Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lower-+.f64 99.9

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 63.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ t_1 := \left(y - x\right) \cdot \left(y - 0.5\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq -2000000000000:\\ \;\;\;\;\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{t\_1} - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{t\_1} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)) (t_1 (* (- y x) (- y 0.5))))
   (if (<= t_0 -5e+140)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 -2000000000000.0)
       (- (/ (* (* (- y 0.5) x) (- x)) t_1) z)
       (if (<= t_0 500.0)
         (fma -0.5 (log y) (- z))
         (- (/ (* (* x x) 0.5) t_1) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double t_1 = (y - x) * (y - 0.5);
	double tmp;
	if (t_0 <= -5e+140) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= -2000000000000.0) {
		tmp = ((((y - 0.5) * x) * -x) / t_1) - z;
	} else if (t_0 <= 500.0) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = (((x * x) * 0.5) / t_1) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	t_1 = Float64(Float64(y - x) * Float64(y - 0.5))
	tmp = 0.0
	if (t_0 <= -5e+140)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= -2000000000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y - 0.5) * x) * Float64(-x)) / t_1) - z);
	elseif (t_0 <= 500.0)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / t_1) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+140], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, -2000000000000.0], N[(N[(N[(N[(N[(y - 0.5), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] / t$95$1), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$1), $MachinePrecision] - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5.00000000000000008e140

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -5.00000000000000008e140 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e12

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 56.9

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

   9. Applied rewrites 56.9%

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

   11. Applied rewrites 56.9%

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

   if -2e12 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.1%

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

   if 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 43.9

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

   9. Applied rewrites 43.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 45.8%

       \[\leadsto \frac{\left(x \cdot x\right) \cdot \color{blue}{0.5}}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -2000000000000:\\ \;\;\;\;\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq 500:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -500000000:\\ \;\;\;\;\frac{\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{y - x}}{y - 0.5} - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (<= t_0 -500000000.0)
     (- (/ (/ (* (* (- y 0.5) x) (- x)) (- y x)) (- y 0.5)) z)
     (if (<= t_0 500.0)
       (* -0.5 (log y))
       (- (/ (* (* x x) 0.5) (* (- y x) (- y 0.5))) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	} else if (t_0 <= 500.0) {
		tmp = -0.5 * log(y);
	} else {
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - ((y + 0.5d0) * log(y))) + y) - z
    if (t_0 <= (-500000000.0d0)) then
        tmp = (((((y - 0.5d0) * x) * -x) / (y - x)) / (y - 0.5d0)) - z
    else if (t_0 <= 500.0d0) then
        tmp = (-0.5d0) * log(y)
    else
        tmp = (((x * x) * 0.5d0) / ((y - x) * (y - 0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * Math.log(y))) + y) - z;
	double tmp;
	if (t_0 <= -500000000.0) {
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	} else if (t_0 <= 500.0) {
		tmp = -0.5 * Math.log(y);
	} else {
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x - ((y + 0.5) * math.log(y))) + y) - z
	tmp = 0
	if t_0 <= -500000000.0:
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z
	elif t_0 <= 500.0:
		tmp = -0.5 * math.log(y)
	else:
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
	tmp = 0.0
	if (t_0 <= -500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y - 0.5) * x) * Float64(-x)) / Float64(y - x)) / Float64(y - 0.5)) - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(-0.5 * log(y));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / Float64(Float64(y - x) * Float64(y - 0.5))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	tmp = 0.0;
	if (t_0 <= -500000000.0)
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	elseif (t_0 <= 500.0)
		tmp = -0.5 * log(y);
	else
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -500000000.0], N[(N[(N[(N[(N[(N[(y - 0.5), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -5e8

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 37.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 29.5

