Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 86.1% → 99.5%

Time: 5.4s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * 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 * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * 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 * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. inv-pow N/A

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

   9. log-pow-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. log-pow-rev N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. unpow1 N/A

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

   16. lift-log.f64 99.5

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. associate--l+ N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lower--.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-log.f64 N/A

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

   16. lift--.f64 86.1

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

3. Applied rewrites 86.1%

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

4. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. inv-pow N/A

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

   9. log-pow-rev N/A

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

   10. lower-fma.f64 N/A

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

   11. log-pow-rev N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. unpow1 N/A

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

   16. lift-log.f64 99.5

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 99.4%

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

Alternative 4: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

2. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 99.1

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

4. Applied rewrites 99.1%

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

Alternative 5: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.2e-28)
   (fma (log y) x (- t))
   (if (<= x 1.15e-62) (- (* (- y) z) t) (- (* (log y) x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.2e-28) {
		tmp = fma(log(y), x, -t);
	} else if (x <= 1.15e-62) {
		tmp = (-y * z) - t;
	} else {
		tmp = (log(y) * x) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.2e-28)
		tmp = fma(log(y), x, Float64(-t));
	elseif (x <= 1.15e-62)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = Float64(Float64(log(y) * x) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.2e-28], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], If[LessEqual[x, 1.15e-62], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999984e-28

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. associate--l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lift--.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 92.5

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

   6. Applied rewrites 92.5%

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

   if -6.19999999999999984e-28 < x < 1.15e-62

   1. Initial program 77.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.4

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

   10. Applied rewrites 86.4%

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

   if 1.15e-62 < x

   1. Initial program 92.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 6: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-62}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -6.2e-28) t_1 (if (<= x 1.15e-62) (- (* (- y) z) 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 <= -6.2e-28) {
		tmp = t_1;
	} else if (x <= 1.15e-62) {
		tmp = (-y * z) - 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 <= (-6.2d-28)) then
        tmp = t_1
    else if (x <= 1.15d-62) then
        tmp = (-y * z) - 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 <= -6.2e-28) {
		tmp = t_1;
	} else if (x <= 1.15e-62) {
		tmp = (-y * z) - 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 <= -6.2e-28:
		tmp = t_1
	elif x <= 1.15e-62:
		tmp = (-y * z) - 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 <= -6.2e-28)
		tmp = t_1;
	elseif (x <= 1.15e-62)
		tmp = Float64(Float64(Float64(-y) * z) - 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 <= -6.2e-28)
		tmp = t_1;
	elseif (x <= 1.15e-62)
		tmp = (-y * z) - 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, -6.2e-28], t$95$1, If[LessEqual[x, 1.15e-62], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999984e-28 or 1.15e-62 < x

   1. Initial program 93.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if -6.19999999999999984e-28 < x < 1.15e-62

   1. Initial program 77.1%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-neg.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 86.4

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

   10. Applied rewrites 86.4%

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

Alternative 7: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-61}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4.8e+143) t_1 (if (<= x 4.5e-61) (- (* (- y) z) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -4.8e+143) {
		tmp = t_1;
	} else if (x <= 4.5e-61) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-4.8d+143)) then
        tmp = t_1
    else if (x <= 4.5d-61) then
        tmp = (-y * z) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -4.8e+143) {
		tmp = t_1;
	} else if (x <= 4.5e-61) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -4.8e+143:
		tmp = t_1
	elif x <= 4.5e-61:
		tmp = (-y * z) - 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 <= -4.8e+143)
		tmp = t_1;
	elseif (x <= 4.5e-61)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -4.8e+143)
		tmp = t_1;
	elseif (x <= 4.5e-61)
		tmp = (-y * z) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.8e+143], t$95$1, If[LessEqual[x, 4.5e-61], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999959e143 or 4.5e-61 < x

   1. Initial program 94.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 70.4

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

   4. Applied rewrites 70.4%

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

   if -4.79999999999999959e143 < x < 4.5e-61

   1. Initial program 79.9%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 78.0

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

   10. Applied rewrites 78.0%

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

Alternative 8: 57.4% accurate, 3.5× 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 86.1%

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

2. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 74.3

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

4. Applied rewrites 74.3%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lower-neg.f64 87.3

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

7. Applied rewrites 87.3%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 57.4

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

10. Applied rewrites 57.4%

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

Alternative 9: 43.7% accurate, 13.4× 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 86.1%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 43.7

      \[\leadsto -t \]

4. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64
  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))