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

Percentage Accurate: 89.5% → 99.8%

Time: 13.3s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log1p(Float64(Float64(-sqrt(y)) * sqrt(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[1 + N[((-N[Sqrt[y], $MachinePrecision]) * N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.6%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. add-sqr-sqrt N/A

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

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

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

   5. lower-log1p.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 110 \lor \neg \left(t\_1 \leq 228\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\frac{1}{y} - -0.5}{z} \cdot \left(y \cdot y\right)\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 110.0) (not (<= t_1 228.0)))
     (- (fma (log y) (- x 1.0) y) t)
     (-
      (* (fma (- (* -0.5 y) 1.0) y (* (/ (- (/ 1.0 y) -0.5) z) (* y y))) z)
      t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= 110.0) || !(t_1 <= 228.0)) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = (fma(((-0.5 * y) - 1.0), y, ((((1.0 / y) - -0.5) / z) * (y * y))) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= 110.0) || !(t_1 <= 228.0))
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.5 * y) - 1.0), y, Float64(Float64(Float64(Float64(1.0 / y) - -0.5) / z) * Float64(y * y))) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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[Or[LessEqual[t$95$1, 110.0], N[Not[LessEqual[t$95$1, 228.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + N[(N[(N[(N[(1.0 / y), $MachinePrecision] - -0.5), $MachinePrecision] / z), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 110 or 228 < (+.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 90.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 89.7%

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

   if 110 < (+.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)))) < 228

   1. Initial program 38.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 94.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 110 \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 228\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\frac{1}{y} - -0.5}{z} \cdot \left(y \cdot y\right)\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 110 \lor \neg \left(t\_1 \leq 228\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\frac{1}{y} - -0.5}{z} \cdot \left(y \cdot y\right)\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 110.0) (not (<= t_1 228.0)))
     (fma (log y) (- x 1.0) (- t))
     (-
      (* (fma (- (* -0.5 y) 1.0) y (* (/ (- (/ 1.0 y) -0.5) z) (* y y))) z)
      t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= 110.0) || !(t_1 <= 228.0)) {
		tmp = fma(log(y), (x - 1.0), -t);
	} else {
		tmp = (fma(((-0.5 * y) - 1.0), y, ((((1.0 / y) - -0.5) / z) * (y * y))) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= 110.0) || !(t_1 <= 228.0))
		tmp = fma(log(y), Float64(x - 1.0), Float64(-t));
	else
		tmp = Float64(Float64(fma(Float64(Float64(-0.5 * y) - 1.0), y, Float64(Float64(Float64(Float64(1.0 / y) - -0.5) / z) * Float64(y * y))) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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[Or[LessEqual[t$95$1, 110.0], N[Not[LessEqual[t$95$1, 228.0]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y + N[(N[(N[(N[(1.0 / y), $MachinePrecision] - -0.5), $MachinePrecision] / z), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 110 or 228 < (+.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 90.6%

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

   5. Applied rewrites 89.4%

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

   if 110 < (+.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)))) < 228

   1. Initial program 38.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 94.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 110 \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 228\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y - 1, y, \frac{\frac{1}{y} - -0.5}{z} \cdot \left(y \cdot y\right)\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 99.7%

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

Alternative 5: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{+21}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.05e+21)
   (- (* (log y) x) t)
   (if (<= t 2.5e-30)
     (fma (- 1.0 z) y (* (log y) (- x 1.0)))
     (fma (log y) (- x 1.0) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.05e+21) {
		tmp = (log(y) * x) - t;
	} else if (t <= 2.5e-30) {
		tmp = fma((1.0 - z), y, (log(y) * (x - 1.0)));
	} else {
		tmp = fma(log(y), (x - 1.0), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.05e+21)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (t <= 2.5e-30)
		tmp = fma(Float64(1.0 - z), y, Float64(log(y) * Float64(x - 1.0)));
	else
		tmp = fma(log(y), Float64(x - 1.0), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.05e+21], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 2.5e-30], N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.05e21

   1. Initial program 98.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.3%

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

   if -2.05e21 < t < 2.49999999999999986e-30

   1. Initial program 78.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 98.8%

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

   if 2.49999999999999986e-30 < t

   1. Initial program 91.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.2%

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

Alternative 6: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3700000 \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, \left(-\log y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3700000.0) (not (<= x 1.8e-21)))
   (fma (log y) (- x 1.0) (- t))
   (fma (- (- z 1.0)) y (- (- (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3700000.0) || !(x <= 1.8e-21)) {
		tmp = fma(log(y), (x - 1.0), -t);
	} else {
		tmp = fma(-(z - 1.0), y, (-log(y) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3700000.0) || !(x <= 1.8e-21))
		tmp = fma(log(y), Float64(x - 1.0), Float64(-t));
	else
		tmp = fma(Float64(-Float64(z - 1.0)), y, Float64(Float64(-log(y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3700000.0], N[Not[LessEqual[x, 1.8e-21]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + (-t)), $MachinePrecision], N[((-N[(z - 1.0), $MachinePrecision]) * y + N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e6 or 1.79999999999999995e-21 < x

