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

Percentage Accurate: 88.5% → 99.6%

Time: 8.9s

Alternatives: 26

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. associate--l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lower--.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. associate--l+ N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lower--.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

function code(x, y, z, t)
	return fma(Float64(fma(Float64(-0.5 * y), Float64(z - 1.0), 1.0) - z), y, Float64(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] + 1.0), $MachinePrecision] - z), $MachinePrecision] * y + N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.5%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lift-*.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 6: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-y, z, \log y \cdot x\right) - t\\ \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (- y) z (* (log y) x)) t)))
   (if (<= x -9000000.0)
     t_1
     (if (<= x 2.2e-6) (- (fma (- 1.0 z) y (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-y, z, (log(y) * x)) - t;
	double tmp;
	if (x <= -9000000.0) {
		tmp = t_1;
	} else if (x <= 2.2e-6) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(-y), z, Float64(log(y) * x)) - t)
	tmp = 0.0
	if (x <= -9000000.0)
		tmp = t_1;
	elseif (x <= 2.2e-6)
		tmp = Float64(fma(Float64(1.0 - 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[((-y) * z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -9000000.0], t$95$1, If[LessEqual[x, 2.2e-6], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9e6 or 2.2000000000000001e-6 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.5%

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

   if -9e6 < x < 2.2000000000000001e-6

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.9

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

   8. Applied rewrites 10.9%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 98.2%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 97.8%

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

Alternative 7: 95.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15500000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 15000000000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -15500000.0)
   (- (* (log y) x) t)
   (if (<= x 15000000000000.0)
     (- (fma (- 1.0 z) y (- (log y))) t)
     (- (* (log y) (- x 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -15500000.0) {
		tmp = (log(y) * x) - t;
	} else if (x <= 15000000000000.0) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = (log(y) * (x - 1.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -15500000.0)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (x <= 15000000000000.0)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -15500000.0], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 15000000000000.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e7

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if -1.55e7 < x < 1.5e13

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.8

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

   8. Applied rewrites 10.8%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 97.2%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 96.9%

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

   if 1.5e13 < x

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 8: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15500000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 15000000000000:\\ \;\;\;\;\mathsf{fma}\left(-y, z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -15500000.0)
   (- (* (log y) x) t)
   (if (<= x 15000000000000.0)
     (- (fma (- y) z (- (log y))) t)
     (- (* (log y) (- x 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -15500000.0) {
		tmp = (log(y) * x) - t;
	} else if (x <= 15000000000000.0) {
		tmp = fma(-y, z, -log(y)) - t;
	} else {
		tmp = (log(y) * (x - 1.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -15500000.0)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (x <= 15000000000000.0)
		tmp = Float64(fma(Float64(-y), z, Float64(-log(y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -15500000.0], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 15000000000000.0], N[(N[((-y) * z + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e7

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.8

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

   5. Applied rewrites 92.8%

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

   if -1.55e7 < x < 1.5e13

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lift-log.f64 96.7

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

   11. Applied rewrites 96.7%

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

   if 1.5e13 < x

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 9: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -9 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-22}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -\left(z - 1\right)\right), y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -9e+70)
     t_1
     (if (<= x 6.6e-22)
       (- (- y (log y)) t)
       (if (<= x 1.3e+67)
         (fma (fma (* -0.5 y) z (- (- z 1.0))) 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 <= -9e+70) {
		tmp = t_1;
	} else if (x <= 6.6e-22) {
		tmp = (y - log(y)) - t;
	} else if (x <= 1.3e+67) {
		tmp = fma(fma((-0.5 * y), z, -(z - 1.0)), 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 <= -9e+70)
		tmp = t_1;
	elseif (x <= 6.6e-22)
		tmp = Float64(Float64(y - log(y)) - t);
	elseif (x <= 1.3e+67)
		tmp = fma(fma(Float64(-0.5 * y), z, Float64(-Float64(z - 1.0))), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9e+70], t$95$1, If[LessEqual[x, 6.6e-22], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.3e+67], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + (-N[(z - 1.0), $MachinePrecision])), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e70 or 1.3e67 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -8.9999999999999999e70 < x < 6.6000000000000002e-22

