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

Percentage Accurate: 88.1% → 99.7%

Time: 7.5s

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: 88.1% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-262} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -4e-262) (not (<= t_0 0.0))) t_0 (- (fma z (/ x y) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -4e-262) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -fma(z, (x / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -4e-262) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(-fma(z, Float64(x / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e-262], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.00000000000000005e-262 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

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

   if -4.00000000000000005e-262 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 10.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. frac-2neg N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 19.8

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

   7. Applied rewrites 19.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto -z \]
   9. Step-by-step derivation

   10. Applied rewrites 80.5%

       \[\leadsto -z \]

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 99.9

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

   13. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -4 \cdot 10^{-262} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-307}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e-54)
   (+ y x)
   (if (<= z -3.15e-307)
     (- (fma x (/ z y) z))
     (if (<= z 3.9e+35) (- (fma z (/ x y) z)) (+ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-54) {
		tmp = y + x;
	} else if (z <= -3.15e-307) {
		tmp = -fma(x, (z / y), z);
	} else if (z <= 3.9e+35) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e-54)
		tmp = Float64(y + x);
	elseif (z <= -3.15e-307)
		tmp = Float64(-fma(x, Float64(z / y), z));
	elseif (z <= 3.9e+35)
		tmp = Float64(-fma(z, Float64(x / y), z));
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e-54], N[(y + x), $MachinePrecision], If[LessEqual[z, -3.15e-307], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[z, 3.9e+35], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999e-54 or 3.8999999999999999e35 < 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 82.3

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

   5. Applied rewrites 82.3%

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

   if -1.1499999999999999e-54 < z < -3.1500000000000002e-307

   1. Initial program 77.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. frac-2neg N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 45.1%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 45.3

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

   7. Applied rewrites 45.3%

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

   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 83.9

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

   10. Applied rewrites 83.9%

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

   if -3.1500000000000002e-307 < z < 3.8999999999999999e35

   1. Initial program 69.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. frac-2neg N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 38.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 34.4

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

   7. Applied rewrites 34.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto -z \]
   9. Step-by-step derivation

   10. Applied rewrites 58.6%

       \[\leadsto -z \]

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 77.6

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

   13. Applied rewrites 77.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-307}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54} \lor \neg \left(z \leq 5 \cdot 10^{-26}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e-54) (not (<= z 5e-26))) (+ y x) (/ (* (+ y x) z) (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-54) || !(z <= 5e-26)) {
		tmp = y + x;
	} else {
		tmp = ((y + x) * z) / -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 <= (-1.15d-54)) .or. (.not. (z <= 5d-26))) then
        tmp = y + x
    else
        tmp = ((y + x) * z) / -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-54) || !(z <= 5e-26)) {
		tmp = y + x;
	} else {
		tmp = ((y + x) * z) / -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e-54) or not (z <= 5e-26):
		tmp = y + x
	else:
		tmp = ((y + x) * z) / -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e-54) || !(z <= 5e-26))
		tmp = Float64(y + x);
	else
		tmp = Float64(Float64(Float64(y + x) * z) / Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e-54) || ~((z <= 5e-26)))
		tmp = y + x;
	else
		tmp = ((y + x) * z) / -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e-54], N[Not[LessEqual[z, 5e-26]], $MachinePrecision]], N[(y + x), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] / (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999e-54 or 5.00000000000000019e-26 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -1.1499999999999999e-54 < z < 5.00000000000000019e-26

   1. Initial program 71.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 81.2

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

   5. Applied rewrites 81.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54} \lor \neg \left(z \leq 5 \cdot 10^{-26}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z}{-y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.15e-54)
   (+ y x)
   (if (<= z 3.9e+35) (/ (* (+ y x) z) (- y)) (fma (+ 1.0 (/ x z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.15e-54) {
		tmp = y + x;
	} else if (z <= 3.9e+35) {
		tmp = ((y + x) * z) / -y;
	} else {
		tmp = fma((1.0 + (x / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.15e-54)
		tmp = Float64(y + x);
	elseif (z <= 3.9e+35)
		tmp = Float64(Float64(Float64(y + x) * z) / Float64(-y));
	else
		tmp = fma(Float64(1.0 + Float64(x / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.15e-54], N[(y + x), $MachinePrecision], If[LessEqual[z, 3.9e+35], N[(N[(N[(y + x), $MachinePrecision] * z), $MachinePrecision] / (-y)), $MachinePrecision], N[(N[(1.0 + N[(x / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999e-54

