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

Percentage Accurate: 87.9% → 99.2%

Time: 3.7s

Alternatives: 8

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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-243}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\frac{z \cdot x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{\frac{z - y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-243)
     t_0
     (if (<= t_0 0.0) (- (- (/ (* z x) y)) z) (/ (+ x y) (/ (- z y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-243) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -((z * x) / y) - z;
	} else {
		tmp = (x + y) / ((z - y) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-5d-243)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = -((z * x) / y) - z
    else
        tmp = (x + y) / ((z - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-243) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -((z * x) / y) - z;
	} else {
		tmp = (x + y) / ((z - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-243:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = -((z * x) / y) - z
	else:
		tmp = (x + y) / ((z - y) / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -5e-243)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(z * x) / y)) - z);
	else
		tmp = Float64(Float64(x + y) / Float64(Float64(z - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -5e-243)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = -((z * x) / y) - z;
	else
		tmp = (x + y) / ((z - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-243], t$95$0, If[LessEqual[t$95$0, 0.0], N[((-N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]) - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-243

   1. Initial program 99.9%

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

   if -5e-243 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 17.2%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 17.1

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.4

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

   10. Applied rewrites 95.4%

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

   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. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ t_1 := \frac{x + y}{\frac{z - y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-243}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-\frac{z \cdot x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))) (t_1 (/ (+ x y) (/ (- z y) z))))
   (if (<= t_0 -5e-243) t_1 (if (<= t_0 0.0) (- (- (/ (* z x) y)) z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double t_1 = (x + y) / ((z - y) / z);
	double tmp;
	if (t_0 <= -5e-243) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -((z * x) / y) - z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    t_1 = (x + y) / ((z - y) / z)
    if (t_0 <= (-5d-243)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = -((z * x) / y) - z
    else
        tmp = t_1
    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 t_1 = (x + y) / ((z - y) / z);
	double tmp;
	if (t_0 <= -5e-243) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = -((z * x) / y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	t_1 = (x + y) / ((z - y) / z)
	tmp = 0
	if t_0 <= -5e-243:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = -((z * x) / y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(x + y) / Float64(Float64(z - y) / z))
	tmp = 0.0
	if (t_0 <= -5e-243)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(z * x) / y)) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	t_1 = (x + y) / ((z - y) / z);
	tmp = 0.0;
	if (t_0 <= -5e-243)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = -((z * x) / y) - z;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] / N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-243], t$95$1, If[LessEqual[t$95$0, 0.0], N[((-N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]) - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-243 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. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if -5e-243 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 17.2%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 17.1

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.4

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

   10. Applied rewrites 95.4%

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

Alternative 3: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+80}:\\ \;\;\;\;\left(-\frac{z \cdot x}{y}\right) - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+80)
   (- (- (/ (* z x) y)) z)
   (if (<= y 4e-36) (/ x (- 1.0 (/ y z))) (- (fma x (/ z y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+80) {
		tmp = -((z * x) / y) - z;
	} else if (y <= 4e-36) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+80)
		tmp = Float64(Float64(-Float64(Float64(z * x) / y)) - z);
	elseif (y <= 4e-36)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(-fma(x, Float64(z / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+80], N[((-N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]) - z), $MachinePrecision], If[LessEqual[y, 4e-36], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < -1.39999999999999992e80

   1. Initial program 70.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.0

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

   10. Applied rewrites 76.0%

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

   if -1.39999999999999992e80 < y < 3.9999999999999998e-36

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 71.6%

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

   if 3.9999999999999998e-36 < y

   1. Initial program 78.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 61.2

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

   7. Applied rewrites 61.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto -\left(z + \frac{x \cdot z}{y}\right) \]
   9. 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 68.3

