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

Percentage Accurate: 89.7% → 99.5%

Time: 10.6s

Alternatives: 11

Speedup: 1.3×

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: 89.7% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. inv-pow N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 99.4%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(x - 1\right) - t\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 21000000:\\ \;\;\;\;\left(\left(-y\right) \cdot \left(z - 1\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) (- x 1.0)) t)))
   (if (<= x -4.4e-8)
     t_1
     (if (<= x 21000000.0) (- (- (* (- y) (- z 1.0)) (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * (x - 1.0)) - t;
	double tmp;
	if (x <= -4.4e-8) {
		tmp = t_1;
	} else if (x <= 21000000.0) {
		tmp = ((-y * (z - 1.0)) - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(y) * (x - 1.0d0)) - t
    if (x <= (-4.4d-8)) then
        tmp = t_1
    else if (x <= 21000000.0d0) then
        tmp = ((-y * (z - 1.0d0)) - 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 - 1.0)) - t;
	double tmp;
	if (x <= -4.4e-8) {
		tmp = t_1;
	} else if (x <= 21000000.0) {
		tmp = ((-y * (z - 1.0)) - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * (x - 1.0)) - t
	tmp = 0
	if x <= -4.4e-8:
		tmp = t_1
	elif x <= 21000000.0:
		tmp = ((-y * (z - 1.0)) - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * Float64(x - 1.0)) - t)
	tmp = 0.0
	if (x <= -4.4e-8)
		tmp = t_1;
	elseif (x <= 21000000.0)
		tmp = Float64(Float64(Float64(Float64(-y) * Float64(z - 1.0)) - 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 - 1.0)) - t;
	tmp = 0.0;
	if (x <= -4.4e-8)
		tmp = t_1;
	elseif (x <= 21000000.0)
		tmp = ((-y * (z - 1.0)) - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -4.4e-8], t$95$1, If[LessEqual[x, 21000000.0], N[(N[(N[((-y) * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999997e-8 or 2.1e7 < x

   1. Initial program 93.9%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if -4.3999999999999997e-8 < x < 2.1e7

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 5: 95.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, \left(-y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e+18)
   (- (* (log y) x) t)
   (if (<= t 4.3e-50)
     (fma (- x 1.0) (log y) (* (- y) z))
     (- (* (log y) (- x 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e+18) {
		tmp = (log(y) * x) - t;
	} else if (t <= 4.3e-50) {
		tmp = fma((x - 1.0), log(y), (-y * z));
	} else {
		tmp = (log(y) * (x - 1.0)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e+18)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (t <= 4.3e-50)
		tmp = fma(Float64(x - 1.0), log(y), Float64(Float64(-y) * z));
	else
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e+18], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 4.3e-50], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[((-y) * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.5e18

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 94.4

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

   4. Applied rewrites 94.4%

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

   if -7.5e18 < t < 4.29999999999999997e-50

   1. Initial program 85.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift--.f64 N/A

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

      19. lift--.f64 85.1

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 99.0

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in z around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-*.f64 97.1

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

   9. Applied rewrites 97.1%

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

   if 4.29999999999999997e-50 < t

   1. Initial program 93.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 92.3

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

   4. Applied rewrites 92.3%

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

Alternative 6: 88.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 88.7

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

4. Applied rewrites 88.7%

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

Alternative 7: 87.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x - 1 \leq -4000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= (- x 1.0) -4000.0)
     t_1
     (if (<= (- x 1.0) -1.0) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if ((x - 1.0) <= -4000.0) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.0) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(y) * x) - t
    if ((x - 1.0d0) <= (-4000.0d0)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-1.0d0)) then
        tmp = (y - log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - t;
	double tmp;
	if ((x - 1.0) <= -4000.0) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.0) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	tmp = 0
	if (x - 1.0) <= -4000.0:
		tmp = t_1
	elif (x - 1.0) <= -1.0:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -4000.0)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -1.0)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - t;
	tmp = 0.0;
	if ((x - 1.0) <= -4000.0)
		tmp = t_1;
	elseif ((x - 1.0) <= -1.0)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -4000.0], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4e3 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if -4e3 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 85.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in z around 0

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

      1. log-pow-rev N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. log-pow-rev N/A

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

      4. unpow1 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 83.9

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

   10. Applied rewrites 83.9%

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

Alternative 8: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.5:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -2e+41)
     t_1
     (if (<= (- x 1.0) -0.5) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+41) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if ((x - 1.0d0) <= (-2d+41)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-0.5d0)) then
        tmp = (y - log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+41) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if (x - 1.0) <= -2e+41:
		tmp = t_1
	elif (x - 1.0) <= -0.5:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+41)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.5)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if ((x - 1.0) <= -2e+41)
		tmp = t_1;
	elseif ((x - 1.0) <= -0.5)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+41], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2.00000000000000001e41 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if -2.00000000000000001e41 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 85.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in z around 0

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

      1. log-pow-rev N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 81.8

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. log-pow-rev N/A

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

      4. unpow1 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 81.6

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

   10. Applied rewrites 81.6%

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

Alternative 9: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 20500000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.45e+41) t_1 (if (<= x 20500000.0) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.45e+41) {
		tmp = t_1;
	} else if (x <= 20500000.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-1.45d+41)) then
        tmp = t_1
    else if (x <= 20500000.0d0) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -1.45e+41) {
		tmp = t_1;
	} else if (x <= 20500000.0) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -1.45e+41:
		tmp = t_1
	elif x <= 20500000.0:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -1.45e+41)
		tmp = t_1;
	elseif (x <= 20500000.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -1.45e+41)
		tmp = t_1;
	elseif (x <= 20500000.0)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.45e+41], t$95$1, If[LessEqual[x, 20500000.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999994e41 or 2.05e7 < x

   1. Initial program 94.6%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 72.8

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

   4. Applied rewrites 72.8%

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

   if -1.44999999999999994e41 < x < 2.05e7

   1. Initial program 85.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 82.4

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-log.f64 81.0

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

   7. Applied rewrites 81.0%

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

Alternative 10: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.5:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -2e+41) t_1 (if (<= (- x 1.0) -0.5) (- t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+41) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if ((x - 1.0d0) <= (-2d+41)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-0.5d0)) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+41) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.5) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if (x - 1.0) <= -2e+41:
		tmp = t_1
	elif (x - 1.0) <= -0.5:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+41)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.5)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if ((x - 1.0) <= -2e+41)
		tmp = t_1;
	elseif ((x - 1.0) <= -0.5)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+41], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2.00000000000000001e41 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 72.0

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

   4. Applied rewrites 72.0%

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

   if -2.00000000000000001e41 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 85.5%

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

   2. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.4

         \[\leadsto -t \]

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.2

      \[\leadsto -t \]

4. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2025112 
(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))