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

Percentage Accurate: 89.1% → 99.4%

Time: 6.4s

Alternatives: 13

Speedup: 1.5×

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: 89.1% 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: 99.4% accurate, 0.9× 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 89.1%

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

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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 99.4

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

4. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× 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 89.1%

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

2. Taylor expanded in y around 0

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

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

4. Applied rewrites 99.4%

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

Alternative 3: 99.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 4: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 98.9

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

10. Applied rewrites 98.9%

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

Alternative 5: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-z, y, \log y \cdot x\right) - t\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (- z) y (* (log y) x)) t)))
   (if (<= x -1.02e+14)
     t_1
     (if (<= x 6.6e-13) (- (fma (- z) y (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-z, y, (log(y) * x)) - t;
	double tmp;
	if (x <= -1.02e+14) {
		tmp = t_1;
	} else if (x <= 6.6e-13) {
		tmp = fma(-z, y, -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(-z), y, Float64(log(y) * x)) - t)
	tmp = 0.0
	if (x <= -1.02e+14)
		tmp = t_1;
	elseif (x <= 6.6e-13)
		tmp = Float64(fma(Float64(-z), y, Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-z) * y + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.02e+14], t$95$1, If[LessEqual[x, 6.6e-13], N[(N[((-z) * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02e14 or 6.6000000000000001e-13 < x

   1. Initial program 93.6%

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

   2. Taylor expanded in y around 0

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

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.2

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

   10. Applied rewrites 99.2%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-log.f64 98.1

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

   13. Applied rewrites 98.1%

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

   if -1.02e14 < x < 6.6000000000000001e-13

   1. Initial program 84.9%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   3. 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 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.7

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

   10. Applied rewrites 98.7%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-log.f64 97.6

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

   13. Applied rewrites 97.6%

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

Alternative 6: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(x - 1\right) - t\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) (- x 1.0)) t)))
   (if (<= x -1.4e-15)
     t_1
     (if (<= x 2.5e-24) (- (fma (- z) y (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * (x - 1.0)) - t;
	double tmp;
	if (x <= -1.4e-15) {
		tmp = t_1;
	} else if (x <= 2.5e-24) {
		tmp = fma(-z, y, -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * Float64(x - 1.0)) - t)
	tmp = 0.0
	if (x <= -1.4e-15)
		tmp = t_1;
	elseif (x <= 2.5e-24)
		tmp = Float64(fma(Float64(-z), y, Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.4e-15], t$95$1, If[LessEqual[x, 2.5e-24], N[(N[((-z) * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000007e-15 or 2.4999999999999999e-24 < x

   1. Initial program 93.2%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - t \]
   3. 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 92.3

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

   4. Applied rewrites 92.3%

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

   if -1.40000000000000007e-15 < x < 2.4999999999999999e-24

   1. Initial program 84.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   3. 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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.7

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

   10. Applied rewrites 98.7%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-log.f64 98.7

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

   13. Applied rewrites 98.7%

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

Alternative 7: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+237}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.45e+237) (- (* (- z) y) t) (- (* (log y) (- x 1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.45e+237) {
		tmp = (-z * y) - t;
	} else {
		tmp = (log(y) * (x - 1.0)) - 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 (z <= (-2.45d+237)) then
        tmp = (-z * y) - t
    else
        tmp = (log(y) * (x - 1.0d0)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.45e+237) {
		tmp = (-z * y) - t;
	} else {
		tmp = (Math.log(y) * (x - 1.0)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.45e+237:
		tmp = (-z * y) - t
	else:
		tmp = (math.log(y) * (x - 1.0)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.45e+237)
		tmp = Float64(Float64(Float64(-z) * y) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.45e+237)
		tmp = (-z * y) - t;
	else
		tmp = (log(y) * (x - 1.0)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.45e+237], N[(N[((-z) * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.45e237

   1. Initial program 58.4%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   3. 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 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 67.9

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

   7. Applied rewrites 67.9%

      \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.5

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

   10. Applied rewrites 66.5%

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

   if -2.45e237 < z

   1. Initial program 91.2%

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

   2. Taylor expanded in y around 0

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

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

   4. Applied rewrites 90.1%

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

Alternative 8: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (<= t_2 -10000.0) t_1 (if (<= t_2 1000.0) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (log(y) * x) - t
    t_2 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if (t_2 <= (-10000.0d0)) then
        tmp = t_1
    else if (t_2 <= 1000.0d0) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - t;
	double t_2 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if (t_2 <= -10000.0) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	t_2 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if t_2 <= -10000.0:
		tmp = t_1
	elif t_2 <= 1000.0:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - t;
	t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if (t_2 <= -10000.0)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = 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[LessEqual[t$95$2, -10000.0], t$95$1, If[LessEqual[t$95$2, 1000.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

