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

Percentage Accurate: 89.5% → 99.8%

Time: 12.3s

Alternatives: 18

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: 89.5% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+ (* (- x 1.0) (log y)) (* (- z 1.0) (- (log1p (* (- 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) * (log1p((-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) * (Math.log1p((-y * y)) - Math.log1p(y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (math.log1p((-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(log1p(Float64(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[Log[1 + N[((-y) * y), $MachinePrecision]], $MachinePrecision] - N[Log[1 + y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   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. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 57.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\\ \mathbf{if}\;t\_1 \leq 45 \lor \neg \left(t\_1 \leq 1000\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t)))
   (if (or (<= t_1 45.0) (not (<= t_1 1000.0)))
     (-
      (* (* (- (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) 1.0) y) z)
      t)
     (- (log y)))))

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)))) - t;
	double tmp;
	if ((t_1 <= 45.0) || !(t_1 <= 1000.0)) {
		tmp = ((((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t;
	} else {
		tmp = -log(y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
	tmp = 0.0
	if ((t_1 <= 45.0) || !(t_1 <= 1000.0))
		tmp = Float64(Float64(Float64(Float64(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t);
	else
		tmp = Float64(-log(y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, 45.0], N[Not[LessEqual[t$95$1, 1000.0]], $MachinePrecision]], N[(N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], (-N[Log[y], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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)))) t) < 45 or 1e3 < (-.f64 (+.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)))) t)

   1. Initial program 93.1%

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 53.8%

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

   if 45 < (-.f64 (+.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)))) t) < 1e3

   1. Initial program 86.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in t around 0

      \[\leadsto -1 \cdot \log y \]
   10. Step-by-step derivation

   11. Applied rewrites 84.8%

       \[\leadsto -\log y \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 45 \lor \neg \left(\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 1000\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 45:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t) 45.0)
   (-
    (* (* (- (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) 1.0) y) z)
    t)
   (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t) <= 45.0) {
		tmp = ((((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t;
	} else {
		tmp = -(log(y) + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t) <= 45.0)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t);
	else
		tmp = Float64(-Float64(log(y) + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[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], 45.0], N[(N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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)))) t) < 45

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 54.9%

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

   if 45 < (-.f64 (+.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)))) t)

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 45:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   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. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (*
    (- z 1.0)
    (* (fma (fma (fma -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) * (fma(fma(fma(-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(fma(fma(fma(-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[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 99.6%

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

Alternative 6: 99.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (* (- z 1.0) (* (fma (fma -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) * (fma(fma(-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(fma(fma(-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[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 99.6%

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

Alternative 7: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 99.3%

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

Alternative 8: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+20} \lor \neg \left(x - 1 \leq 50000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -2e+20) (not (<= (- x 1.0) 50000000000.0)))
   (fma (log y) x (- t))
   (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -2e+20) || !((x - 1.0) <= 50000000000.0)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = -(log(y) + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -2e+20) || !(Float64(x - 1.0) <= 50000000000.0))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = Float64(-Float64(log(y) + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+20], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 50000000000.0]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e20 or 5e10 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.2%

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 96.2%

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

   if -2e20 < (-.f64 x #s(literal 1 binary64)) < 5e10

   1. Initial program 87.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+20} \lor \neg \left(x - 1 \leq 50000000000\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   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. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 99.6%

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

11. Taylor expanded in y around 0

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(-1 \cdot z\right) \cdot y\right) - t \]
12. Step-by-step derivation

13. Applied rewrites 99.0%

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

Alternative 10: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-22}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{elif}\;t \leq 1200000:\\ \;\;\;\;\log y \cdot \left(x - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.5e-22)
   (- (* (- y) z) t)
   (if (<= t 1200000.0)
     (* (log y) (- x 1.0))
     (-
      (* (* (- (* (fma (fma -0.25 y -0.3333333333333333) y -0.5) y) 1.0) y) z)
      t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.5e-22) {
		tmp = (-y * z) - t;
	} else if (t <= 1200000.0) {
		tmp = log(y) * (x - 1.0);
	} else {
		tmp = ((((fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.5e-22)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	elseif (t <= 1200000.0)
		tmp = Float64(log(y) * Float64(x - 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5) * y) - 1.0) * y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.5e-22], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1200000.0], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.49999999999999987e-22

   1. Initial program 90.4%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 70.3%

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

   if -4.49999999999999987e-22 < t < 1.2e6

   1. Initial program 90.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 89.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 88.6%

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

   if 1.2e6 < t

   1. Initial program 94.8%

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

   3. Taylor expanded in 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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 80.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 80.3%

       \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+23} \lor \neg \left(x \leq 1.05 \cdot 10^{+45}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.55e+23) (not (<= x 1.05e+45)))
   (* (log y) x)
   (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.55e+23) || !(x <= 1.05e+45)) {
		tmp = log(y) * x;
	} else {
		tmp = -(log(y) + t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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
    real(8) :: tmp
    if ((x <= (-2.55d+23)) .or. (.not. (x <= 1.05d+45))) then
        tmp = log(y) * x
    else
        tmp = -(log(y) + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.55e+23) || !(x <= 1.05e+45)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -(Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.55e+23) or not (x <= 1.05e+45):
		tmp = math.log(y) * x
	else:
		tmp = -(math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.55e+23) || !(x <= 1.05e+45))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(-Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.55e+23) || ~((x <= 1.05e+45)))
		tmp = log(y) * x;
	else
		tmp = -(log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.55e+23], N[Not[LessEqual[x, 1.05e+45]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -2.5500000000000001e23 or 1.04999999999999997e45 < x

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 76.1%

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

   if -2.5500000000000001e23 < x < 1.04999999999999997e45

   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

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \cdot 10^{+23} \lor \neg \left(x \leq 1.05 \cdot 10^{+45}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 90.3%

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

Alternative 13: 46.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 42.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 42.5%

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

Alternative 14: 46.1% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 42.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 42.5%

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

Alternative 15: 46.1% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 42.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 42.5%

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

Alternative 16: 46.0% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 42.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 42.2%

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

Alternative 17: 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 91.4%

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 42.7%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 42.0%

    \[\leadsto \left(-y\right) \cdot z - t \]
12. Add Preprocessing

Alternative 18: 35.7% 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 91.4%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 33.9%

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

Reproduce

herbie shell --seed 2025026 
(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))