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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 10

Speedup: 0.9×

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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.1× 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. +-commutative N/A

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

   12. associate--l+ N/A

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

   13. +-commutative N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lift-log.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.00145:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.00145)
   (fma 0.5 x (fma (log z) y y))
   (fma (- (log z) z) y (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.00145) {
		tmp = fma(0.5, x, fma(log(z), y, y));
	} else {
		tmp = fma((log(z) - z), y, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.00145)
		tmp = fma(0.5, x, fma(log(z), y, y));
	else
		tmp = fma(Float64(log(z) - z), y, Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.00145], N[(0.5 * x + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00145

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 99.2%

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

   if 0.00145 < z

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. distribute-lft-neg-out N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 98.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log z - z, y, 0.5 \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.00145:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(\log z, y, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, -y, 0.5 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 0.00145) (fma 0.5 x (fma (log z) y y)) (fma z (- y) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.00145) {
		tmp = fma(0.5, x, fma(log(z), y, y));
	} else {
		tmp = fma(z, -y, (0.5 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 0.00145)
		tmp = fma(0.5, x, fma(log(z), y, y));
	else
		tmp = fma(z, Float64(-y), Float64(0.5 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 0.00145], N[(0.5 * x + N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision], N[(z * (-y) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00145

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 99.2%

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

   if 0.00145 < z

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. distribute-lft-neg-out N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 98.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.2%

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

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z - z, y, y\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(z, -y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- (log z) z) y y)))
   (if (<= y -2.1e+57) t_0 (if (<= y 1.16e+148) (fma z (- y) (* 0.5 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((log(z) - z), y, y);
	double tmp;
	if (y <= -2.1e+57) {
		tmp = t_0;
	} else if (y <= 1.16e+148) {
		tmp = fma(z, -y, (0.5 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(log(z) - z), y, y)
	tmp = 0.0
	if (y <= -2.1e+57)
		tmp = t_0;
	elseif (y <= 1.16e+148)
		tmp = fma(z, Float64(-y), Float64(0.5 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[z], $MachinePrecision] - z), $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[y, -2.1e+57], t$95$0, If[LessEqual[y, 1.16e+148], N[(z * (-y) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999991e57 or 1.1599999999999999e148 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

         \[\leadsto \left(1 + \left(\log z - z\right)\right) \cdot y \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\log z - z\right) + 1\right) \cdot y \]

      4. distribute-lft1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-log.f64 89.0

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

   4. Applied rewrites 89.0%

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

   if -2.09999999999999991e57 < y < 1.1599999999999999e148

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. distribute-lft-neg-out N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 88.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 84.0%

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

Alternative 6: 75.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log z, y, y\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+274}:\\ \;\;\;\;\mathsf{fma}\left(z, -y, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (log z) y y)))
   (if (<= y -2.3e+196) t_0 (if (<= y 4.6e+274) (fma z (- y) (* 0.5 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(log(z), y, y);
	double tmp;
	if (y <= -2.3e+196) {
		tmp = t_0;
	} else if (y <= 4.6e+274) {
		tmp = fma(z, -y, (0.5 * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(log(z), y, y)
	tmp = 0.0
	if (y <= -2.3e+196)
		tmp = t_0;
	elseif (y <= 4.6e+274)
		tmp = fma(z, Float64(-y), Float64(0.5 * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[z], $MachinePrecision] * y + y), $MachinePrecision]}, If[LessEqual[y, -2.3e+196], t$95$0, If[LessEqual[y, 4.6e+274], N[(z * (-y) + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999998e196 or 4.60000000000000006e274 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto y + \color{blue}{y \cdot \log z} \]
   6. 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. lift-log.f64 N/A

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

      4. lift-fma.f64 46.5

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

   7. Applied rewrites 46.5%

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

   if -2.2999999999999998e196 < y < 4.60000000000000006e274

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-commutative N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. distribute-lft-neg-out N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 84.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 84.5

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

   8. Applied rewrites 84.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 79.1%

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

Alternative 7: 75.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 82.4

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

6. Applied rewrites 82.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. *-commutative N/A

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

   6. cancel-sign-sub-inv N/A

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

   7. distribute-lft-neg-out N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

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

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

   11. lift-neg.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lift--.f64 82.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 82.4

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

8. Applied rewrites 82.4%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 75.9%

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

Alternative 8: 75.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 82.4

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

6. Applied rewrites 82.4%

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

7. Taylor expanded in z around inf

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

9. Applied rewrites 75.9%

   \[\leadsto x \cdot 0.5 - \color{blue}{z} \cdot y \]
10. Add Preprocessing

Alternative 9: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(1 - z\right) + \log z\right)\\ t_1 := \left(-y\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+41}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ (- 1.0 z) (log z)))) (t_1 (* (- y) z)))
   (if (<= t_0 -1e+59) t_1 (if (<= t_0 4e+41) (* 0.5 x) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * ((1.0 - z) + log(z));
	double t_1 = -y * z;
	double tmp;
	if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= 4e+41) {
		tmp = 0.5 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * ((1.0d0 - z) + log(z))
    t_1 = -y * z
    if (t_0 <= (-1d+59)) then
        tmp = t_1
    else if (t_0 <= 4d+41) then
        tmp = 0.5d0 * x
    else
        tmp = t_1
    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 t_1 = -y * z;
	double tmp;
	if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= 4e+41) {
		tmp = 0.5 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * ((1.0 - z) + math.log(z))
	t_1 = -y * z
	tmp = 0
	if t_0 <= -1e+59:
		tmp = t_1
	elif t_0 <= 4e+41:
		tmp = 0.5 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(1.0 - z) + log(z)))
	t_1 = Float64(Float64(-y) * z)
	tmp = 0.0
	if (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= 4e+41)
		tmp = Float64(0.5 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * ((1.0 - z) + log(z));
	t_1 = -y * z;
	tmp = 0.0;
	if (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= 4e+41)
		tmp = 0.5 * x;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[((-y) * z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+59], t$95$1, If[LessEqual[t$95$0, 4e+41], N[(0.5 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < -9.99999999999999972e58 or 4.00000000000000002e41 < (*.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. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 55.1

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

   4. Applied rewrites 55.1%

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

   if -9.99999999999999972e58 < (*.f64 y (+.f64 (-.f64 #s(literal 1 binary64) z) (log.f64 z))) < 4.00000000000000002e41

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

Alternative 10: 40.9% accurate, 5.2× 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. Taylor expanded in x around inf

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

   1. lower-*.f64 40.9

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

4. Applied rewrites 40.9%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64
  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))