System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Percentage Accurate: 99.9% → 100.0%

Time: 4.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(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 * 0.5d0) + (y * ((1.0d0 - z) + log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + Math.log(z)));
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * ((1.0 - z) + math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(Float64(1.0 - z) + log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * ((1.0 - z) + log(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \left(\mathsf{expm1}\left(-\log z\right) \cdot z\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* x 0.5) (fma (log z) y (* (* (expm1 (- (log z))) z) y))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + fma(log(z), y, ((expm1(-log(z)) * z) * y));
}

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + fma(log(z), y, Float64(Float64(expm1(Float64(-log(z))) * z) * y)))
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y + N[(N[(N[(Exp[(-N[Log[z], $MachinePrecision])] - 1), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 99.9

       \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \color{blue}{\left(1 - z\right)} \cdot y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\mathsf{fma}\left(\log z, y, \left(1 - z\right) \cdot y\right)} \]

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. *-commutative N/A

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

   6. lower-expm1.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \mathsf{fma}\left(\log z, y, \left(\mathsf{expm1}\left(-1 \cdot \log z\right) \cdot z\right) \cdot y\right) \]

   7. mul-1-neg N/A

      \[\leadsto x \cdot \frac{1}{2} + \mathsf{fma}\left(\log z, y, \left(\mathsf{expm1}\left(\mathsf{neg}\left(\log z\right)\right) \cdot z\right) \cdot y\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto x \cdot \frac{1}{2} + \mathsf{fma}\left(\log z, y, \left(\mathsf{expm1}\left(-\log z\right) \cdot z\right) \cdot y\right) \]

   9. lift-log.f64 100.0

      \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \left(\mathsf{expm1}\left(-\log z\right) \cdot z\right) \cdot y\right) \]

7. Applied rewrites 100.0%

   \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \color{blue}{\left(\mathsf{expm1}\left(-\log z\right) \cdot z\right)} \cdot y\right) \]
8. Add Preprocessing

Alternative 2: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(1 - z\right) + \log z\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+95} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (- 1.0 z) (log z)))))
   (if (or (<= t_0 -5e+95) (not (<= t_0 2e+101))) (* (- y) z) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((1.0 - z) + log(z));
	double tmp;
	if ((t_0 <= -5e+95) || !(t_0 <= 2e+101)) {
		tmp = -y * z;
	} else {
		tmp = 0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = y * ((1.0d0 - z) + log(z))
    if ((t_0 <= (-5d+95)) .or. (.not. (t_0 <= 2d+101))) then
        tmp = -y * z
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * ((1.0 - z) + Math.log(z));
	double tmp;
	if ((t_0 <= -5e+95) || !(t_0 <= 2e+101)) {
		tmp = -y * z;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((1.0 - z) + math.log(z))
	tmp = 0
	if (t_0 <= -5e+95) or not (t_0 <= 2e+101):
		tmp = -y * z
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(1.0 - z) + log(z)))
	tmp = 0.0
	if ((t_0 <= -5e+95) || !(t_0 <= 2e+101))
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((1.0 - z) + log(z));
	tmp = 0.0;
	if ((t_0 <= -5e+95) || ~((t_0 <= 2e+101)))
		tmp = -y * z;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(N[(1.0 - z), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+95], N[Not[LessEqual[t$95$0, 2e+101]], $MachinePrecision]], N[((-y) * z), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -5.00000000000000025e95 or 2e101 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z)))

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.0

         \[\leadsto \left(-y\right) \cdot z \]

   5. Applied rewrites 64.0%

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

   if -5.00000000000000025e95 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 2e101

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 70.9

         \[\leadsto 0.5 \cdot \color{blue}{x} \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(\left(1 - z\right) + \log z\right) \leq -5 \cdot 10^{+95} \lor \neg \left(y \cdot \left(\left(1 - z\right) + \log z\right) \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \left(1 - z\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* x 0.5) (fma (log z) y (* (- 1.0 z) y))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + fma(log(z), y, ((1.0 - z) * y));
}

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + fma(log(z), y, Float64(Float64(1.0 - z) * y)))
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] * y + N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift--.f64 99.9

       \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(\log z, y, \color{blue}{\left(1 - z\right)} \cdot y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\mathsf{fma}\left(\log z, y, \left(1 - z\right) \cdot y\right)} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70000000 \lor \neg \left(y \leq 2.6 \cdot 10^{+92}\right):\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -70000000.0) (not (<= y 2.6e+92)))
   (fma (- (log z) z) y y)
   (+ (* x 0.5) (* y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -70000000.0) || !(y <= 2.6e+92)) {
		tmp = fma((log(z) - z), y, y);
	} else {
		tmp = (x * 0.5) + (y * -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -70000000.0) || !(y <= 2.6e+92))
		tmp = fma(Float64(log(z) - z), y, y);
	else
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -70000000.0], N[Not[LessEqual[y, 2.6e+92]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7e7 or 2.5999999999999999e92 < y