   1. Initial program 94.1%

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

   5. Applied rewrites 93.2%

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

   if -3.7e6 < x < 1.79999999999999995e-21

   1. Initial program 78.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3700000 \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, \left(-\log y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3700000 \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3700000.0) (not (<= x 1.8e-21)))
   (fma (log y) (- x 1.0) (- t))
   (- (fma (- 1.0 z) y (- (log y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3700000.0) || !(x <= 1.8e-21)) {
		tmp = fma(log(y), (x - 1.0), -t);
	} else {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3700000.0) || !(x <= 1.8e-21))
		tmp = fma(log(y), Float64(x - 1.0), Float64(-t));
	else
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3700000.0], N[Not[LessEqual[x, 1.8e-21]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e6 or 1.79999999999999995e-21 < x

   1. Initial program 94.1%

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

   5. Applied rewrites 93.2%

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

   if -3.7e6 < x < 1.79999999999999995e-21

   1. Initial program 78.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3700000 \lor \neg \left(x \leq 1.8 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

5. Applied rewrites 99.4%

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

Alternative 9: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4800000.0) (not (<= x 55000000000.0)))
   (- (* (log y) x) t)
   (fma (- (- z 1.0)) y (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4800000.0) || !(x <= 55000000000.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = fma(-(z - 1.0), y, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4800000.0) || !(x <= 55000000000.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = fma(Float64(-Float64(z - 1.0)), y, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4800000.0], N[Not[LessEqual[x, 55000000000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[((-N[(z - 1.0), $MachinePrecision]) * y + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e6 or 5.5e10 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.5%

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

   if -4.8e6 < x < 5.5e10

   1. Initial program 78.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 63.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000 \lor \neg \left(x \leq 55000000000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+89} \lor \neg \left(x \leq 4.5 \cdot 10^{+37}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.1e+89) (not (<= x 4.5e+37)))
   (* (log y) x)
   (fma (- (- z 1.0)) y (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.1e+89) || !(x <= 4.5e+37)) {
		tmp = log(y) * x;
	} else {
		tmp = fma(-(z - 1.0), y, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.1e+89) || !(x <= 4.5e+37))
		tmp = Float64(log(y) * x);
	else
		tmp = fma(Float64(-Float64(z - 1.0)), y, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.1e+89], N[Not[LessEqual[x, 4.5e+37]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-N[(z - 1.0), $MachinePrecision]) * y + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.09999999999999985e89 or 4.49999999999999962e37 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 75.6%

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

   if -4.09999999999999985e89 < x < 4.49999999999999962e37

   1. Initial program 80.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing
   3. 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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 63.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+89} \lor \neg \left(x \leq 4.5 \cdot 10^{+37}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\left(z - 1\right), y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+16} \lor \neg \left(t \leq 16500\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.9e+16) (not (<= t 16500.0))) (- t) (* (- 1.0 z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e+16) || !(t <= 16500.0)) {
		tmp = -t;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-3.9d+16)) .or. (.not. (t <= 16500.0d0))) then
        tmp = -t
    else
        tmp = (1.0d0 - z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e+16) || !(t <= 16500.0)) {
		tmp = -t;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.9e+16) or not (t <= 16500.0):
		tmp = -t
	else:
		tmp = (1.0 - z) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.9e+16) || !(t <= 16500.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(1.0 - z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.9e+16) || ~((t <= 16500.0)))
		tmp = -t;
	else
		tmp = (1.0 - z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.9e+16], N[Not[LessEqual[t, 16500.0]], $MachinePrecision]], (-t), N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9e16 or 16500 < t

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 66.7%

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

   if -3.9e16 < t < 16500

   1. Initial program 79.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 23.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+16} \lor \neg \left(t \leq 16500\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+16} \lor \neg \left(t \leq 16500\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.9e+16) (not (<= t 16500.0))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e+16) || !(t <= 16500.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-3.9d+16)) .or. (.not. (t <= 16500.0d0))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.9e+16) || !(t <= 16500.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.9e+16) or not (t <= 16500.0):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.9e+16) || !(t <= 16500.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.9e+16) || ~((t <= 16500.0)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.9e+16], N[Not[LessEqual[t, 16500.0]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9e16 or 16500 < t

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 66.7%

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

   if -3.9e16 < t < 16500

   1. Initial program 79.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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. add-sqr-sqrt N/A

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

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

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

      5. lower-log1p.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 23.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 22.9%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+16} \lor \neg \left(t \leq 16500\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.6% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. add-sqr-sqrt N/A

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

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

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

   5. lower-log1p.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in t around inf

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

10. Applied rewrites 48.0%

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

Alternative 14: 35.4% 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 86.6%

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

3. Taylor expanded in t around inf

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

5. Applied rewrites 34.8%

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

Reproduce

herbie shell --seed 2025021 
(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))