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

   9. Applied rewrites 29.5%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 32.0%

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

   if -5e8 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in z around 0

      \[\leadsto \frac{-1}{2} \cdot \log y \]
   10. Step-by-step derivation

   11. Applied rewrites 91.4%

       \[\leadsto -0.5 \cdot \log y \]

   if 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 63.3

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

   9. Applied rewrites 63.3%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 64.8%

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

Alternative 4: 48.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2200000:\\ \;\;\;\;\frac{\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{y - x}}{y - 0.5} - z\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+20} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2200000.0)
   (- (/ (/ (* (* (- y 0.5) x) (- x)) (- y x)) (- y 0.5)) z)
   (if (or (<= x 7.2e+20) (not (<= x 1.35e+154)))
     (fma -0.5 (log y) (- z))
     (- (/ (* (* x x) 0.5) (* (- y x) (- y 0.5))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2200000.0) {
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	} else if ((x <= 7.2e+20) || !(x <= 1.35e+154)) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2200000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y - 0.5) * x) * Float64(-x)) / Float64(y - x)) / Float64(y - 0.5)) - z);
	elseif ((x <= 7.2e+20) || !(x <= 1.35e+154))
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / Float64(Float64(y - x) * Float64(y - 0.5))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2200000.0], N[(N[(N[(N[(N[(N[(y - 0.5), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[x, 7.2e+20], N[Not[LessEqual[x, 1.35e+154]], $MachinePrecision]], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e6

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 29.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 28.9

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

   9. Applied rewrites 28.9%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 33.2%

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

   if -2.2e6 < x < 7.2e20 or 1.35000000000000003e154 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 86.7

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 59.6%

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

   if 7.2e20 < x < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 60.5

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

   9. Applied rewrites 60.5%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 67.4%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2200000:\\ \;\;\;\;\frac{\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{y - x}}{y - 0.5} - z\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+20} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) - \log y \cdot y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e-8) (- (fma -0.5 (log y) x) z) (- (- (+ y x) (* (log y) y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.8e-8) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((y + x) - (log(y) * y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.8e-8)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(y + x) - Float64(log(y) * y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.8e-8], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.8000000000000003e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.8000000000000003e-8 < y

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.8e-8) (- (fma -0.5 (log y) x) z) (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.8e-8) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.8e-8)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.8e-8], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.8000000000000003e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.8000000000000003e-8 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.68:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.68) (- (fma -0.5 (log y) x) z) (- y (fma (+ 0.5 y) (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.68) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.68)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.68], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.680000000000000049

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.680000000000000049 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

Alternative 8: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+26) (- (fma -0.5 (log y) x) z) (- (* (- 1.0 (log y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+26) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e+26)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e+26], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.49999999999999941e26

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 7.49999999999999941e26 < y

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 80.3

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

   7. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\left(1 - \log y\right) \cdot y} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 10: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e+136) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e+136) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.2e+136)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.2e+136], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.20000000000000011e136

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if 7.20000000000000011e136 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\left(1 - \log y\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+215} \lor \neg \left(z \leq 1.5 \cdot 10^{+121}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{y - x}}{y - 0.5} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e+215) (not (<= z 1.5e+121)))
   (- z)
   (- (/ (/ (* (* (- y 0.5) x) (- x)) (- y x)) (- y 0.5)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e+215) || !(z <= 1.5e+121)) {
		tmp = -z;
	} else {
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.5d+215)) .or. (.not. (z <= 1.5d+121))) then
        tmp = -z
    else
        tmp = (((((y - 0.5d0) * x) * -x) / (y - x)) / (y - 0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e+215) || !(z <= 1.5e+121)) {
		tmp = -z;
	} else {
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e+215) or not (z <= 1.5e+121):
		tmp = -z
	else:
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e+215) || !(z <= 1.5e+121))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y - 0.5) * x) * Float64(-x)) / Float64(y - x)) / Float64(y - 0.5)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.5e+215) || ~((z <= 1.5e+121)))
		tmp = -z;
	else
		tmp = (((((y - 0.5) * x) * -x) / (y - x)) / (y - 0.5)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e+215], N[Not[LessEqual[z, 1.5e+121]], $MachinePrecision]], (-z), N[(N[(N[(N[(N[(N[(y - 0.5), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(y - 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999997e215 or 1.5000000000000001e121 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -6.4999999999999997e215 < z < 1.5000000000000001e121