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-lft-identity N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sub-sign-inv N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lift-log.f64 99.8

          \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\log z - z\right) \cdot y + y \]

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -7e7 < y < 2.5999999999999999e92

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 87.7

         \[\leadsto x \cdot 0.5 + y \cdot \left(-z\right) \]

   5. Applied rewrites 87.7%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70000000 \lor \neg \left(y \leq 2.6 \cdot 10^{+92}\right):\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-188} \lor \neg \left(x \leq 4.4 \cdot 10^{-231}\right):\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.65e-188) (not (<= x 4.4e-231)))
   (+ (* x 0.5) (* y (- z)))
   (fma (log z) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.65e-188) || !(x <= 4.4e-231)) {
		tmp = (x * 0.5) + (y * -z);
	} else {
		tmp = fma(log(z), y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.65e-188) || !(x <= 4.4e-231))
		tmp = Float64(Float64(x * 0.5) + Float64(y * Float64(-z)));
	else
		tmp = fma(log(z), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.65e-188], N[Not[LessEqual[x, 4.4e-231]], $MachinePrecision]], N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.65000000000000007e-188 or 4.40000000000000018e-231 < x

   1. Initial program 99.9%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.8

         \[\leadsto x \cdot 0.5 + y \cdot \left(-z\right) \]

   5. Applied rewrites 83.8%

      \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]

   if -2.65000000000000007e-188 < x < 4.40000000000000018e-231

   1. Initial program 99.6%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-lft-identity N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sub-sign-inv N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lift-log.f64 99.5

          \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(y \cdot \log z + \frac{1}{2} \cdot x\right) + y \]

      4. *-commutative N/A

         \[\leadsto \left(\log z \cdot y + \frac{1}{2} \cdot x\right) + y \]

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 68.0

         \[\leadsto \mathsf{fma}\left(\log z, y, 0.5 \cdot x\right) + y \]

   7. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log z, y, 0.5 \cdot x\right) + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{y \cdot \log z} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \log z + y \]

      2. *-commutative N/A

         \[\leadsto \log z \cdot y + y \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log z, y, y\right) \]

      4. lift-log.f64 64.9

         \[\leadsto \mathsf{fma}\left(\log z, y, y\right) \]

   10. Applied rewrites 64.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-188} \lor \neg \left(x \leq 4.4 \cdot 10^{-231}\right):\\ \;\;\;\;x \cdot 0.5 + y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.62) (fma 0.5 x (fma (log z) y y)) (+ (fma (- z) y (* 0.5 x)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.62) {
		tmp = fma(0.5, x, fma(log(z), y, y));
	} else {
		tmp = fma(-z, y, (0.5 * x)) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.62)
		tmp = fma(0.5, x, fma(log(z), y, y));
	else
		tmp = Float64(fma(Float64(-z), y, Float64(0.5 * x)) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.62], N[(0.5 * x + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision], N[(N[((-z) * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.619999999999999996

   1. Initial program 99.8%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 97.5

         \[\leadsto \mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right) \]

   5. Applied rewrites 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)} \]

   if 0.619999999999999996 < z

   1. Initial program 100.0%

      \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. distribute-rgt-in N/A

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

      10. *-lft-identity N/A

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

      11. lower-+.f64 N/A

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

      12. *-lft-identity N/A

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

      13. fp-cancel-sub-sign-inv N/A

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

      14. metadata-eval N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. fp-cancel-sub-sign-inv N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 N/A

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

      20. lift-log.f64 100.0

          \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y} \]

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, \frac{1}{2} \cdot x\right) + y \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

         \[\leadsto \mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y \]

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.62:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z - z, y, y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x 0.5 (fma (- (log z) z) y y)))

C

double code(double x, double y, double z) {
	return fma(x, 0.5, fma((log(z) - z), y, y));
}

Julia

function code(x, y, z)
	return fma(x, 0.5, fma(Float64(log(z) - z), y, y))
end

Wolfram

code[x_, y_, z_] := N[(x * 0.5 + N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-+.f64 N/A

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

   12. *-lft-identity N/A

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

   13. fp-cancel-sub-sign-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. fp-cancel-sub-sign-inv N/A

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

   18. *-lft-identity N/A

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

   19. lower--.f64 N/A

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

   20. lift-log.f64 99.9

       \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 99.9

       \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\color{blue}{\log z - z}, y, y\right)\right) \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(\log z - z, y, y\right)\right)} \]
7. Add Preprocessing

Alternative 8: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log z - z, y, \mathsf{fma}\left(0.5, x, y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- (log z) z) y (fma 0.5 x y)))

