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

Percentage Accurate: 88.2% → 99.1%

Time: 3.3s

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: 88.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, z, \frac{z \cdot x + z \cdot z}{-y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-246)
     (/ (+ x y) (/ (- z y) z))
     (if (<= t_0 0.0) (fma -1.0 z (/ (+ (* z x) (* z z)) (- y))) 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 <= -1e-246) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = fma(-1.0, z, (((z * x) + (z * z)) / -y));
	} 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 <= -1e-246)
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	elseif (t_0 <= 0.0)
		tmp = fma(-1.0, z, Float64(Float64(Float64(z * x) + Float64(z * z)) / Float64(-y)));
	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, -1e-246], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(-1.0 * z + N[(N[(N[(z * x), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision]), $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))) < -9.99999999999999956e-247

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -9.99999999999999956e-247 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 21.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, z, \frac{z \cdot x + z \cdot z}{-y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-246} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-246) (not (<= t_0 0.0)))
     (/ (+ x y) (/ (- z y) z))
     (* (- z) (/ (+ y x) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-246) || !(t_0 <= 0.0)) {
		tmp = (x + y) / ((z - y) / z);
	} else {
		tmp = -z * ((y + x) / y);
	}
	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 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-246)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = (x + y) / ((z - y) / z)
    else
        tmp = -z * ((y + x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-246) || !(t_0 <= 0.0)) {
		tmp = (x + y) / ((z - y) / z);
	} else {
		tmp = -z * ((y + x) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-246) or not (t_0 <= 0.0):
		tmp = (x + y) / ((z - y) / z)
	else:
		tmp = -z * ((y + x) / y)
	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 <= -1e-246) || !(t_0 <= 0.0))
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	else
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -1e-246) || ~((t_0 <= 0.0)))
		tmp = (x + y) / ((z - y) / z);
	else
		tmp = -z * ((y + x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-246], 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[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -9.99999999999999956e-247 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 z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -9.99999999999999956e-247 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 21.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-246} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -1e-246)
     (/ (+ x y) (/ (- z y) z))
     (if (<= t_0 0.0) (* (- z) (/ (+ y x) y)) 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 <= -1e-246) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = -z * ((y + x) / y);
	} 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 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-1d-246)) then
        tmp = (x + y) / ((z - y) / z)
    else if (t_0 <= 0.0d0) then
        tmp = -z * ((y + x) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -1e-246) {
		tmp = (x + y) / ((z - y) / z);
	} else if (t_0 <= 0.0) {
		tmp = -z * ((y + x) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -1e-246:
		tmp = (x + y) / ((z - y) / z)
	elif t_0 <= 0.0:
		tmp = -z * ((y + x) / y)
	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 <= -1e-246)
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -1e-246)
		tmp = (x + y) / ((z - y) / z);
	elseif (t_0 <= 0.0)
		tmp = -z * ((y + x) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-246], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[((-z) * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $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))) < -9.99999999999999956e-247

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -9.99999999999999956e-247 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 21.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

   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. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-246}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+175}:\\ \;\;\;\;\frac{y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -300000.0)
   (* (- z) (/ (+ y x) y))
   (if (<= y 1.2e+97)
     (+ y x)
     (if (<= y 1.35e+175) (/ y (/ (- z y) z)) (- (fma x (/ z y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -300000.0) {
		tmp = -z * ((y + x) / y);
	} else if (y <= 1.2e+97) {
		tmp = y + x;
	} else if (y <= 1.35e+175) {
		tmp = y / ((z - y) / z);
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -300000.0)
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	elseif (y <= 1.2e+97)
		tmp = Float64(y + x);
	elseif (y <= 1.35e+175)
		tmp = Float64(y / Float64(Float64(z - y) / z));
	else
		tmp = Float64(-fma(x, Float64(z / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -300000.0], N[((-z) * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+97], N[(y + x), $MachinePrecision], If[LessEqual[y, 1.35e+175], N[(y / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if y < -3e5

   1. Initial program 68.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 71.0

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

   5. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -3e5 < y < 1.2e97

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if 1.2e97 < y < 1.35e175

   1. Initial program 95.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.5%

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

   if 1.35e175 < y

   1. Initial program 73.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.8

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

   5. Applied rewrites 65.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 93.2

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

   8. Applied rewrites 93.2%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+175}:\\ \;\;\;\;\frac{y}{\frac{z - y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+69} \lor \neg \left(z \leq 7.2 \cdot 10^{-50}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e+69) (not (<= z 7.2e-50))) (+ y x) (- (fma x (/ z y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e+69) || !(z <= 7.2e-50)) {
		tmp = y + x;
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e+69) || !(z <= 7.2e-50))
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(x, Float64(z / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e+69], N[Not[LessEqual[z, 7.2e-50]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -1.3000000000000001e69 or 7.19999999999999958e-50 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -1.3000000000000001e69 < z < 7.19999999999999958e-50

