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

Percentage Accurate: 88.0% → 99.0%

Time: 8.9s

Alternatives: 10

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

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: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-225}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -4e-225)
     (/ (+ x y) (/ (- z y) z))
     (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -4e-225) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = t_0;
	}
	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 <= -4e-225)
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = t_0;
	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[LessEqual[t$95$0, -4e-225], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -3.9999999999999998e-225

   1. Initial program 99.9%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if -3.9999999999999998e-225 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 10.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 100.0%

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

   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 100.0%

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

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

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-225} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-225) (not (<= t_0 0.0)))
     (/ (+ x y) (/ (- z y) z))
     (- (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 <= -4e-225) || !(t_0 <= 0.0)) {
		tmp = (x + y) / ((z - y) / z);
	} 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 <= -4e-225) || !(t_0 <= 0.0))
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	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, -4e-225], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], (-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))) < -3.9999999999999998e-225 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   3. 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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 99.9%

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

   if -3.9999999999999998e-225 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 10.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 86.3%

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

   7. Applied rewrites 100.0%

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

   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 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-225} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t\_0}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+65}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x + z}{y}, z\right)\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)))
   (if (<= y -5.2e+65)
     (- (fma z (/ (+ x z) y) z))
     (if (<= y -2.35e-42)
       t_1
       (if (<= y 4.8e-81)
         (/ x t_0)
         (if (<= y 1.55e+133) t_1 (- (fma z (/ x y) z))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -5.2e+65) {
		tmp = -fma(z, ((x + z) / y), z);
	} else if (y <= -2.35e-42) {
		tmp = t_1;
	} else if (y <= 4.8e-81) {
		tmp = x / t_0;
	} else if (y <= 1.55e+133) {
		tmp = t_1;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (y <= -5.2e+65)
		tmp = Float64(-fma(z, Float64(Float64(x + z) / y), z));
	elseif (y <= -2.35e-42)
		tmp = t_1;
	elseif (y <= 4.8e-81)
		tmp = Float64(x / t_0);
	elseif (y <= 1.55e+133)
		tmp = t_1;
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, If[LessEqual[y, -5.2e+65], (-N[(z * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, -2.35e-42], t$95$1, If[LessEqual[y, 4.8e-81], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 1.55e+133], t$95$1, (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.20000000000000005e65

   1. Initial program 68.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 84.0%

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

   if -5.20000000000000005e65 < y < -2.35e-42 or 4.7999999999999998e-81 < y < 1.55e133

   1. Initial program 96.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if -2.35e-42 < y < 4.7999999999999998e-81

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 88.5

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

   5. Applied rewrites 88.5%

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

   if 1.55e133 < y

   1. Initial program 55.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 89.2%

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

   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 89.3%

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

Alternative 4: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-40}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x + z}{y}, z\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;\frac{x + y}{1}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e-40)
   (- (fma z (/ (+ x z) y) z))
   (if (<= y 4.5e-71)
     (/ x (- 1.0 (/ y z)))
     (if (<= y 1.25e+66) (/ (+ x y) 1.0) (- (fma z (/ x y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e-40) {
		tmp = -fma(z, ((x + z) / y), z);
	} else if (y <= 4.5e-71) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 1.25e+66) {
		tmp = (x + y) / 1.0;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e-40)
		tmp = Float64(-fma(z, Float64(Float64(x + z) / y), z));
	elseif (y <= 4.5e-71)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 1.25e+66)
		tmp = Float64(Float64(x + y) / 1.0);
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e-40], (-N[(z * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, 4.5e-71], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+66], N[(N[(x + y), $MachinePrecision] / 1.0), $MachinePrecision], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.7999999999999999e-40

   1. Initial program 77.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 75.1%

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

   if -3.7999999999999999e-40 < y < 4.5000000000000002e-71

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 87.7

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

   5. Applied rewrites 87.7%

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

   if 4.5000000000000002e-71 < y < 1.24999999999999998e66

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 66.5%

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

   if 1.24999999999999998e66 < y

   1. Initial program 62.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 84.5%

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

   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 84.8%

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

Alternative 5: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x + z}{y}, z\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(y + x, \frac{y}{z}, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-39)
   (- (fma z (/ (+ x z) y) z))
   (if (<= y 1.25e+66) (+ (fma (+ y x) (/ y z) y) x) (- (fma z (/ x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-39) {
		tmp = -fma(z, ((x + z) / y), z);
	} else if (y <= 1.25e+66) {
		tmp = fma((y + x), (y / z), y) + x;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-39)
		tmp = Float64(-fma(z, Float64(Float64(x + z) / y), z));
	elseif (y <= 1.25e+66)
		tmp = Float64(fma(Float64(y + x), Float64(y / z), y) + x);
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-39], (-N[(z * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, 1.25e+66], N[(N[(N[(y + x), $MachinePrecision] * N[(y / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999986e-39

