Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.6% → 99.7%

Time: 19.0s

Alternatives: 22

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (*
    (- z 1.0)
    (-
     (* (- (* (- (* (* y y) -0.3333333333333333) 0.5) (* y y)) 1.0) (* y y))
     (log1p y))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (((((((y * y) * -0.3333333333333333) - 0.5) * (y * y)) - 1.0) * (y * y)) - log1p(y)))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * (((((((y * y) * -0.3333333333333333) - 0.5) * (y * y)) - 1.0) * (y * y)) - Math.log1p(y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (((((((y * y) * -0.3333333333333333) - 0.5) * (y * y)) - 1.0) * (y * y)) - math.log1p(y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(y * y) * -0.3333333333333333) - 0.5) * Float64(y * y)) - 1.0) * Float64(y * y)) - log1p(y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - N[Log[1 + y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. lower--.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-log1p.f64 99.7

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - y \cdot y\right) - \color{blue}{\mathsf{log1p}\left(y\right)}\right)\right) - t \]

4. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left(1 - y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)}\right) - t \]

5. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot {\color{blue}{y}}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   6. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left({y}^{2} \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left({y}^{2} \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   9. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   11. pow2 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   13. pow2 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   14. lift-*.f64 99.8

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

7. Applied rewrites 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
8. Add Preprocessing

Alternative 2: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -5000000 \lor \neg \left(t\_1 \leq 1000\right):\\ \;\;\;\;\mathsf{fma}\left(-y, z, \log y \cdot x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 -5000000.0) (not (<= t_1 1000.0)))
     (- (fma (- y) z (* (log y) x)) t)
     (- (fma (- y) (- z 1.0) (- (log y))) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -5000000.0) || !(t_1 <= 1000.0)) {
		tmp = fma(-y, z, (log(y) * x)) - t;
	} else {
		tmp = fma(-y, (z - 1.0), -log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -5000000.0) || !(t_1 <= 1000.0))
		tmp = Float64(fma(Float64(-y), z, Float64(log(y) * x)) - t);
	else
		tmp = Float64(fma(Float64(-y), Float64(z - 1.0), Float64(-log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000000.0], N[Not[LessEqual[t$95$1, 1000.0]], $MachinePrecision]], N[(N[((-y) * z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[((-y) * N[(z - 1.0), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5e6 or 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-y, z, \log y \cdot \left(x - 1\right)\right) - t \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(-y, z, \log y \cdot x\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 97.4%

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

   if -5e6 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1e3

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(-y, z - 1, \mathsf{neg}\left(\log y\right)\right) - t \]

      8. lower-neg.f64 N/A

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

      9. lift-log.f64 99.6

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

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(-y, \color{blue}{z - 1}, -\log y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq -5000000 \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 1000\right):\\ \;\;\;\;\mathsf{fma}\left(-y, z, \log y \cdot x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -5000000 \lor \neg \left(t\_1 \leq 1000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 -5000000.0) (not (<= t_1 1000.0)))
     (- (* (log y) x) t)
     (- (fma (log y) -1.0 y) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -5000000.0) || !(t_1 <= 1000.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = fma(log(y), -1.0, y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -5000000.0) || !(t_1 <= 1000.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(fma(log(y), -1.0, y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000000.0], N[Not[LessEqual[t$95$1, 1000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -1.0 + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5e6 or 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \log y \cdot \color{blue}{x} - t \]

      2. lower-*.f64 N/A

         \[\leadsto \log y \cdot \color{blue}{x} - t \]

      3. lift-log.f64 92.6

         \[\leadsto \log y \cdot x - t \]

   5. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\log y \cdot x} - t \]

   if -5e6 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1e3

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 86.6

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

   8. Applied rewrites 86.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\log y, -1, y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 86.2%

       \[\leadsto \mathsf{fma}\left(\log y, -1, y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq -5000000 \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 1000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -1, y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot -0.5 - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (* (- z 1.0) (- (* (- (* (* y y) -0.5) 1.0) (* y y)) (log1p y))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (((((y * y) * -0.5) - 1.0) * (y * y)) - log1p(y)))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * (((((y * y) * -0.5) - 1.0) * (y * y)) - Math.log1p(y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (((((y * y) * -0.5) - 1.0) * (y * y)) - math.log1p(y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(Float64(Float64(Float64(Float64(y * y) * -0.5) - 1.0) * Float64(y * y)) - log1p(y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.5), $MachinePrecision] - 1.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - N[Log[1 + y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. lower--.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-log1p.f64 99.7

