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

Percentage Accurate: 89.6% → 99.8%

Time: 12.3s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 150:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{elif}\;t\_2 \leq 665:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (<= t_2 -1e+48)
     t_1
     (if (<= t_2 150.0)
       (- (fma (- y) z y) t)
       (if (<= t_2 665.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -1e+48) {
		tmp = t_1;
	} else if (t_2 <= 150.0) {
		tmp = fma(-y, z, y) - t;
	} else if (t_2 <= 665.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -1e+48)
		tmp = t_1;
	elseif (t_2 <= 150.0)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	elseif (t_2 <= 665.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+48], t$95$1, If[LessEqual[t$95$2, 150.0], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 665.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.00000000000000004e48 or 665 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 97.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -1.00000000000000004e48 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 150

   1. Initial program 68.4%

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

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 76.4%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 76.4%

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

   if 150 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 665

   1. Initial program 89.4%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log1p.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. log-rec N/A

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

      5. lower-*.f64 N/A

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

      6. log-rec N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 89.4%

       \[\leadsto \log y \cdot -1 - t \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 89.4%

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

Alternative 3: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 150:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{elif}\;t\_2 \leq 10^{+142}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (<= t_2 -1e+102)
     t_1
     (if (<= t_2 150.0)
       (- (* (* (- (* -0.5 y) 1.0) z) y) t)
       (if (<= t_2 1e+142) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -1e+102) {
		tmp = t_1;
	} else if (t_2 <= 150.0) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else if (t_2 <= 1e+142) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if (t_2 <= (-1d+102)) then
        tmp = t_1
    else if (t_2 <= 150.0d0) then
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    else if (t_2 <= 1d+142) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if (t_2 <= -1e+102) {
		tmp = t_1;
	} else if (t_2 <= 150.0) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else if (t_2 <= 1e+142) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if t_2 <= -1e+102:
		tmp = t_1
	elif t_2 <= 150.0:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	elif t_2 <= 1e+142:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -1e+102)
		tmp = t_1;
	elseif (t_2 <= 150.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	elseif (t_2 <= 1e+142)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if (t_2 <= -1e+102)
		tmp = t_1;
	elseif (t_2 <= 150.0)
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	elseif (t_2 <= 1e+142)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+102], t$95$1, If[LessEqual[t$95$2, 150.0], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 1e+142], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -9.99999999999999977e101 or 1.00000000000000005e142 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log1p.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   7. Applied rewrites 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), y \cdot y, -0.5\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right) - t} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right)} - t \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(x - 1\right) \cdot \log y\right)} - t \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(\left(x - 1\right) \cdot \log y - t\right)} \]

   9. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), -0.5\right) \cdot y, y, -1\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(y\right), z - 1, \log y \cdot \left(x - 1\right) - t\right)} \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. remove-double-neg N/A

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

       3. mul-1-neg N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. log-rec N/A

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

       6. lower-*.f64 N/A

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

       7. log-rec N/A

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

       8. distribute-rgt-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. remove-double-neg N/A

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

       11. lower-log.f64 76.8

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

   12. Applied rewrites 76.8%

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

   if -9.99999999999999977e101 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 150

   1. Initial program 78.0%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log1p.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. remove-double-neg N/A

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

      16. mul-1-neg N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 71.9%

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

   if 150 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1.00000000000000005e142

   1. Initial program 92.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log1p.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. log-rec N/A

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

      5. lower-*.f64 N/A

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

      6. log-rec N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower-log.f64 N/A

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

      11. lower--.f64 92.6

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 81.5%

       \[\leadsto \log y \cdot -1 - t \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 81.5%

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

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), -0.5\right) \cdot y, y, -1\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(y\right), z - 1, \log y \cdot \left(x - 1\right) - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  (-
   (*
    (*
     (fma
      (* (fma (* y y) (fma (* -0.25 y) y -0.3333333333333333) -0.5) y)
      y
      -1.0)
     y)
    y)
   (log1p y))
  (- z 1.0)
  (- (* (log y) (- x 1.0)) t)))

C

double code(double x, double y, double z, double t) {
	return fma((((fma((fma((y * y), fma((-0.25 * y), y, -0.3333333333333333), -0.5) * y), y, -1.0) * y) * y) - log1p(y)), (z - 1.0), ((log(y) * (x - 1.0)) - t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(Float64(Float64(fma(Float64(fma(Float64(y * y), fma(Float64(-0.25 * y), y, -0.3333333333333333), -0.5) * y), y, -1.0) * y) * y) - log1p(y)), Float64(z - 1.0), Float64(Float64(log(y) * Float64(x - 1.0)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(-0.25 * y), $MachinePrecision] * y + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] * y), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] - N[Log[1 + y], $MachinePrecision]), $MachinePrecision] * N[(z - 1.0), $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

7. Applied rewrites 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), y \cdot y, -0.5\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right) - t} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right)} - t \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(x - 1\right) \cdot \log y\right)} - t \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(\left(x - 1\right) \cdot \log y - t\right)} \]

9. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), -0.5\right) \cdot y, y, -1\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(y\right), z - 1, \log y \cdot \left(x - 1\right) - t\right)} \]
10. Add Preprocessing

Alternative 5: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

Alternative 6: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

   10. log-rec N/A

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

   11. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 7: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval N/A

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

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

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 8: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. remove-double-neg N/A

