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

Percentage Accurate: 87.9% → 99.8%

Time: 3.1s

Alternatives: 9

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-294) (not (<= t_0 0.0)))
     t_0
     (fma -1.0 z (/ (+ (* z x) (* z z)) (- y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-294) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = fma(-1.0, z, (((z * x) + (z * z)) / -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 <= -5e-294) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = fma(-1.0, z, Float64(Float64(Float64(z * x) + Float64(z * z)) / Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-294], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(-1.0 * z + N[(N[(N[(z * x), $MachinePrecision] + N[(z * z), $MachinePrecision]), $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))) < -5.0000000000000003e-294 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

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

   1. Initial program 5.9%

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

   3. Taylor expanded in y around inf

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

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-294) (not (<= t_0 0.0))) t_0 (* (- 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 <= -5e-294) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} 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 <= (-5d-294)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    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 <= -5e-294) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} 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 <= -5e-294) or not (t_0 <= 0.0):
		tmp = t_0
	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 <= -5e-294) || !(t_0 <= 0.0))
		tmp = t_0;
	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 <= -5e-294) || ~((t_0 <= 0.0)))
		tmp = t_0;
	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, -5e-294], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, 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))) < -5.0000000000000003e-294 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

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

   1. Initial program 5.9%

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

   3. Taylor expanded in 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{-\frac{\left(y + x\right) \cdot z}{y}} \]
   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 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.8%

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

Alternative 3: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 7000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+20) (not (<= z 7000000000.0)))
   (+ y x)
   (* (- z) (/ (+ y x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+20) || !(z <= 7000000000.0)) {
		tmp = y + x;
	} 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) :: tmp
    if ((z <= (-5.2d+20)) .or. (.not. (z <= 7000000000.0d0))) then
        tmp = y + x
    else
        tmp = -z * ((y + x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+20) || !(z <= 7000000000.0)) {
		tmp = y + x;
	} else {
		tmp = -z * ((y + x) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.2e+20) or not (z <= 7000000000.0):
		tmp = y + x
	else:
		tmp = -z * ((y + x) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+20) || !(z <= 7000000000.0))
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.2e+20) || ~((z <= 7000000000.0)))
		tmp = y + x;
	else
		tmp = -z * ((y + x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e+20], N[Not[LessEqual[z, 7000000000.0]], $MachinePrecision]], N[(y + x), $MachinePrecision], N[((-z) * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e20 or 7e9 < 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -5.2e20 < z < 7e9

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(x + y\right)}{y}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{-\frac{\left(y + x\right) \cdot z}{y}} \]
   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 78.4

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

   7. Applied rewrites 78.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 7000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 7000000000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2e+20)
   (fma (+ 1.0 (/ x z)) y x)
   (if (<= z 7000000000.0) (* (- z) (/ (+ y x) y)) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2e+20) {
		tmp = fma((1.0 + (x / z)), y, x);
	} else if (z <= 7000000000.0) {
		tmp = -z * ((y + x) / y);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2e+20)
		tmp = fma(Float64(1.0 + Float64(x / z)), y, x);
	elseif (z <= 7000000000.0)
		tmp = Float64(Float64(-z) * Float64(Float64(y + x) / y));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2e+20], N[(N[(1.0 + N[(x / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 7000000000.0], N[((-z) * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e20

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 80.2

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

   5. Applied rewrites 80.2%

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

   if -5.2e20 < z < 7e9

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(x + y\right)}{y}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{-\frac{\left(y + x\right) \cdot z}{y}} \]
   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 78.4

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

   7. Applied rewrites 78.4%

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

   if 7e9 < 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 83.4

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

   5. Applied rewrites 83.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 7000000000:\\ \;\;\;\;\left(-z\right) \cdot \frac{y + x}{y}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 7000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+20) (not (<= z 7000000000.0)))
   (+ y x)
   (- (fma x (/ z y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+20) || !(z <= 7000000000.0)) {
		tmp = y + x;
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+20) || !(z <= 7000000000.0))
		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, -5.2e+20], N[Not[LessEqual[z, 7000000000.0]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e20 or 7e9 < 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -5.2e20 < z < 7e9

   1. Initial program 72.4%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(x + y\right)}{y}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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 75.6

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

   8. Applied rewrites 75.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+20} \lor \neg \left(z \leq 7000000000\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+185} \lor \neg \left(y \leq 6.4 \cdot 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.5e+185) (not (<= y 6.4e+55))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.5e+185) || !(y <= 6.4e+55)) {
		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 <= (-5.5d+185)) .or. (.not. (y <= 6.4d+55))) 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 <= -5.5e+185) || !(y <= 6.4e+55)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.5e+185) or not (y <= 6.4e+55):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.5e+185) || !(y <= 6.4e+55))
		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 <= -5.5e+185) || ~((y <= 6.4e+55)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.5e+185], N[Not[LessEqual[y, 6.4e+55]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999996e185 or 6.4000000000000005e55 < y

   1. Initial program 64.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 67.0

         \[\leadsto -z \]

   5. Applied rewrites 67.0%

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

   if -5.4999999999999996e185 < y < 6.4000000000000005e55

   1. Initial program 97.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+185} \lor \neg \left(y \leq 6.4 \cdot 10^{+55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+27} \lor \neg \left(y \leq 2.1 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.4e+27) (not (<= y 2.1e-52))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+27) || !(y <= 2.1e-52)) {
		tmp = -z;
	} 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 ((y <= (-3.4d+27)) .or. (.not. (y <= 2.1d-52))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+27) || !(y <= 2.1e-52)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.4e+27) or not (y <= 2.1e-52):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.4e+27) || !(y <= 2.1e-52))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.4e+27) || ~((y <= 2.1e-52)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.4e+27], N[Not[LessEqual[y, 2.1e-52]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e27 or 2.0999999999999999e-52 < y

   1. Initial program 75.4%

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

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

      2. lower-neg.f64 53.7

         \[\leadsto -z \]

   5. Applied rewrites 53.7%

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

   if -3.4e27 < y < 2.0999999999999999e-52

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 61.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+27} \lor \neg \left(y \leq 2.1 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-126}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.35e-184) x (if (<= x 1.15e-126) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.35e-184) {
		tmp = x;
	} else if (x <= 1.15e-126) {
		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 <= (-2.35d-184)) then
        tmp = x
    else if (x <= 1.15d-126) 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 <= -2.35e-184) {
		tmp = x;
	} else if (x <= 1.15e-126) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.35e-184:
		tmp = x
	elif x <= 1.15e-126:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.35e-184)
		tmp = x;
	elseif (x <= 1.15e-126)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.35e-184)
		tmp = x;
	elseif (x <= 1.15e-126)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.35e-184], x, If[LessEqual[x, 1.15e-126], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3500000000000001e-184 or 1.15000000000000005e-126 < x

   1. Initial program 86.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 43.5%

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

   if -2.3500000000000001e-184 < x < 1.15000000000000005e-126

   1. Initial program 89.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.3%

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

Alternative 9: 34.6% 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 87.4%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 35.4%

   \[\leadsto \color{blue}{x} \]
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 2025057 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

  (/ (+ x y) (- 1.0 (/ y z))))