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - y \cdot y\right) - \color{blue}{\mathsf{log1p}\left(y\right)}\right)\right) - t \]

4. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left(1 - y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)}\right) - t \]

5. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{-1}{2} \cdot {y}^{2} - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot {y}^{2} - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot {y}^{2} - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\frac{-1}{2} \cdot {y}^{2} - 1\right) \cdot {\color{blue}{y}}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \frac{-1}{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \frac{-1}{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   6. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{-1}{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{-1}{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   8. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{-1}{2} - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   9. lift-*.f64 99.7

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot y\right) \cdot -0.5 - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

7. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(y \cdot y\right) \cdot -0.5 - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
8. Add Preprocessing

Alternative 5: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(-y\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (* (- z 1.0) (- (* (* (- y) y) y) (log1p (fma y y y)))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (((-y * y) * y) - log1p(fma(y, y, y))))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(Float64(Float64(Float64(-y) * y) * y) - log1p(fma(y, y, y))))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[((-y) * y), $MachinePrecision] * y), $MachinePrecision] - N[Log[1 + N[(y * y + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. flip3-- N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \color{blue}{\left(\frac{{1}^{3} - {y}^{3}}{1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)}\right)}\right) - t \]

   4. log-div N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

   5. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left({1}^{3} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)}\right) - t \]

   6. lower-log.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\log \left({1}^{3} - {y}^{3}\right)} - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(\color{blue}{1} - {y}^{3}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   8. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \color{blue}{\left(1 - {y}^{3}\right)} - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - \color{blue}{{y}^{3}}\right) - \log \left(1 \cdot 1 + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   10. metadata-eval N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - {y}^{3}\right) - \log \left(\color{blue}{1} + \left(y \cdot y + 1 \cdot y\right)\right)\right)\right) - t \]

   11. lower-log1p.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - {y}^{3}\right) - \color{blue}{\mathsf{log1p}\left(y \cdot y + 1 \cdot y\right)}\right)\right) - t \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - {y}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, 1 \cdot y\right)}\right)\right)\right) - t \]

   13. lower-*.f64 99.8

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - {y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, \color{blue}{1 \cdot y}\right)\right)\right)\right) - t \]

4. Applied rewrites 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left(1 - {y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)}\right) - t \]

5. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{-1 \cdot {y}^{3}} - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\mathsf{neg}\left({y}^{3}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   2. lower-neg.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-{y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   3. lift-pow.f64 99.7

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-{y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

7. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(-{y}^{3}\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-{y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   2. unpow3 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-\left(y \cdot y\right) \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   3. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-{y}^{2} \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-{y}^{2} \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   5. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-\left(y \cdot y\right) \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

   6. lift-*.f64 99.7

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-\left(y \cdot y\right) \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-\left(y \cdot y\right) \cdot y\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, 1 \cdot y\right)\right)\right)\right) - t \]

10. Final simplification 99.7%

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(-y\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)\right) - t \]
11. Add Preprocessing

Alternative 6: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 1, \log y, \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, 0.3333333333333333\right) \cdot y - 0.5, y, 1\right) \cdot y\right) \cdot \left(z - 1\right) - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  (- x 1.0)
  (log y)
  (-
   (*
    (-
     (* (- (* (- (* (* y y) -0.3333333333333333) 0.5) (* y y)) 1.0) (* y y))
     (* (fma (- (* (fma -0.25 y 0.3333333333333333) y) 0.5) y 1.0) y))
    (- z 1.0))
   t)))

C

double code(double x, double y, double z, double t) {
	return fma((x - 1.0), log(y), (((((((((y * y) * -0.3333333333333333) - 0.5) * (y * y)) - 1.0) * (y * y)) - (fma(((fma(-0.25, y, 0.3333333333333333) * y) - 0.5), y, 1.0) * y)) * (z - 1.0)) - t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(x - 1.0), log(y), Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y * y) * -0.3333333333333333) - 0.5) * Float64(y * y)) - 1.0) * Float64(y * y)) - Float64(fma(Float64(Float64(fma(-0.25, y, 0.3333333333333333) * y) - 0.5), y, 1.0) * y)) * Float64(z - 1.0)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(-0.25 * y + 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. lower--.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-log1p.f64 99.7