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 82.4

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

   8. Applied rewrites 82.4%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. log-pow-rev N/A

          \[\leadsto \left(y - \log \left({y}^{1}\right)\right) - t \]

       4. unpow1 N/A

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

       5. lower--.f64 N/A

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

       6. lift-log.f64 77.5

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

   11. Applied rewrites 77.5%

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

   if 6.6000000000000002e-22 < x < 1.3e67

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 54.8

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

   11. Applied rewrites 54.8%

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

Alternative 10: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -9 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-22}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -\left(z - 1\right)\right), y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -9e+70)
     t_1
     (if (<= x 6.6e-22)
       (- (- (log y)) t)
       (if (<= x 1.3e+67)
         (fma (fma (* -0.5 y) z (- (- z 1.0))) 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 <= -9e+70) {
		tmp = t_1;
	} else if (x <= 6.6e-22) {
		tmp = -log(y) - t;
	} else if (x <= 1.3e+67) {
		tmp = fma(fma((-0.5 * y), z, -(z - 1.0)), 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 <= -9e+70)
		tmp = t_1;
	elseif (x <= 6.6e-22)
		tmp = Float64(Float64(-log(y)) - t);
	elseif (x <= 1.3e+67)
		tmp = fma(fma(Float64(-0.5 * y), z, Float64(-Float64(z - 1.0))), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9e+70], t$95$1, If[LessEqual[x, 6.6e-22], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], If[LessEqual[x, 1.3e+67], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + (-N[(z - 1.0), $MachinePrecision])), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e70 or 1.3e67 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -8.9999999999999999e70 < x < 6.6000000000000002e-22

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.7

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

   8. Applied rewrites 10.7%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 94.4%

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

   12. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 77.4

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

   14. Applied rewrites 77.4%

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

   if 6.6000000000000002e-22 < x < 1.3e67

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 54.8

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

   11. Applied rewrites 54.8%

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

Alternative 11: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq -1 \cdot 10^{+121}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{elif}\;z - 1 \leq 10^{+178}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -z\right), y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) -1e+121)
   (- (* (- y) z) t)
   (if (<= (- z 1.0) 1e+178)
     (- (- (log y)) t)
     (fma (fma (* -0.5 y) z (- z)) y (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -1e+121) {
		tmp = (-y * z) - t;
	} else if ((z - 1.0) <= 1e+178) {
		tmp = -log(y) - t;
	} else {
		tmp = fma(fma((-0.5 * y), z, -z), y, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -1e+121)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	elseif (Float64(z - 1.0) <= 1e+178)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = fma(fma(Float64(-0.5 * y), z, Float64(-z)), y, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], -1e+121], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+178], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], N[(N[(N[(-0.5 * y), $MachinePrecision] * z + (-z)), $MachinePrecision] * y + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -1.00000000000000004e121

   1. Initial program 67.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 62.6

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

   8. Applied rewrites 62.6%

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

   if -1.00000000000000004e121 < (-.f64 z #s(literal 1 binary64)) < 1.0000000000000001e178

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 5.1

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

   8. Applied rewrites 5.1%

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

   9. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 62.9%

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

   12. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lift-log.f64 N/A

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

       3. lift-neg.f64 59.9

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

   14. Applied rewrites 59.9%

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

   if 1.0000000000000001e178 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 60.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 67.3

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

   11. Applied rewrites 67.3%

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

   12. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 67.3

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

   14. Applied rewrites 67.3%

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

Alternative 12: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -9000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -9000000.0) t_1 (if (<= x 2.2e-6) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -9000000.0) {
		tmp = t_1;
	} else if (x <= 2.2e-6) {
		tmp = (y - 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) - t
    if (x <= (-9000000.0d0)) then
        tmp = t_1
    else if (x <= 2.2d-6) then
        tmp = (y - 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 tmp;
	if (x <= -9000000.0) {
		tmp = t_1;
	} else if (x <= 2.2e-6) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	tmp = 0
	if x <= -9000000.0:
		tmp = t_1
	elif x <= 2.2e-6:
		tmp = (y - 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)
	tmp = 0.0
	if (x <= -9000000.0)
		tmp = t_1;
	elseif (x <= 2.2e-6)
		tmp = Float64(Float64(y - 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;
	tmp = 0.0;
	if (x <= -9000000.0)
		tmp = t_1;
	elseif (x <= 2.2e-6)
		tmp = (y - 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]}, If[LessEqual[x, -9000000.0], t$95$1, If[LessEqual[x, 2.2e-6], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9e6 or 2.2000000000000001e-6 < x

