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

Percentage Accurate: 89.2% → 99.7%

Time: 3.5s

Alternatives: 10

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-267} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\left(-z\right) - x}{y}, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-267) (not (<= t_0 0.0)))
     t_0
     (fma z (/ (- (- 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 <= -5e-267) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = fma(z, ((-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 <= -5e-267) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = fma(z, Float64(Float64(Float64(-z) - x) / y), Float64(-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, -5e-267], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(N[((-z) - x), $MachinePrecision] / 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))) < -4.9999999999999999e-267 or 0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

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

   if -4.9999999999999999e-267 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 10.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

       1. associate--l+ N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. div-sub N/A

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

       6. pow2 N/A

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

       7. +-commutative N/A

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-267} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\left(-z\right) - x}{y}, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.15e-13)
   (+ y x)
   (if (<= z 6.8e+14) (- (/ (* (- z) x) y) z) (fma (+ 1.0 (/ x z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.15e-13) {
		tmp = y + x;
	} else if (z <= 6.8e+14) {
		tmp = ((-z * x) / y) - z;
	} else {
		tmp = fma((1.0 + (x / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.15e-13)
		tmp = Float64(y + x);
	elseif (z <= 6.8e+14)
		tmp = Float64(Float64(Float64(Float64(-z) * x) / y) - z);
	else
		tmp = fma(Float64(1.0 + Float64(x / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.15e-13], N[(y + x), $MachinePrecision], If[LessEqual[z, 6.8e+14], N[(N[(N[((-z) * x), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 + N[(x / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1499999999999999e-13

   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -2.1499999999999999e-13 < z < 6.8e14

   1. Initial program 71.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if 6.8e14 < 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 71.8

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

   5. Applied rewrites 71.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-13}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-z\right) \cdot x}{y} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y}{z}, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.15e-13)
   (+ y x)
   (if (<= z 6.8e+14) (- (/ (* (- z) x) y) z) (+ (fma y (/ y z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.15e-13) {
		tmp = y + x;
	} else if (z <= 6.8e+14) {
		tmp = ((-z * x) / y) - z;
	} else {
		tmp = fma(y, (y / z), y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.15e-13)
		tmp = Float64(y + x);
	elseif (z <= 6.8e+14)
		tmp = Float64(Float64(Float64(Float64(-z) * x) / y) - z);
	else
		tmp = Float64(fma(y, Float64(y / z), y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.15e-13], N[(y + x), $MachinePrecision], If[LessEqual[z, 6.8e+14], N[(N[(N[((-z) * x), $MachinePrecision] / y), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(y / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1499999999999999e-13

   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -2.1499999999999999e-13 < z < 6.8e14

   1. Initial program 71.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if 6.8e14 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 71.7

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

   8. Applied rewrites 71.7%

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

Alternative 4: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+53}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+14}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y}{z}, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.66e+53)
   (+ y x)
   (if (<= z 6.8e+14) (- (fma x (/ z y) z)) (+ (fma y (/ y z) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.66e+53) {
		tmp = y + x;
	} else if (z <= 6.8e+14) {
		tmp = -fma(x, (z / y), z);
	} else {
		tmp = fma(y, (y / z), y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.66e+53)
		tmp = Float64(y + x);
	elseif (z <= 6.8e+14)
		tmp = Float64(-fma(x, Float64(z / y), z));
	else
		tmp = Float64(fma(y, Float64(y / z), y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.66e+53], N[(y + x), $MachinePrecision], If[LessEqual[z, 6.8e+14], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision]), N[(N[(y * N[(y / z), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.65999999999999999e53

   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 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.65999999999999999e53 < z < 6.8e14

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.6

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

   8. Applied rewrites 80.6%

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

   if 6.8e14 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in x around 0

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

      1. lift-/.f64 71.7

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

   8. Applied rewrites 71.7%

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

Alternative 5: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+53} \lor \neg \left(z \leq 6.8 \cdot 10^{+14}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.66e+53) (not (<= z 6.8e+14))) (+ y x) (- (fma x (/ z y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.66e+53) || !(z <= 6.8e+14)) {
		tmp = y + x;
	} else {
		tmp = -fma(x, (z / y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.66e+53) || !(z <= 6.8e+14))
		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, -1.66e+53], N[Not[LessEqual[z, 6.8e+14]], $MachinePrecision]], N[(y + x), $MachinePrecision], (-N[(x * N[(z / y), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -1.65999999999999999e53 or 6.8e14 < 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 77.8