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log \left(1 - y \cdot y\right) - \color{blue}{\mathsf{log1p}\left(y\right)}\right)\right) - t \]

4. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log \left(1 - y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)}\right) - t \]

5. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{{y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left({y}^{2} \cdot \left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) - 1\right) \cdot {\color{blue}{y}}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   6. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot {y}^{2} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left({y}^{2} \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left({y}^{2} \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   9. pow2 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot {y}^{2} - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   11. pow2 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot {y}^{2} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   13. pow2 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   14. lift-*.f64 99.8

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot \color{blue}{y}\right) - \mathsf{log1p}\left(y\right)\right)\right) - t \]

7. Applied rewrites 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

8. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}\right)\right)}\right)\right) - t \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \left(1 + y \cdot \left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}\right)\right) \cdot \color{blue}{y}\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \left(1 + y \cdot \left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}\right)\right) \cdot \color{blue}{y}\right)\right) - t \]

   3. +-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \left(y \cdot \left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}\right) + 1\right) \cdot y\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \left(\left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}\right) \cdot y + 1\right) \cdot y\right)\right) - t \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

   6. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(y \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\left(\frac{1}{3} + \frac{-1}{4} \cdot y\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

   9. +-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\left(\frac{-1}{4} \cdot y + \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

   10. lower-fma.f64 99.7

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, 0.3333333333333333\right) \cdot y - 0.5, y, 1\right) \cdot y\right)\right) - t \]

10. Applied rewrites 99.7%

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, 0.3333333333333333\right) \cdot y - 0.5, y, 1\right) \cdot y}\right)\right) - t \]
11. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, y, \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t} \]

    2. lift-+.f64 N/A

       \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, y, \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right)} - t \]

    3. lift--.f64 N/A

       \[\leadsto \left(\color{blue}{\left(x - 1\right)} \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, y, \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\color{blue}{\left(x - 1\right) \cdot \log y} + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, y, \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

    5. lift-log.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \color{blue}{\log y} + \left(z - 1\right) \cdot \left(\left(\left(\left(y \cdot y\right) \cdot \frac{-1}{3} - \frac{1}{2}\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4}, y, \frac{1}{3}\right) \cdot y - \frac{1}{2}, y, 1\right) \cdot y\right)\right) - t \]

12. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, \log y, \left(\left(\left(\left(y \cdot y\right) \cdot -0.3333333333333333 - 0.5\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \left(y \cdot y\right) - \mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, 0.3333333333333333\right) \cdot y - 0.5, y, 1\right) \cdot y\right) \cdot \left(z - 1\right) - t\right)} \]
13. Add Preprocessing

Alternative 7: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (*
    (- z 1.0)
    (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t;
}

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * ((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 1.0d0) * y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{y}\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot \color{blue}{y}\right)\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   6. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   9. lower--.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   10. lower-*.f64 99.7

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

5. Applied rewrites 99.7%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right)}\right) - t \]
6. Add Preprocessing

Alternative 8: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(-0.3333333333333333 \cdot y - 0.5\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (fma
   (fma (* (- (* -0.3333333333333333 y) 0.5) z) y (- (- z 1.0)))
   y
   (* (log y) (- x 1.0)))
  t))

C

double code(double x, double y, double z, double t) {
	return fma(fma((((-0.3333333333333333 * y) - 0.5) * z), y, -(z - 1.0)), y, (log(y) * (x - 1.0))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(fma(fma(Float64(Float64(Float64(-0.3333333333333333 * y) - 0.5) * z), y, Float64(-Float64(z - 1.0))), y, Float64(log(y) * Float64(x - 1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(N[(-0.3333333333333333 * y), $MachinePrecision] - 0.5), $MachinePrecision] * z), $MachinePrecision] * y + (-N[(z - 1.0), $MachinePrecision])), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) \cdot y + \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right), \color{blue}{y}, \log y \cdot \left(x - 1\right)\right) - t \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \left(z - 1\right) \cdot y, -0.5 \cdot \left(z - 1\right)\right), y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right)} - t \]

