Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Percentage Accurate: 99.9% → 100.0%

Time: 5.5s

Alternatives: 8

Speedup: 1.1×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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 + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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 + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z 5.0 (* (+ z y) x)))

C

double code(double x, double y, double z) {
	return fma(z, 5.0, ((z + y) * x));
}

Julia

function code(x, y, z)
	return fma(z, 5.0, Float64(Float64(z + y) * x))
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0 + N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

   4. lower-fma.f64 100.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

   7. lower-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

   8. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   10. lower-+.f64 100.0

       \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
5. Add Preprocessing

Alternative 2: 60.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+154}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-26}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+194}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.9e+154)
   (* x z)
   (if (<= x -6e-26)
     (* y x)
     (if (<= x 2.7e-95) (* 5.0 z) (if (<= x 4.5e+194) (* y x) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e+154) {
		tmp = x * z;
	} else if (x <= -6e-26) {
		tmp = y * x;
	} else if (x <= 2.7e-95) {
		tmp = 5.0 * z;
	} else if (x <= 4.5e+194) {
		tmp = y * x;
	} else {
		tmp = x * 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 (x <= (-2.9d+154)) then
        tmp = x * z
    else if (x <= (-6d-26)) then
        tmp = y * x
    else if (x <= 2.7d-95) then
        tmp = 5.0d0 * z
    else if (x <= 4.5d+194) then
        tmp = y * x
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e+154) {
		tmp = x * z;
	} else if (x <= -6e-26) {
		tmp = y * x;
	} else if (x <= 2.7e-95) {
		tmp = 5.0 * z;
	} else if (x <= 4.5e+194) {
		tmp = y * x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.9e+154:
		tmp = x * z
	elif x <= -6e-26:
		tmp = y * x
	elif x <= 2.7e-95:
		tmp = 5.0 * z
	elif x <= 4.5e+194:
		tmp = y * x
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.9e+154)
		tmp = Float64(x * z);
	elseif (x <= -6e-26)
		tmp = Float64(y * x);
	elseif (x <= 2.7e-95)
		tmp = Float64(5.0 * z);
	elseif (x <= 4.5e+194)
		tmp = Float64(y * x);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.9e+154)
		tmp = x * z;
	elseif (x <= -6e-26)
		tmp = y * x;
	elseif (x <= 2.7e-95)
		tmp = 5.0 * z;
	elseif (x <= 4.5e+194)
		tmp = y * x;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.9e+154], N[(x * z), $MachinePrecision], If[LessEqual[x, -6e-26], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.7e-95], N[(5.0 * z), $MachinePrecision], If[LessEqual[x, 4.5e+194], N[(y * x), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999979e154 or 4.4999999999999998e194 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. lower-+.f64 66.8

         \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 66.8%

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

   if -2.89999999999999979e154 < x < -6.00000000000000023e-26 or 2.7e-95 < x < 4.4999999999999998e194

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -6.00000000000000023e-26 < x < 2.7e-95

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.0

         \[\leadsto \color{blue}{5 \cdot z} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26} \lor \neg \left(x \leq 2.7 \cdot 10^{-95}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6e-26) (not (<= x 2.7e-95)))
   (* (+ y z) x)
   (fma z 5.0 (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-26) || !(x <= 2.7e-95)) {
		tmp = (y + z) * x;
	} else {
		tmp = fma(z, 5.0, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6e-26) || !(x <= 2.7e-95))
		tmp = Float64(Float64(y + z) * x);
	else
		tmp = fma(z, 5.0, Float64(x * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6e-26], N[Not[LessEqual[x, 2.7e-95]], $MachinePrecision]], N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision], N[(z * 5.0 + N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000023e-26 or 2.7e-95 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + z \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \color{blue}{z \cdot 5}\right) + y \cdot x \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

      11. lower-*.f64 98.0

          \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{y \cdot x}\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

   5. Taylor expanded in x around -inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 93.9

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 93.9

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

   10. Applied rewrites 93.9%

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

   if -6.00000000000000023e-26 < x < 2.7e-95

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. lower-+.f64 79.0

         \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.0%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{5}, x \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26} \lor \neg \left(x \leq 2.7 \cdot 10^{-95}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6e-26)
   (fma z x (* x y))
   (if (<= x 2.7e-95) (fma z 5.0 (* x z)) (* (+ y z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6e-26) {
		tmp = fma(z, x, (x * y));
	} else if (x <= 2.7e-95) {
		tmp = fma(z, 5.0, (x * z));
	} else {
		tmp = (y + z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6e-26)
		tmp = fma(z, x, Float64(x * y));
	elseif (x <= 2.7e-95)
		tmp = fma(z, 5.0, Float64(x * z));
	else
		tmp = Float64(Float64(y + z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6e-26], N[(z * x + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-95], N[(z * 5.0 + N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000023e-26

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + z \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \color{blue}{z \cdot 5}\right) + y \cdot x \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

      11. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{y \cdot x}\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

