + Float64(z * z)) / y), -1.0, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e+20], N[Not[LessEqual[z, 1.8e+26]], $MachinePrecision]], N[(N[(y * N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(z * N[(N[(z * x + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e20 or 1.80000000000000012e26 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 82.7

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

   5. Applied rewrites 82.7%

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

   if -5.2e20 < z < 1.80000000000000012e26

   1. Initial program 73.3%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 78.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 1.8 \cdot 10^{+26}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, x, z \cdot z\right)}{y}, z \cdot x\right) + z \cdot z}{y}, -1, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 2900000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-z\right) \cdot \left(\frac{x}{{y}^{3}} + {y}^{-2}\right) - {y}^{-1}\right) - \frac{x}{y \cdot y}, z, \frac{y + x}{-y}\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+20) (not (<= z 2900000000.0)))
   (+ (fma y (/ (+ y x) z) y) x)
   (*
    (fma
     (-
      (- (* (- z) (+ (/ x (pow y 3.0)) (pow y -2.0))) (pow y -1.0))
      (/ x (* y y)))
     z
     (/ (+ y x) (- y)))
    z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+20) || !(z <= 2900000000.0)) {
		tmp = fma(y, ((y + x) / z), y) + x;
	} else {
		tmp = fma((((-z * ((x / pow(y, 3.0)) + pow(y, -2.0))) - pow(y, -1.0)) - (x / (y * y))), z, ((y + x) / -y)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+20) || !(z <= 2900000000.0))
		tmp = Float64(fma(y, Float64(Float64(y + x) / z), y) + x);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-z) * Float64(Float64(x / (y ^ 3.0)) + (y ^ -2.0))) - (y ^ -1.0)) - Float64(x / Float64(y * y))), z, Float64(Float64(y + x) / Float64(-y))) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e+20], N[Not[LessEqual[z, 2900000000.0]], $MachinePrecision]], N[(N[(y * N[(N[(y + x), $MachinePrecision] / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(N[((-z) * N[(N[(x / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[y, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e20 or 2.9e9 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -5.2e20 < z < 2.9e9

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right) \cdot \color{blue}{z} \]

   5. Applied rewrites 71.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 2900000000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y + x}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-z\right) \cdot \left(\frac{x}{{y}^{3}} + {y}^{-2}\right) - {y}^{-1}\right) - \frac{x}{y \cdot y}, z, \frac{y + x}{-y}\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(-z\right) \cdot \left(\frac{x}{{y}^{3}} + {y}^{-2}\right) - {y}^{-1}\right) - \frac{x}{y \cdot y}, z, \frac{y + x}{-y}\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (*
  (fma
   (-
    (- (* (- z) (+ (/ x (pow y 3.0)) (pow y -2.0))) (pow y -1.0))
    (/ x (* y y)))
   z
   (/ (+ y x) (- y)))
  z))

C

double code(double x, double y, double z) {
	return fma((((-z * ((x / pow(y, 3.0)) + pow(y, -2.0))) - pow(y, -1.0)) - (x / (y * y))), z, ((y + x) / -y)) * z;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(Float64(Float64(Float64(-z) * Float64(Float64(x / (y ^ 3.0)) + (y ^ -2.0))) - (y ^ -1.0)) - Float64(x / Float64(y * y))), z, Float64(Float64(y + x) / Float64(-y))) * z)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[((-z) * N[(N[(x / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[y, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision] - N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{z \cdot \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right) \cdot \color{blue}{z} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-1 \cdot \frac{x + y}{y} + z \cdot \left(-1 \cdot \left(z \cdot \left(\frac{1}{{y}^{2}} + \frac{x}{{y}^{3}}\right)\right) - \left(\frac{1}{y} + \frac{x}{{y}^{2}}\right)\right)\right) \cdot \color{blue}{z} \]

5. Applied rewrites 42.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-z\right) \cdot \left(\frac{x}{{y}^{3}} + {y}^{-2}\right) - {y}^{-1}\right) - \frac{x}{y \cdot y}, z, -\frac{y + x}{y}\right) \cdot z} \]

6. Final simplification 42.7%

   \[\leadsto \mathsf{fma}\left(\left(\left(-z\right) \cdot \left(\frac{x}{{y}^{3}} + {y}^{-2}\right) - {y}^{-1}\right) - \frac{x}{y \cdot y}, z, \frac{y + x}{-y}\right) \cdot z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))