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)))) < -1e4 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 93.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} - t \]
   3. 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 90.2

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

   4. Applied rewrites 90.2%

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

   if -1e4 < (+.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 85.2%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - t \]
   3. 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 84.8

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto \left(-\log y\right) - t \]

      3. lift-log.f64 83.9

         \[\leadsto \left(-\log y\right) - t \]

   7. Applied rewrites 83.9%

      \[\leadsto \left(-\log y\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -3.95 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -3.95e+14) t_1 (if (<= x 2.4e+97) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -3.95e+14) {
		tmp = t_1;
	} else if (x <= 2.4e+97) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-3.95d+14)) then
        tmp = t_1
    else if (x <= 2.4d+97) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -3.95e+14) {
		tmp = t_1;
	} else if (x <= 2.4e+97) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -3.95e+14:
		tmp = t_1
	elif x <= 2.4e+97:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -3.95e+14)
		tmp = t_1;
	elseif (x <= 2.4e+97)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -3.95e+14)
		tmp = t_1;
	elseif (x <= 2.4e+97)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.95e+14], t$95$1, If[LessEqual[x, 2.4e+97], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.95e14 or 2.4e97 < x

   1. Initial program 95.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   3. 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 73.6

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

   4. Applied rewrites 73.6%

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

   if -3.95e14 < x < 2.4e97

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log y \cdot \left(x - 1\right)} - t \]
   3. 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 83.7

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto \left(-\log y\right) - t \]

      3. lift-log.f64 76.0

         \[\leadsto \left(-\log y\right) - t \]

   7. Applied rewrites 76.0%

      \[\leadsto \left(-\log y\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -2.8e+170) t_1 (if (<= x 2.4e+97) (- (* (- z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -2.8e+170) {
		tmp = t_1;
	} else if (x <= 2.4e+97) {
		tmp = (-z * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-2.8d+170)) then
        tmp = t_1
    else if (x <= 2.4d+97) then
        tmp = (-z * y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -2.8e+170) {
		tmp = t_1;
	} else if (x <= 2.4e+97) {
		tmp = (-z * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -2.8e+170:
		tmp = t_1
	elif x <= 2.4e+97:
		tmp = (-z * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -2.8e+170)
		tmp = t_1;
	elseif (x <= 2.4e+97)
		tmp = Float64(Float64(Float64(-z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -2.8e+170)
		tmp = t_1;
	elseif (x <= 2.4e+97)
		tmp = (-z * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.8e+170], t$95$1, If[LessEqual[x, 2.4e+97], N[(N[((-z) * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000015e170 or 2.4e97 < x

   1. Initial program 97.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   3. 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 81.5

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

   4. Applied rewrites 81.5%

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

   if -2.80000000000000015e170 < x < 2.4e97

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   3. 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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 57.8

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

   7. Applied rewrites 57.8%

      \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

       \[\leadsto \left(-z\right) \cdot y - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-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 = (-z * y) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
3. 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 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 47.1

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

7. Applied rewrites 47.1%

   \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]

8. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 46.7

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

10. Applied rewrites 46.7%

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

Alternative 12: 43.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.9e-18) (- t) (if (<= t 5.2e+16) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-18) {
		tmp = -t;
	} else if (t <= 5.2e+16) {
		tmp = -y * z;
	} else {
		tmp = -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 (t <= (-2.9d-18)) then
        tmp = -t
    else if (t <= 5.2d+16) then
        tmp = -y * z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.9e-18) {
		tmp = -t;
	} else if (t <= 5.2e+16) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.9e-18:
		tmp = -t
	elif t <= 5.2e+16:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.9e-18)
		tmp = Float64(-t);
	elseif (t <= 5.2e+16)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.9e-18)
		tmp = -t;
	elseif (t <= 5.2e+16)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.9e-18], (-t), If[LessEqual[t, 5.2e+16], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e-18 or 5.2e16 < t

   1. Initial program 94.0%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.3

         \[\leadsto -t \]

   4. Applied rewrites 68.3%

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

   if -2.9e-18 < t < 5.2e16

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 3.9

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

   4. Applied rewrites 3.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 17.6

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

   7. Applied rewrites 17.6%

      \[\leadsto \left(-y\right) \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 36.3% accurate, 16.1× 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 89.1%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 36.3

      \[\leadsto -t \]

4. Applied rewrites 36.3%

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

Reproduce

herbie shell --seed 2025119 
(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))