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -9e6 < x < 2.2000000000000001e-6

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 82.0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. metadata-eval N/A

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

       3. log-pow-rev N/A

          \[\leadsto \left(y - \log \left({y}^{1}\right)\right) - t \]

       4. unpow1 N/A

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

       5. lower--.f64 N/A

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

       6. lift-log.f64 80.8

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

   11. Applied rewrites 80.8%

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

Alternative 13: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around 0

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

   1. lower--.f64 99.2

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

8. Applied rewrites 99.2%

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

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.1%

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

Alternative 15: 87.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift--.f64 87.5

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

8. Applied rewrites 87.5%

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

Alternative 16: 87.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   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 87.4

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

5. Applied rewrites 87.4%

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

Alternative 17: 47.0% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 47.0

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

11. Applied rewrites 47.0%

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

12. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. +-commutative N/A

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

    4. associate--l+ N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. *-commutative N/A

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

    12. lift-*.f64 N/A

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

    13. lift--.f64 N/A

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

    14. lift--.f64 47.0

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

14. Applied rewrites 47.0%

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

Alternative 18: 47.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 47.0

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

11. Applied rewrites 47.0%

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

Alternative 19: 46.9% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lift--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift--.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 47.0

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

11. Applied rewrites 47.0%

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

12. Taylor expanded in z around inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 46.9

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

14. Applied rewrites 46.9%

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

Alternative 20: 43.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -11000000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1920000:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -11000000.0) (- t) (if (<= t 1920000.0) (fma (- y) z y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -11000000.0) {
		tmp = -t;
	} else if (t <= 1920000.0) {
		tmp = fma(-y, z, y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -11000000.0)
		tmp = Float64(-t);
	elseif (t <= 1920000.0)
		tmp = fma(Float64(-y), z, y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -11000000.0], (-t), If[LessEqual[t, 1920000.0], N[((-y) * z + y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1e7 or 1.92e6 < t

   1. Initial program 93.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.7

         \[\leadsto -t \]

   5. Applied rewrites 69.7%

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

   if -1.1e7 < t < 1.92e6

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.9

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

   8. Applied rewrites 10.9%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 18.6

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

   11. Applied rewrites 18.6%

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

   12. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. mul-1-neg N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-neg.f64 18.6

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

   14. Applied rewrites 18.6%

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

Alternative 21: 43.2% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -11000000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1920000:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -11000000.0) (- t) (if (<= t 1920000.0) (* (- z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -11000000.0) {
		tmp = -t;
	} else if (t <= 1920000.0) {
		tmp = -z * y;
	} 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 <= (-11000000.0d0)) then
        tmp = -t
    else if (t <= 1920000.0d0) then
        tmp = -z * y
    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 <= -11000000.0) {
		tmp = -t;
	} else if (t <= 1920000.0) {
		tmp = -z * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -11000000.0:
		tmp = -t
	elif t <= 1920000.0:
		tmp = -z * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -11000000.0)
		tmp = Float64(-t);
	elseif (t <= 1920000.0)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -11000000.0)
		tmp = -t;
	elseif (t <= 1920000.0)
		tmp = -z * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -11000000.0], (-t), If[LessEqual[t, 1920000.0], N[((-z) * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1e7 or 1.92e6 < t

   1. Initial program 93.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.7

         \[\leadsto -t \]

   5. Applied rewrites 69.7%

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

   if -1.1e7 < t < 1.92e6

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.9

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

   8. Applied rewrites 10.9%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 18.6

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

   11. Applied rewrites 18.6%

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

   12. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 18.1

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

   14. Applied rewrites 18.1%

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

Alternative 22: 46.8% accurate, 18.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return fma((1.0 - z), y, -t);
}