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 28.2

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

   9. Applied rewrites 28.2%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

   11. Applied rewrites 29.7%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+215} \lor \neg \left(z \leq 1.5 \cdot 10^{+121}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{y - x}}{y - 0.5} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+55)
   (- (/ (* (* (- y 0.5) x) (- x)) (* (- y x) (- y 0.5))) z)
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+55) {
		tmp = ((((y - 0.5) * x) * -x) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.4d+55) then
        tmp = ((((y - 0.5d0) * x) * -x) / ((y - x) * (y - 0.5d0))) - z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+55) {
		tmp = ((((y - 0.5) * x) * -x) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+55:
		tmp = ((((y - 0.5) * x) * -x) / ((y - x) * (y - 0.5))) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+55)
		tmp = Float64(Float64(Float64(Float64(Float64(y - 0.5) * x) * Float64(-x)) / Float64(Float64(y - x) * Float64(y - 0.5))) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+55)
		tmp = ((((y - 0.5) * x) * -x) / ((y - x) * (y - 0.5))) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+55], N[(N[(N[(N[(N[(y - 0.5), $MachinePrecision] * x), $MachinePrecision] * (-x)), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000021e55

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 50.2

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

   9. Applied rewrites 50.2%

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

   11. Applied rewrites 50.2%

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

   if 4.40000000000000021e55 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.4

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

   5. Applied rewrites 21.4%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\left(y - 0.5\right) \cdot x\right) \cdot \left(-x\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{\left(\left(-x\right) \cdot x\right) \cdot \left(y - 0.5\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+55)
   (- (/ (* (* (- x) x) (- y 0.5)) (* (- y x) (- y 0.5))) z)
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+55) {
		tmp = (((-x * x) * (y - 0.5)) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.4d+55) then
        tmp = (((-x * x) * (y - 0.5d0)) / ((y - x) * (y - 0.5d0))) - z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+55) {
		tmp = (((-x * x) * (y - 0.5)) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+55:
		tmp = (((-x * x) * (y - 0.5)) / ((y - x) * (y - 0.5))) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+55)
		tmp = Float64(Float64(Float64(Float64(Float64(-x) * x) * Float64(y - 0.5)) / Float64(Float64(y - x) * Float64(y - 0.5))) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+55)
		tmp = (((-x * x) * (y - 0.5)) / ((y - x) * (y - 0.5))) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+55], N[(N[(N[(N[((-x) * x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000021e55

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 100.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 50.2

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

   9. Applied rewrites 50.2%

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

   if 4.40000000000000021e55 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.4

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

   5. Applied rewrites 21.4%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 35.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.5}{\left(y - x\right) \cdot \left(y - 0.5\right)} - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.35e+154) (- (/ (* (* x x) 0.5) (* (- y x) (- y 0.5))) z) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = (((x * x) * 0.5d0) / ((y - x) * (y - 0.5d0))) - z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.35e+154:
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(Float64(Float64(x * x) * 0.5) / Float64(Float64(y - x) * Float64(y - 0.5))) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = (((x * x) * 0.5) / ((y - x) * (y - 0.5))) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.35e+154], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] * N[(y - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. associate-*r/ N/A

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

      14. frac-add N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(x + y\right) \cdot \left(y - x\right), y - 0.5, \left(y - x\right) \cdot \left(\left(-\log y\right) \cdot \left(y \cdot y - 0.25\right)\right)\right)}{\left(y - x\right) \cdot \left(y - 0.5\right)}} - z \]

   7. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unpow2 N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 40.2

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

   9. Applied rewrites 40.2%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 40.8%

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

   if 1.35000000000000003e154 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 18.1

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

   5. Applied rewrites 18.1%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 30.0% accurate, 39.3× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 31.9

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

5. Applied rewrites 31.9%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025006 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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