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right), y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   4. lower-*.f64 99.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(-0.3333333333333333 \cdot y - 0.5\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(-0.3333333333333333 \cdot y - 0.5\right) \cdot z, y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right) - t \]
9. Add Preprocessing

Alternative 9: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift--.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 10: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * ((((-0.5d0) * y) - 1.0d0) * y))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 11: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 1.7 \cdot 10^{-10}\right):\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e-6) (not (<= x 1.7e-10)))
   (- (* (log y) (- x 1.0)) t)
   (- (fma (- y) (- z 1.0) (- (log y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e-6) || !(x <= 1.7e-10)) {
		tmp = (log(y) * (x - 1.0)) - t;
	} else {
		tmp = fma(-y, (z - 1.0), -log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e-6) || !(x <= 1.7e-10))
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	else
		tmp = Float64(fma(Float64(-y), Float64(z - 1.0), Float64(-log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e-6], N[Not[LessEqual[x, 1.7e-10]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[((-y) * N[(z - 1.0), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-6 or 1.70000000000000007e-10 < x

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if -3.1e-6 < x < 1.70000000000000007e-10

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(-y, z - 1, \mathsf{neg}\left(\log y\right)\right) - t \]

      8. lower-neg.f64 N/A

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

      9. lift-log.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(-y, \color{blue}{z - 1}, -\log y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-6} \lor \neg \left(x \leq 1.7 \cdot 10^{-10}\right):\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 13: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-58} \lor \neg \left(x \leq 0.00106\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.9e-58) (not (<= x 0.00106)))
   (- (* (log y) x) t)
   (- (fma (- y) z y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.9e-58) || !(x <= 0.00106)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = fma(-y, z, y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.9e-58) || !(x <= 0.00106))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(fma(Float64(-y), z, y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.9e-58], N[Not[LessEqual[x, 0.00106]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8999999999999999e-58 or 0.00105999999999999996 < x

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \log y \cdot \color{blue}{x} - t \]

      2. lower-*.f64 N/A

         \[\leadsto \log y \cdot \color{blue}{x} - t \]

      3. lift-log.f64 91.8

         \[\leadsto \log y \cdot x - t \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\log y \cdot x} - t \]

   if -2.8999999999999999e-58 < x < 0.00105999999999999996

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift--.f64 54.2

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-neg.f64 54.2

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

   11. Applied rewrites 54.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-58} \lor \neg \left(x \leq 0.00106\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(-y, z, \log y \cdot \left(x - 1\right)\right) - t \]
7. Step-by-step derivation

8. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(-y, z, \log y \cdot \left(x - 1\right)\right) - t \]
9. Add Preprocessing

Alternative 15: 65.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+110} \lor \neg \left(x \leq 12000000000000\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e+110) (not (<= x 12000000000000.0)))
   (* (log y) x)
   (- (* (- y) (* (fma (fma 0.3333333333333333 y 0.5) y 1.0) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e+110) || !(x <= 12000000000000.0)) {
		tmp = log(y) * x;
	} else {
		tmp = (-y * (fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e+110) || !(x <= 12000000000000.0))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(-y) * Float64(fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e+110], N[Not[LessEqual[x, 12000000000000.0]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[((-y) * N[(N[(N[(0.3333333333333333 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999978e110 or 1.2e13 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 79.4

         \[\leadsto \log y \cdot x \]

   5. Applied rewrites 79.4%

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

   if -4.99999999999999978e110 < x < 1.2e13

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) \cdot y + \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right), \color{blue}{y}, \log y \cdot \left(x - 1\right)\right) - t \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \left(z - 1\right) \cdot y, -0.5 \cdot \left(z - 1\right)\right), y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right)} - t \]

   6. Taylor expanded in z around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \left(z \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right)\right)} - t \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot y\right) \cdot \left(z \cdot \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)}\right) - t \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(z \cdot \left(\color{blue}{1} + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right) - t \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(z \cdot \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)}\right) - t \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(-y\right) \cdot \left(z \cdot \left(\color{blue}{1} + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right) - t \]