Julia

function code(x, y, z, t)
	return fma(Float64(1.0 - z), y, Float64(-t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 - z), $MachinePrecision] * y + (-t)), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \color{blue}{\left(\log y \cdot \left(x - 1\right) - t\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y + \left(\color{blue}{\log y \cdot \left(x - 1\right)} - t\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right), \color{blue}{y}, \log y \cdot \left(x - 1\right) - t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right) + -1 \cdot \left(z - 1\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right) + -1 \cdot \left(z - 1\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, \mathsf{neg}\left(\left(z - 1\right)\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   11. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   14. lift-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   15. lift--.f64 99.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

9. Taylor expanded in t around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z, -\left(z - 1\right)\right), y, -1 \cdot t\right) \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z, -\left(z - 1\right)\right), y, \mathsf{neg}\left(t\right)\right) \]

    2. lower-neg.f64 47.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -\left(z - 1\right)\right), y, -t\right) \]

11. Applied rewrites 47.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z, -\left(z - 1\right)\right), y, -t\right) \]

12. Taylor expanded in y around 0

    \[\leadsto \mathsf{fma}\left(1 - z, y, -t\right) \]
13. Step-by-step derivation

    1. lift--.f64 46.8

       \[\leadsto \mathsf{fma}\left(1 - z, y, -t\right) \]

14. Applied rewrites 46.8%

    \[\leadsto \mathsf{fma}\left(1 - z, y, -t\right) \]
15. Add Preprocessing

Alternative 23: 46.6% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \left(-y\right) \cdot z - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* (- y) z) t))

C

double code(double x, double y, double z, double t) {
	return (-y * z) - t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-y * z) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-y * z) - t;
}

Python

def code(x, y, z, t):
	return (-y * z) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-y) * z) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-y * z) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot y\right) \cdot \left(z - 1\right) + \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   2. mul-1-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(z - 1\right) + \log \color{blue}{y} \cdot \left(x - 1\right)\right) - t \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{z - 1}, \log y \cdot \left(x - 1\right)\right) - t \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, \color{blue}{z} - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - \color{blue}{1}, \log y \cdot \left(x - 1\right)\right) - t \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   7. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   8. lift--.f64 99.2

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right)} - t \]

6. Taylor expanded in z around inf

   \[\leadsto -1 \cdot \color{blue}{\left(y \cdot z\right)} - t \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot y\right) \cdot z - t \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z - t \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot z - t \]

   4. lift-neg.f64 46.6

      \[\leadsto \left(-y\right) \cdot z - t \]

8. Applied rewrites 46.6%

   \[\leadsto \left(-y\right) \cdot \color{blue}{z} - t \]
9. Add Preprocessing

Alternative 24: 35.8% accurate, 56.5× speedup

Math

\[\begin{array}{l} \\ y - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- y t))

C

double code(double x, double y, double z, double t) {
	return y - t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return y - t;
}

Python

def code(x, y, z, t):
	return y - t

Julia

function code(x, y, z, t)
	return Float64(y - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(y - t), $MachinePrecision]

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(-1 \cdot y\right) \cdot \left(z - 1\right) + \color{blue}{\log y} \cdot \left(x - 1\right)\right) - t \]

   2. mul-1-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(z - 1\right) + \log \color{blue}{y} \cdot \left(x - 1\right)\right) - t \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{z - 1}, \log y \cdot \left(x - 1\right)\right) - t \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, \color{blue}{z} - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   5. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - \color{blue}{1}, \log y \cdot \left(x - 1\right)\right) - t \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   7. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

   8. lift--.f64 99.2

      \[\leadsto \mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right) - t \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-y, z - 1, \log y \cdot \left(x - 1\right)\right)} - t \]

6. Taylor expanded in z around 0

   \[\leadsto \left(y + \color{blue}{\log y \cdot \left(x - 1\right)}\right) - t \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\log y \cdot \left(x - 1\right) + y\right) - t \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log y, x - \color{blue}{1}, y\right) - t \]