   1. Initial program 77.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 75.1%

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

   if -1.99999999999999986e-39 < y < 1.24999999999999998e66

   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

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

      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 73.9

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

   5. Applied rewrites 73.9%

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

   if 1.24999999999999998e66 < y

   1. Initial program 62.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 84.5%

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

   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 84.8%

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

Alternative 6: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-39} \lor \neg \left(y \leq 1.25 \cdot 10^{+66}\right):\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e-39) (not (<= y 1.25e+66)))
   (- (fma z (/ x y) z))
   (/ (+ x y) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e-39) || !(y <= 1.25e+66)) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = (x + y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2e-39) || !(y <= 1.25e+66))
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(Float64(x + y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2e-39], N[Not[LessEqual[y, 1.25e+66]], $MachinePrecision]], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(N[(x + y), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999986e-39 or 1.24999999999999998e66 < y

   1. Initial program 70.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 79.3%

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

   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 78.7%

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

   if -1.99999999999999986e-39 < y < 1.24999999999999998e66

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 73.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-39} \lor \neg \left(y \leq 1.25 \cdot 10^{+66}\right):\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-39}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x + z}{y}, z\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;\frac{x + y}{1}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-39)
   (- (fma z (/ (+ x z) y) z))
   (if (<= y 1.25e+66) (/ (+ x y) 1.0) (- (fma z (/ x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-39) {
		tmp = -fma(z, ((x + z) / y), z);
	} else if (y <= 1.25e+66) {
		tmp = (x + y) / 1.0;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-39)
		tmp = Float64(-fma(z, Float64(Float64(x + z) / y), z));
	elseif (y <= 1.25e+66)
		tmp = Float64(Float64(x + y) / 1.0);
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-39], (-N[(z * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[y, 1.25e+66], N[(N[(x + y), $MachinePrecision] / 1.0), $MachinePrecision], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999986e-39

   1. Initial program 77.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 75.1%

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

   if -1.99999999999999986e-39 < y < 1.24999999999999998e66

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 73.8%

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

   if 1.24999999999999998e66 < y

   1. Initial program 62.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 84.5%

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

   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 84.8%

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

Alternative 8: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+103} \lor \neg \left(y \leq 1.25 \cdot 10^{+66}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.1e+103) (not (<= y 1.25e+66))) (- z) (/ (+ x y) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.1e+103) || !(y <= 1.25e+66)) {
		tmp = -z;
	} else {
		tmp = (x + y) / 1.0;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y <= (-2.1d+103)) .or. (.not. (y <= 1.25d+66))) then
        tmp = -z
    else
        tmp = (x + y) / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.1e+103) || !(y <= 1.25e+66)) {
		tmp = -z;
	} else {
		tmp = (x + y) / 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.1e+103) or not (y <= 1.25e+66):
		tmp = -z
	else:
		tmp = (x + y) / 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.1e+103) || !(y <= 1.25e+66))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x + y) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.1e+103) || ~((y <= 1.25e+66)))
		tmp = -z;
	else
		tmp = (x + y) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.1e+103], N[Not[LessEqual[y, 1.25e+66]], $MachinePrecision]], (-z), N[(N[(x + y), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1000000000000002e103 or 1.24999999999999998e66 < y

   1. Initial program 63.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 71.3

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

   5. Applied rewrites 71.3%

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

   if -2.1000000000000002e103 < y < 1.24999999999999998e66

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+103} \lor \neg \left(y \leq 1.25 \cdot 10^{+66}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;z \cdot \left(-1 - \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+66}:\\ \;\;\;\;\frac{x + y}{1}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+103)
   (* z (- -1.0 (/ z y)))
   (if (<= y 1.25e+66) (/ (+ x y) 1.0) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+103) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= 1.25e+66) {
		tmp = (x + y) / 1.0;
	} else {
		tmp = -z;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= (-2.1d+103)) then
        tmp = z * ((-1.0d0) - (z / y))
    else if (y <= 1.25d+66) then
        tmp = (x + y) / 1.0d0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+103) {
		tmp = z * (-1.0 - (z / y));
	} else if (y <= 1.25e+66) {
		tmp = (x + y) / 1.0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+103:
		tmp = z * (-1.0 - (z / y))
	elif y <= 1.25e+66:
		tmp = (x + y) / 1.0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+103)
		tmp = Float64(z * Float64(-1.0 - Float64(z / y)));
	elseif (y <= 1.25e+66)
		tmp = Float64(Float64(x + y) / 1.0);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+103)
		tmp = z * (-1.0 - (z / y));
	elseif (y <= 1.25e+66)
		tmp = (x + y) / 1.0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+103], N[(z * N[(-1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+66], N[(N[(x + y), $MachinePrecision] / 1.0), $MachinePrecision], (-z)]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1000000000000002e103

   1. Initial program 65.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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. associate-+r- N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. associate--l- N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. div-add N/A

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

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

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

      11. *-lft-identity N/A

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

      12. metadata-eval N/A

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

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

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

      14. distribute-lft-out N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.9%

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

   if -2.1000000000000002e103 < y < 1.24999999999999998e66

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.5%

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

   if 1.24999999999999998e66 < y

   1. Initial program 62.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

Alternative 10: 34.9% 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

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 = -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 86.0%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 37.6

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

5. Applied rewrites 37.6%

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

Developer Target 1: 93.4% 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

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
    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 2024352 
(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))))