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

   5. Applied rewrites 77.8%

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

   if -1.65999999999999999e53 < z < 6.8e14

   1. Initial program 73.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 80.6

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

   8. Applied rewrites 80.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.66 \cdot 10^{+53} \lor \neg \left(z \leq 6.8 \cdot 10^{+14}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \frac{z}{y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{z \cdot x}{-y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+34)
   (- z)
   (if (<= y -1.9e-58) (/ (* z x) (- y)) (if (<= y 8.2e+51) (+ y x) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+34) {
		tmp = -z;
	} else if (y <= -1.9e-58) {
		tmp = (z * x) / -y;
	} else if (y <= 8.2e+51) {
		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 <= (-5.2d+34)) then
        tmp = -z
    else if (y <= (-1.9d-58)) then
        tmp = (z * x) / -y
    else if (y <= 8.2d+51) 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 <= -5.2e+34) {
		tmp = -z;
	} else if (y <= -1.9e-58) {
		tmp = (z * x) / -y;
	} else if (y <= 8.2e+51) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e+34:
		tmp = -z
	elif y <= -1.9e-58:
		tmp = (z * x) / -y
	elif y <= 8.2e+51:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e+34)
		tmp = Float64(-z);
	elseif (y <= -1.9e-58)
		tmp = Float64(Float64(z * x) / Float64(-y));
	elseif (y <= 8.2e+51)
		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 <= -5.2e+34)
		tmp = -z;
	elseif (y <= -1.9e-58)
		tmp = (z * x) / -y;
	elseif (y <= 8.2e+51)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e+34], (-z), If[LessEqual[y, -1.9e-58], N[(N[(z * x), $MachinePrecision] / (-y)), $MachinePrecision], If[LessEqual[y, 8.2e+51], N[(y + x), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999995e34 or 8.20000000000000021e51 < y

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.6

         \[\leadsto -z \]

   5. Applied rewrites 68.6%

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

   if -5.19999999999999995e34 < y < -1.8999999999999999e-58

   1. Initial program 95.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 50.1

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

   8. Applied rewrites 50.1%

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

   if -1.8999999999999999e-58 < y < 8.20000000000000021e51

   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 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{z \cdot x}{-y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{y} \cdot z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+51}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.2e+34)
   (- z)
   (if (<= y -1.9e-58) (* (/ (- x) y) z) (if (<= y 8.2e+51) (+ y x) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.2e+34) {
		tmp = -z;
	} else if (y <= -1.9e-58) {
		tmp = (-x / y) * z;
	} else if (y <= 8.2e+51) {
		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 <= (-5.2d+34)) then
        tmp = -z
    else if (y <= (-1.9d-58)) then
        tmp = (-x / y) * z
    else if (y <= 8.2d+51) 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 <= -5.2e+34) {
		tmp = -z;
	} else if (y <= -1.9e-58) {
		tmp = (-x / y) * z;
	} else if (y <= 8.2e+51) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.2e+34:
		tmp = -z
	elif y <= -1.9e-58:
		tmp = (-x / y) * z
	elif y <= 8.2e+51:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.2e+34)
		tmp = Float64(-z);
	elseif (y <= -1.9e-58)
		tmp = Float64(Float64(Float64(-x) / y) * z);
	elseif (y <= 8.2e+51)
		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 <= -5.2e+34)
		tmp = -z;
	elseif (y <= -1.9e-58)
		tmp = (-x / y) * z;
	elseif (y <= 8.2e+51)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.2e+34], (-z), If[LessEqual[y, -1.9e-58], N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 8.2e+51], N[(y + x), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999995e34 or 8.20000000000000021e51 < y

   1. Initial program 67.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.6

         \[\leadsto -z \]

   5. Applied rewrites 68.6%

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

   if -5.19999999999999995e34 < y < -1.8999999999999999e-58

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 71.5

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

   8. Applied rewrites 71.5%

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

   9. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac N/A

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

       3. lower-/.f64 N/A

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

       4. lower-neg.f64 46.6

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

   11. Applied rewrites 46.6%

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

   if -1.8999999999999999e-58 < y < 8.20000000000000021e51

   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 68.6

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

   5. Applied rewrites 68.6%

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

Alternative 8: 68.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+67} \lor \neg \left(y \leq 8.2 \cdot 10^{+51}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.7e+67) (not (<= y 8.2e+51))) (- z) (+ y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.7e+67) || !(y <= 8.2e+51)) {
		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 <= (-3.7d+67)) .or. (.not. (y <= 8.2d+51))) 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 <= -3.7e+67) || !(y <= 8.2e+51)) {
		tmp = -z;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.7e+67) or not (y <= 8.2e+51):
		tmp = -z
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.7e+67) || !(y <= 8.2e+51))
		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 <= -3.7e+67) || ~((y <= 8.2e+51)))
		tmp = -z;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.7e+67], N[Not[LessEqual[y, 8.2e+51]], $MachinePrecision]], (-z), N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6999999999999997e67 or 8.20000000000000021e51 < y

   1. Initial program 65.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 70.8

         \[\leadsto -z \]

   5. Applied rewrites 70.8%

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

   if -3.6999999999999997e67 < y < 8.20000000000000021e51

   1. Initial program 98.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 60.0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+67} \lor \neg \left(y \leq 8.2 \cdot 10^{+51}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-50} \lor \neg \left(y \leq 8.8 \cdot 10^{-89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-50) (not (<= y 8.8e-89))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-50) || !(y <= 8.8e-89)) {
		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 <= (-2.2d-50)) .or. (.not. (y <= 8.8d-89))) 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 <= -2.2e-50) || !(y <= 8.8e-89)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e-50) or not (y <= 8.8e-89):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-50) || !(y <= 8.8e-89))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e-50) || ~((y <= 8.8e-89)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-50], N[Not[LessEqual[y, 8.8e-89]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -2.1999999999999999e-50 or 8.80000000000000048e-89 < y

   1. Initial program 76.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 55.9

         \[\leadsto -z \]

   5. Applied rewrites 55.9%

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

   if -2.1999999999999999e-50 < y < 8.80000000000000048e-89

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 62.8%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-50} \lor \neg \left(y \leq 8.8 \cdot 10^{-89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.9% 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 83.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 27.4%

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

Reproduce

herbie shell --seed 2025085 
(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))))