   3. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log y, x - 1, y\right) - t \]

   4. lift--.f64 87.5

      \[\leadsto \mathsf{fma}\left(\log y, x - 1, y\right) - t \]

8. Applied rewrites 87.5%

   \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{x - 1}, y\right) - t \]

9. Taylor expanded in y around inf

   \[\leadsto y - t \]
10. Step-by-step derivation

11. Applied rewrites 35.8%

    \[\leadsto y - t \]
12. Add Preprocessing

Alternative 25: 35.6% accurate, 75.3× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- t))

C

double code(double x, double y, double z, double t) {
	return -t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -t
end function

Java

public static double code(double x, double y, double z, double t) {
	return -t;
}

Python

def code(x, y, z, t):
	return -t

Julia

function code(x, y, z, t)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(t\right) \]

   2. lower-neg.f64 35.6

      \[\leadsto -t \]

5. Applied rewrites 35.6%

   \[\leadsto \color{blue}{-t} \]
6. Add Preprocessing

Alternative 26: 2.9% accurate, 226.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 y)

C

double code(double x, double y, double z, double t) {
	return y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y
end function

Java

public static double code(double x, double y, double z, double t) {
	return y;
}

Python

def code(x, y, z, t):
	return y

Julia

function code(x, y, z, t)
	return y
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = y;
end

Wolfram

code[x_, y_, z_, t_] := y

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \color{blue}{\left(\log y \cdot \left(x - 1\right) - t\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) \cdot y + \left(\color{blue}{\log y \cdot \left(x - 1\right)} - t\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right), \color{blue}{y}, \log y \cdot \left(x - 1\right) - t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right) + -1 \cdot \left(z - 1\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot y\right) \cdot \left(z - 1\right) + -1 \cdot \left(z - 1\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   8. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -1 \cdot \left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, \mathsf{neg}\left(\left(z - 1\right)\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   11. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   14. lift-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2} \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

   15. lift--.f64 99.5

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right) \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5 \cdot y, z - 1, -\left(z - 1\right)\right), y, \log y \cdot \left(x - 1\right) - t\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{1}{y}\right) - \frac{z}{y}\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{1}{y}\right) - \frac{z}{y}\right) \cdot {y}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{1}{y}\right) - \frac{z}{y}\right) \cdot {y}^{\color{blue}{2}} \]

   3. associate--l+ N/A

      \[\leadsto \left(\frac{-1}{2} \cdot \left(z - 1\right) + \left(\frac{1}{y} - \frac{z}{y}\right)\right) \cdot {y}^{2} \]

   4. div-sub N/A

      \[\leadsto \left(\frac{-1}{2} \cdot \left(z - 1\right) + \frac{1 - z}{y}\right) \cdot {y}^{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z - 1, \frac{1 - z}{y}\right) \cdot {y}^{2} \]

   6. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z - 1, \frac{1 - z}{y}\right) \cdot {y}^{2} \]

   7. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z - 1, \frac{1 - z}{y}\right) \cdot {y}^{2} \]

   8. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z - 1, \frac{1 - z}{y}\right) \cdot {y}^{2} \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z - 1, \frac{1 - z}{y}\right) \cdot \left(y \cdot y\right) \]

   10. lower-*.f64 8.1

       \[\leadsto \mathsf{fma}\left(-0.5, z - 1, \frac{1 - z}{y}\right) \cdot \left(y \cdot y\right) \]

8. Applied rewrites 8.1%

   \[\leadsto \mathsf{fma}\left(-0.5, z - 1, \frac{1 - z}{y}\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto y \cdot \left(1 - \color{blue}{z}\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(1 - z\right) \cdot y \]

    2. lower-*.f64 N/A

       \[\leadsto \left(1 - z\right) \cdot y \]

    3. lift--.f64 13.7

       \[\leadsto \left(1 - z\right) \cdot y \]

11. Applied rewrites 13.7%

    \[\leadsto \left(1 - z\right) \cdot y \]

12. Taylor expanded in z around 0

    \[\leadsto y \]
13. Step-by-step derivation

14. Applied rewrites 2.9%

    \[\leadsto y \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(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))