   5. Taylor expanded in x around -inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 98.1

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.2%

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

   if -6.00000000000000023e-26 < x < 2.7e-95

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. lower-+.f64 79.0

         \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.0%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{5}, x \cdot z\right) \]

   if 2.7e-95 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + z \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \color{blue}{z \cdot 5}\right) + y \cdot x \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

      11. lower-*.f64 96.4

          \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{y \cdot x}\right) \]

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

   5. Taylor expanded in x around -inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 90.4

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

   10. Applied rewrites 90.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x \cdot y\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26} \lor \neg \left(x \leq 2.7 \cdot 10^{-95}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6e-26) (not (<= x 2.7e-95))) (* (+ y z) x) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-26) || !(x <= 2.7e-95)) {
		tmp = (y + z) * x;
	} else {
		tmp = 5.0 * 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 ((x <= (-6d-26)) .or. (.not. (x <= 2.7d-95))) then
        tmp = (y + z) * x
    else
        tmp = 5.0d0 * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6e-26) || !(x <= 2.7e-95)) {
		tmp = (y + z) * x;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6e-26) or not (x <= 2.7e-95):
		tmp = (y + z) * x
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6e-26) || !(x <= 2.7e-95))
		tmp = Float64(Float64(y + z) * x);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -6e-26) || ~((x <= 2.7e-95)))
		tmp = (y + z) * x;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6e-26], N[Not[LessEqual[x, 2.7e-95]], $MachinePrecision]], N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.00000000000000023e-26 or 2.7e-95 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + z \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      6. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(z \cdot x + \color{blue}{z \cdot 5}\right) + y \cdot x \]

      8. distribute-lft-out N/A

         \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

      11. lower-*.f64 98.0

          \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{y \cdot x}\right) \]

   4. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

   5. Taylor expanded in x around -inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 93.9

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-*l* N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 93.9

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

   10. Applied rewrites 93.9%

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

   if -6.00000000000000023e-26 < x < 2.7e-95

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.0

         \[\leadsto \color{blue}{5 \cdot z} \]

   5. Applied rewrites 79.0%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-26} \lor \neg \left(x \leq 2.7 \cdot 10^{-95}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-122} \lor \neg \left(z \leq 3.2 \cdot 10^{-96}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.8e-122) (not (<= z 3.2e-96))) (* (+ 5.0 x) z) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.8e-122) || !(z <= 3.2e-96)) {
		tmp = (5.0 + x) * 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 ((z <= (-6.8d-122)) .or. (.not. (z <= 3.2d-96))) then
        tmp = (5.0d0 + x) * 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 ((z <= -6.8e-122) || !(z <= 3.2e-96)) {
		tmp = (5.0 + x) * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.8e-122) or not (z <= 3.2e-96):
		tmp = (5.0 + x) * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.8e-122) || !(z <= 3.2e-96))
		tmp = Float64(Float64(5.0 + x) * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.8e-122) || ~((z <= 3.2e-96)))
		tmp = (5.0 + x) * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.8e-122], N[Not[LessEqual[z, 3.2e-96]], $MachinePrecision]], N[(N[(5.0 + x), $MachinePrecision] * z), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7999999999999996e-122 or 3.20000000000000012e-96 < z

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. lower-+.f64 78.7

         \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

   5. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

   if -6.7999999999999996e-122 < z < 3.20000000000000012e-96

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-122} \lor \neg \left(z \leq 3.2 \cdot 10^{-96}\right):\\ \;\;\;\;\left(5 + x\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 3500\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 3500.0))) (* x z) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 3500.0)) {
		tmp = x * z;
	} else {
		tmp = 5.0 * 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 ((x <= (-5.0d0)) .or. (.not. (x <= 3500.0d0))) then
        tmp = x * z
    else
        tmp = 5.0d0 * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.0) || !(x <= 3500.0)) {
		tmp = x * z;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.0) or not (x <= 3500.0):
		tmp = x * z
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 3500.0))
		tmp = Float64(x * z);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.0) || ~((x <= 3500.0)))
		tmp = x * z;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 3500.0]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 3500 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. lower-+.f64 49.0

         \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 48.4%

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

   if -5 < x < 3500

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 71.6

         \[\leadsto \color{blue}{5 \cdot z} \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 3500\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x * 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 * z
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(x * z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x * z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

   4. lower-+.f64 60.8

      \[\leadsto \color{blue}{\left(5 + x\right)} \cdot z \]

5. Applied rewrites 60.8%

   \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 26.2%

   \[\leadsto x \cdot \color{blue}{z} \]
9. Add Preprocessing

Developer Target 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 5\right) \cdot z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (+ x 5.0) z) (* x y)))

C

double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

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 + 5.0d0) * z) + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Python

def code(x, y, z):
	return ((x + 5.0) * z) + (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + 5.0) * z) + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + 5.0) * z) + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + 5.0), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024363 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (+ x 5) z) (* x y)))

  (+ (* x (+ y z)) (* z 5.0)))