   1. Initial program 80.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 70.8

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

   8. Applied rewrites 70.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+69} \lor \neg \left(z \leq 7.2 \cdot 10^{-50}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+127}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -300000.0)
   (* (- z) (/ (+ y x) y))
   (if (<= y 2.4e+127) (+ y x) (- (fma x (/ z y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -300000.0) {
		tmp = -z * ((y + x) / y);
	} else if (y <= 2.4e+127) {
		tmp = y + x;
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -300000.0)
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	elseif (y <= 2.4e+127)
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(x, Float64(z / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -300000.0], N[((-z) * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e+127], N[(y + x), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -3e5

   1. Initial program 68.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 71.0

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

   5. Applied rewrites 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 78.5

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

   7. Applied rewrites 78.5%

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

   if -3e5 < y < 2.4000000000000002e127

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 2.4000000000000002e127 < y

   1. Initial program 79.0%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 60.4

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 82.2

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

   8. Applied rewrites 82.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -300000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+127}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -156:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-123}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -156.0) (- z) (if (<= y -5e-123) y (if (<= y 1.2e+97) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -156.0) {
		tmp = -z;
	} else if (y <= -5e-123) {
		tmp = y;
	} else if (y <= 1.2e+97) {
		tmp = x;
	} 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 <= (-156.0d0)) then
        tmp = -z
    else if (y <= (-5d-123)) then
        tmp = y
    else if (y <= 1.2d+97) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -156.0) {
		tmp = -z;
	} else if (y <= -5e-123) {
		tmp = y;
	} else if (y <= 1.2e+97) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -156.0:
		tmp = -z
	elif y <= -5e-123:
		tmp = y
	elif y <= 1.2e+97:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -156.0)
		tmp = Float64(-z);
	elseif (y <= -5e-123)
		tmp = y;
	elseif (y <= 1.2e+97)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -156.0)
		tmp = -z;
	elseif (y <= -5e-123)
		tmp = y;
	elseif (y <= 1.2e+97)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -156.0], (-z), If[LessEqual[y, -5e-123], y, If[LessEqual[y, 1.2e+97], x, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -156 or 1.2e97 < y

   1. Initial program 76.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.3

         \[\leadsto -z \]

   5. Applied rewrites 62.3%

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

   if -156 < y < -5.0000000000000003e-123

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

      \[\leadsto y \]

   if -5.0000000000000003e-123 < y < 1.2e97

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 61.8%

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

Alternative 8: 67.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+116} \lor \neg \left(y \leq 3.3 \cdot 10^{+127}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.52e+116) (not (<= y 3.3e+127))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.52e+116) || !(y <= 3.3e+127)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	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 <= (-1.52d+116)) .or. (.not. (y <= 3.3d+127))) then
        tmp = -z
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.52e+116) || !(y <= 3.3e+127)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.52e+116) or not (y <= 3.3e+127):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.52e+116) || !(y <= 3.3e+127))
		tmp = Float64(-z);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.52e+116) || ~((y <= 3.3e+127)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.52e+116], N[Not[LessEqual[y, 3.3e+127]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.52e116 or 3.29999999999999977e127 < y

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.8

         \[\leadsto -z \]

   5. Applied rewrites 77.8%

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

   if -1.52e116 < y < 3.29999999999999977e127

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 71.1

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

   5. Applied rewrites 71.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{+116} \lor \neg \left(y \leq 3.3 \cdot 10^{+127}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-174}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-140) x (if (<= x 1.5e-174) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-140) {
		tmp = x;
	} else if (x <= 1.5e-174) {
		tmp = y;
	} else {
		tmp = x;
	}
	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 (x <= (-2.3d-140)) then
        tmp = x
    else if (x <= 1.5d-174) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-140) {
		tmp = x;
	} else if (x <= 1.5e-174) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e-140:
		tmp = x
	elif x <= 1.5e-174:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-140)
		tmp = x;
	elseif (x <= 1.5e-174)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-140)
		tmp = x;
	elseif (x <= 1.5e-174)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-140], x, If[LessEqual[x, 1.5e-174], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e-140 or 1.50000000000000011e-174 < x

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 47.7%

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

   if -2.3000000000000001e-140 < x < 1.50000000000000011e-174

   1. Initial program 91.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto y \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

      \[\leadsto y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 34.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

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
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 38.3%

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

Developer Target 1: 93.9% 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 2025051 
(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))))