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.1499999999999999e-54 < z < 3.8999999999999999e35

   1. Initial program 73.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 79.4

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

   5. Applied rewrites 79.4%

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

   if 3.8999999999999999e35 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(1 - -1 \cdot \frac{x}{z}\right)} \]
   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 87.1

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

   5. Applied rewrites 87.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot z}{-y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54} \lor \neg \left(z \leq 5 \cdot 10^{-26}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e-54) (not (<= z 5e-26))) (+ y x) (- (fma x (/ z y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-54) || !(z <= 5e-26)) {
		tmp = y + x;
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e-54) || !(z <= 5e-26))
		tmp = Float64(y + x);
	else
		tmp = Float64(-fma(x, Float64(z / y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e-54], N[Not[LessEqual[z, 5e-26]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999e-54 or 5.00000000000000019e-26 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -1.1499999999999999e-54 < z < 5.00000000000000019e-26

   1. Initial program 71.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. +-commutative N/A

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

      7. frac-2neg N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lift--.f64 N/A

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

   4. Applied rewrites 39.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 40.5

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

   7. Applied rewrites 40.5%

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

   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 79.0

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

   10. Applied rewrites 79.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-54} \lor \neg \left(z \leq 5 \cdot 10^{-26}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+58} \lor \neg \left(y \leq 8.2 \cdot 10^{+39}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3e+58) (not (<= y 8.2e+39))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3e+58) || !(y <= 8.2e+39)) {
		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 <= (-3d+58)) .or. (.not. (y <= 8.2d+39))) 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 <= -3e+58) || !(y <= 8.2e+39)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3e+58) or not (y <= 8.2e+39):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3e+58) || !(y <= 8.2e+39))
		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 <= -3e+58) || ~((y <= 8.2e+39)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3e+58], N[Not[LessEqual[y, 8.2e+39]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000002e58 or 8.20000000000000008e39 < y

   1. Initial program 67.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 69.5

         \[\leadsto -z \]

   5. Applied rewrites 69.5%

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

   if -3.0000000000000002e58 < y < 8.20000000000000008e39

   1. Initial program 98.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+58} \lor \neg \left(y \leq 8.2 \cdot 10^{+39}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+40} \lor \neg \left(y \leq 9.8 \cdot 10^{-40}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.02e+40) (not (<= y 9.8e-40))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.02e+40) || !(y <= 9.8e-40)) {
		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 <= (-1.02d+40)) .or. (.not. (y <= 9.8d-40))) 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 <= -1.02e+40) || !(y <= 9.8e-40)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.02e+40) or not (y <= 9.8e-40):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.02e+40) || !(y <= 9.8e-40))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.02e+40) || ~((y <= 9.8e-40)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.02e+40], N[Not[LessEqual[y, 9.8e-40]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e40 or 9.7999999999999995e-40 < y

   1. Initial program 73.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.7

         \[\leadsto -z \]

   5. Applied rewrites 62.7%

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

   if -1.02e40 < y < 9.7999999999999995e-40

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 63.8%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+40} \lor \neg \left(y \leq 9.8 \cdot 10^{-40}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-148}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.3e-225) x (if (<= x 2.55e-148) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-225) {
		tmp = x;
	} else if (x <= 2.55e-148) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.3d-225)) then
        tmp = x
    else if (x <= 2.55d-148) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.3e-225) {
		tmp = x;
	} else if (x <= 2.55e-148) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.3e-225:
		tmp = x
	elif x <= 2.55e-148:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.3e-225)
		tmp = x;
	elseif (x <= 2.55e-148)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.3e-225)
		tmp = x;
	elseif (x <= 2.55e-148)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.3e-225], x, If[LessEqual[x, 2.55e-148], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-225 or 2.55e-148 < x

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 42.9%

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

   if -2.2999999999999999e-225 < x < 2.55e-148

   1. Initial program 85.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 43.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-225}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{-148}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.5% 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 86.3%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 38.2%

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

6. Final simplification 38.2%

   \[\leadsto x \]
7. Add Preprocessing

Developer Target 1: 93.8% 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 2025043 
(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))))