C

double code(double x, double y, double z) {
	return fma((log(z) - z), y, fma(0.5, x, y));
}

Julia

function code(x, y, z)
	return fma(Float64(log(z) - z), y, fma(0.5, x, y))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-+.f64 N/A

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

   12. *-lft-identity N/A

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

   13. fp-cancel-sub-sign-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. fp-cancel-sub-sign-inv N/A

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

   18. *-lft-identity N/A

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

   19. lower--.f64 N/A

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

   20. lift-log.f64 99.9

       \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lower-fma.f64 99.9

       \[\leadsto \mathsf{fma}\left(\log z - z, y, \mathsf{fma}\left(0.5, x, y\right)\right) \]

9. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log z - z, y, \mathsf{fma}\left(0.5, x, y\right)\right)} \]

10. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(\log z - z, y, \mathsf{fma}\left(0.5, x, y\right)\right) \]
11. Add Preprocessing

Alternative 9: 75.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 + y \cdot \left(-z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (- z))))

C

double code(double x, double y, double z) {
	return (x * 0.5) + (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 * 0.5d0) + (y * -z)
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.5) + (y * -z);
}

Python

def code(x, y, z):
	return (x * 0.5) + (y * -z)

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.5) + Float64(y * Float64(-z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.5) + (y * -z);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.5), $MachinePrecision] + N[(y * (-z)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 77.9

      \[\leadsto x \cdot 0.5 + y \cdot \left(-z\right) \]

5. Applied rewrites 77.9%

   \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(-z\right)} \]
6. Add Preprocessing

Alternative 10: 74.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-z, y, \mathsf{fma}\left(0.5, x, y\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (- z) y (fma 0.5 x y)))

C

double code(double x, double y, double z) {
	return fma(-z, y, fma(0.5, x, y));
}

Julia

function code(x, y, z)
	return fma(Float64(-z), y, fma(0.5, x, y))
end

Wolfram

code[x_, y_, z_] := N[((-z) * y + N[(0.5 * x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. +-commutative N/A

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

   8. associate--l+ N/A

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

   9. distribute-rgt-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-+.f64 N/A

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

   12. *-lft-identity N/A

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

   13. fp-cancel-sub-sign-inv N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. fp-cancel-sub-sign-inv N/A

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

   18. *-lft-identity N/A

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

   19. lower--.f64 N/A

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

   20. lift-log.f64 99.9

       \[\leadsto x \cdot 0.5 + \left(y + \left(\color{blue}{\log z} - z\right) \cdot y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto x \cdot 0.5 + \color{blue}{\left(y + \left(\log z - z\right) \cdot y\right)} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y \]

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right) + y} \]

8. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(-1 \cdot z, y, \frac{1}{2} \cdot x\right) + y \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 76.8

      \[\leadsto \mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y \]

10. Applied rewrites 76.8%

    \[\leadsto \mathsf{fma}\left(-z, y, 0.5 \cdot x\right) + y \]
11. Step-by-step derivation

    1. associate-+r+ 76.8

       \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)} + y \]

    2. *-commutative 76.8

       \[\leadsto \mathsf{fma}\left(-\color{blue}{z}, y, 0.5 \cdot x\right) + y \]

    3. +-commutative 76.8

       \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)} + y \]

    4. associate-+r+ 76.8

       \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, 0.5 \cdot x\right)} + y \]

    5. lift-+.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-fma.f64 N/A

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

    8. associate-+l+ N/A

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

    9. lower-fma.f64 N/A

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

    10. lift-fma.f64 76.8

        \[\leadsto \mathsf{fma}\left(-z, y, \mathsf{fma}\left(0.5, x, y\right)\right) \]

12. Applied rewrites 76.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-z, y, \mathsf{fma}\left(0.5, x, y\right)\right)} \]
13. Add Preprocessing

Alternative 11: 40.2% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 x))

C

double code(double x, double y, double z) {
	return 0.5 * 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 = 0.5d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * x;
}

Python

def code(x, y, z):
	return 0.5 * x

Julia

function code(x, y, z)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 45.6

      \[\leadsto 0.5 \cdot \color{blue}{x} \]

5. Applied rewrites 45.6%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

6. Final simplification 45.6%

   \[\leadsto 0.5 \cdot x \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (* 0.5 x)) (* y (- z (log z)))))

C

double code(double x, double y, double z) {
	return (y + (0.5 * x)) - (y * (z - log(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 = (y + (0.5d0 * x)) - (y * (z - log(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (y + (0.5 * x)) - (y * (z - Math.log(z)));
}

Python

def code(x, y, z):
	return (y + (0.5 * x)) - (y * (z - math.log(z)))

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(0.5 * x)) - Float64(y * Float64(z - log(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (0.5 * x)) - (y * (z - log(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] - N[(y * N[(z - N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (* 1/2 x)) (* y (- z (log z)))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))