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

   10. Applied rewrites 68.3%

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

Alternative 4: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma x (/ z y) z))))
   (if (<= y -1.85e+80) t_0 (if (<= y 4e-36) (/ x (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -fma(x, (z / y), z);
	double tmp;
	if (y <= -1.85e+80) {
		tmp = t_0;
	} else if (y <= 4e-36) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(x, Float64(z / y), z))
	tmp = 0.0
	if (y <= -1.85e+80)
		tmp = t_0;
	elseif (y <= 4e-36)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -1.85e+80], t$95$0, If[LessEqual[y, 4e-36], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.84999999999999998e80 or 3.9999999999999998e-36 < y

   1. Initial program 75.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.8

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -\left(z + \frac{x \cdot z}{y}\right) \]
   9. 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 71.8

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

   10. Applied rewrites 71.8%

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

   if -1.84999999999999998e80 < y < 3.9999999999999998e-36

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 71.6%

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

Alternative 5: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-84}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (fma x (/ z y) z))))
   (if (<= y -7.8e+83) t_0 (if (<= y 1.75e-84) (+ y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -fma(x, (z / y), z);
	double tmp;
	if (y <= -7.8e+83) {
		tmp = t_0;
	} else if (y <= 1.75e-84) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(-fma(x, Float64(z / y), z))
	tmp = 0.0
	if (y <= -7.8e+83)
		tmp = t_0;
	elseif (y <= 1.75e-84)
		tmp = Float64(y + x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -7.8e+83], t$95$0, If[LessEqual[y, 1.75e-84], N[(y + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000003e83 or 1.7500000000000001e-84 < y

   1. Initial program 76.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in z around 0

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

      1. div-add-rev N/A

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

      2. frac-add N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto -\left(z + \frac{x \cdot z}{y}\right) \]
   9. 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 69.4

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

   10. Applied rewrites 69.4%

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

   if -7.8000000000000003e83 < y < 1.7500000000000001e-84

   1. Initial program 99.1%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 72.8

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

   4. Applied rewrites 72.8%

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

Alternative 6: 67.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+128}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+83) (- z) (if (<= y 7.6e+128) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+83) {
		tmp = -z;
	} else if (y <= 7.6e+128) {
		tmp = y + 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 <= (-7.8d+83)) then
        tmp = -z
    else if (y <= 7.6d+128) then
        tmp = y + 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 <= -7.8e+83) {
		tmp = -z;
	} else if (y <= 7.6e+128) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.8e+83:
		tmp = -z
	elif y <= 7.6e+128:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+83)
		tmp = Float64(-z);
	elseif (y <= 7.6e+128)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.8e+83)
		tmp = -z;
	elseif (y <= 7.6e+128)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+83], (-z), If[LessEqual[y, 7.6e+128], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8000000000000003e83 or 7.5999999999999998e128 < y

   1. Initial program 68.1%

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

   2. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.2

         \[\leadsto -z \]

   4. Applied rewrites 70.2%

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

   if -7.8000000000000003e83 < y < 7.5999999999999998e128

   1. Initial program 97.8%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 66.4

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

   4. Applied rewrites 66.4%

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

Alternative 7: 57.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+80) (- z) (if (<= y 4e-36) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+80) {
		tmp = -z;
	} else if (y <= 4e-36) {
		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 <= (-1.4d+80)) then
        tmp = -z
    else if (y <= 4d-36) 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 <= -1.4e+80) {
		tmp = -z;
	} else if (y <= 4e-36) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+80:
		tmp = -z
	elif y <= 4e-36:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+80)
		tmp = Float64(-z);
	elseif (y <= 4e-36)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.4e+80)
		tmp = -z;
	elseif (y <= 4e-36)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+80], (-z), If[LessEqual[y, 4e-36], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999992e80 or 3.9999999999999998e-36 < y

   1. Initial program 75.0%

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

   2. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.8

         \[\leadsto -z \]

   4. Applied rewrites 60.8%

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

   if -1.39999999999999992e80 < y < 3.9999999999999998e-36

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 54.5%

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

Alternative 8: 34.0% accurate, 13.1× 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 87.9%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 34.0%

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

Reproduce

herbie shell --seed 2025115 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64
  (/ (+ x y) (- 1.0 (/ y z))))