      5. *-commutative N/A

         \[\leadsto \left(-y\right) \cdot \left(\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right) \cdot z\right) - t \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-y\right) \cdot \left(\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right) \cdot z\right) - t \]

      7. +-commutative N/A

         \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right) + 1\right) \cdot z\right) - t \]

      8. *-commutative N/A

         \[\leadsto \left(-y\right) \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{3} \cdot y\right) \cdot y + 1\right) \cdot z\right) - t \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{3} \cdot y, y, 1\right) \cdot z\right) - t \]

      10. +-commutative N/A

          \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3} \cdot y + \frac{1}{2}, y, 1\right) \cdot z\right) - t \]

      11. lower-fma.f64 56.1

          \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t \]

   8. Applied rewrites 56.1%

      \[\leadsto \left(-y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+110} \lor \neg \left(x \leq 12000000000000\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 90.3

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

8. Applied rewrites 90.3%

   \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{x - 1}, y\right) - t \]
9. Add Preprocessing

Alternative 17: 87.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 90.1

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

5. Applied rewrites 90.1%

   \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - t \]
6. Add Preprocessing

Alternative 18: 46.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (- y) (* (fma (fma 0.3333333333333333 y 0.5) y 1.0) z)) t))

C

double code(double x, double y, double z, double t) {
	return (-y * (fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-y) * Float64(fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-y) * N[(N[(N[(0.3333333333333333 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right)\right) \cdot y + \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(z - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{-1}{3} \cdot \left(y \cdot \left(z - 1\right)\right)\right), \color{blue}{y}, \log y \cdot \left(x - 1\right)\right) - t \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, \left(z - 1\right) \cdot y, -0.5 \cdot \left(z - 1\right)\right), y, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right)\right)} - t \]

6. Taylor expanded in z around -inf

   \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \left(z \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right)\right)} - t \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot y\right) \cdot \left(z \cdot \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)}\right) - t \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(z \cdot \left(\color{blue}{1} + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right) - t \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(z \cdot \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)}\right) - t \]

   4. lift-neg.f64 N/A

      \[\leadsto \left(-y\right) \cdot \left(z \cdot \left(\color{blue}{1} + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right)\right) - t \]

   5. *-commutative N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right) \cdot z\right) - t \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right)\right) \cdot z\right) - t \]

   7. +-commutative N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(y \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot y\right) + 1\right) \cdot z\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(-y\right) \cdot \left(\left(\left(\frac{1}{2} + \frac{1}{3} \cdot y\right) \cdot y + 1\right) \cdot z\right) - t \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{3} \cdot y, y, 1\right) \cdot z\right) - t \]

   10. +-commutative N/A

       \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3} \cdot y + \frac{1}{2}, y, 1\right) \cdot z\right) - t \]

   11. lower-fma.f64 41.8

       \[\leadsto \left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t \]

8. Applied rewrites 41.8%

   \[\leadsto \left(-y\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right)} - t \]
9. Add Preprocessing

Alternative 19: 46.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift--.f64 41.7

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

8. Applied rewrites 41.7%

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

9. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. mul-1-neg N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-neg.f64 41.7

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

11. Applied rewrites 41.7%

    \[\leadsto \mathsf{fma}\left(-y, z, y\right) - t \]
12. Add Preprocessing

Alternative 20: 45.8% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-y * z) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in z around inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lift-neg.f64 41.6

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

8. Applied rewrites 41.6%

   \[\leadsto \left(-y\right) \cdot \color{blue}{z} - t \]
9. Add Preprocessing

Alternative 21: 35.2% accurate, 56.5× speedup

Math

\[\begin{array}{l} \\ y - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- y t))

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y - t
end function

Java

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

Python

def code(x, y, z, t):
	return y - t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y - t), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 90.3

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

8. Applied rewrites 90.3%

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

9. Taylor expanded in y around inf

   \[\leadsto y - t \]
10. Step-by-step derivation

11. Applied rewrites 33.3%

    \[\leadsto y - t \]
12. Add Preprocessing

Alternative 22: 35.0% accurate, 75.3× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- t))

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -t
end function

Java

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

Python

def code(x, y, z, t):
	return -t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 91.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 33.1

      \[\leadsto -t \]

5. Applied rewrites 33.1%

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

Reproduce

herbie shell --seed 2025043 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))