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

   16. mul-1-neg N/A

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

7. Applied rewrites 99.3%

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

Alternative 9: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

   10. log-rec N/A

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

   11. lower-fma.f64 N/A

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

   12. mul-1-neg N/A

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

   13. log-rec N/A

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

   14. remove-double-neg N/A

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

   15. lower-log.f64 N/A

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

   16. lower--.f64 N/A

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

5. Applied rewrites 99.3%

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

Alternative 10: 95.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1000:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+39}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1000.0)
   (- (* (log y) (- x 1.0)) t)
   (if (<= x 1.85e+39)
     (- (- (fma (- z 1.0) y (log y))) t)
     (- (* (log y) x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1000.0) {
		tmp = (log(y) * (x - 1.0)) - t;
	} else if (x <= 1.85e+39) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = (log(y) * x) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1000.0)
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	elseif (x <= 1.85e+39)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = Float64(Float64(log(y) * x) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1000.0], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.85e+39], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e3

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower--.f64 96.6

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

   5. Applied rewrites 96.6%

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

   if -1e3 < x < 1.85000000000000006e39

   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

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.8%

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

   if 1.85000000000000006e39 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 11: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+144} \lor \neg \left(x - 1 \leq 5 \cdot 10^{+98}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -4e+144) (not (<= (- x 1.0) 5e+98)))
   (* (log y) x)
   (- (* (* (- (* (fma -0.3333333333333333 y -0.5) y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -4e+144) || !((x - 1.0) <= 5e+98)) {
		tmp = log(y) * x;
	} else {
		tmp = ((((fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -4e+144) || !(Float64(x - 1.0) <= 5e+98))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -4e+144], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 5e+98]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.00000000000000009e144 or 4.9999999999999998e98 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 99.3%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. lower-log1p.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log1p.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-1}{4} \cdot {y}^{2} - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot {y}^{2}} - \mathsf{log1p}\left(y\right)\right)\right) - t \]

   7. Applied rewrites 99.6%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), y \cdot y, -0.5\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right)} - \mathsf{log1p}\left(y\right)\right)\right) - t \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right) - t} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right)\right)} - t \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(x - 1\right) \cdot \log y\right)} - t \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{\left(z - 1\right) \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y, y, \frac{-1}{3}\right), y \cdot y, \frac{-1}{2}\right) \cdot y\right) \cdot y - 1\right) \cdot \left(y \cdot y\right) - \mathsf{log1p}\left(y\right)\right) + \left(\left(x - 1\right) \cdot \log y - t\right)} \]

   9. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(-0.25 \cdot y, y, -0.3333333333333333\right), -0.5\right) \cdot y, y, -1\right) \cdot y\right) \cdot y - \mathsf{log1p}\left(y\right), z - 1, \log y \cdot \left(x - 1\right) - t\right)} \]

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. remove-double-neg N/A

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

       3. mul-1-neg N/A

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

       4. distribute-rgt-neg-in N/A

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

       5. log-rec N/A

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

       6. lower-*.f64 N/A

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

       7. log-rec N/A

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

       8. distribute-rgt-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. remove-double-neg N/A

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

       11. lower-log.f64 83.8

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

   12. Applied rewrites 83.8%

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

   if -4.00000000000000009e144 < (-.f64 x #s(literal 1 binary64)) < 4.9999999999999998e98

   1. Initial program 87.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+144} \lor \neg \left(x - 1 \leq 5 \cdot 10^{+98}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

   10. log-rec N/A

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

   11. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 98.5%

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

Alternative 13: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+179}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.5e+179)
   (- (* (* (- (* (fma -0.3333333333333333 y -0.5) y) 1.0) z) y) t)
   (- (fma (log y) (- x 1.0) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+179) {
		tmp = ((((fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * y) - t;
	} else {
		tmp = fma(log(y), (x - 1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.5e+179)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * y) - t);
	else
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.5e+179], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e179

   1. Initial program 50.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.9%

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

   if -2.5e179 < z

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 99.0

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

   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

   8. Applied rewrites 94.4%

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

Alternative 14: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+179}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.5e+179)
   (- (* (* (- (* (fma -0.3333333333333333 y -0.5) y) 1.0) z) y) t)
   (- (* (log y) (- x 1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+179) {
		tmp = ((((fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * y) - t;
	} else {
		tmp = (log(y) * (x - 1.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.5e+179)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(-0.3333333333333333, y, -0.5) * y) - 1.0) * z) * 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[z, -2.5e+179], N[(N[(N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e179

   1. Initial program 50.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. log-rec N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 76.9%

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

   if -2.5e179 < z

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower--.f64 94.0

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

   5. Applied rewrites 94.0%

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

Alternative 15: 46.2% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

   10. log-rec N/A

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

   11. lower-fma.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 52.4%

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

Alternative 16: 46.1% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y, z, t):
	return ((((-0.5 * y) - 1.0) * z) * y) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. lift-log.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. log-div N/A

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

   5. lower--.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. lower-log1p.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log1p.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-fma.f64 N/A

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

   15. remove-double-neg N/A

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

   16. mul-1-neg N/A

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 52.2%

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

Alternative 17: 46.0% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. log-rec N/A

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

   8. mul-1-neg N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. log-rec N/A

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

   12. remove-double-neg N/A

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

   13. lower-log.f64 N/A

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

   14. lower--.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 51.4%

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

9. Taylor expanded in y around inf

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

11. Applied rewrites 51.5%

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

12. Taylor expanded in z around 0

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

14. Applied rewrites 51.5%

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

Alternative 18: 45.8% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. remove-double-neg N/A

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

   7. log-rec N/A

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

   8. mul-1-neg N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. log-rec N/A

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

   12. remove-double-neg N/A

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

   13. lower-log.f64 N/A

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

   14. lower--.f64 98.5

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 51.4%

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

Alternative 19: 35.7% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 41.8

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

5. Applied rewrites 41.8%

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

Reproduce

herbie shell --seed 2025016 
(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))